1 Introduction↩︎

Modern wireless communication systems rely or will rely on the large scale deployment of vehicular and in particular non-terrestrial base stations (BS). Due to the mobility of such BSs, it is extremely important to asses the performance at all epochs of this process and find out ways to mitigate challenges posed by certain types of “good” or “bad” epochs or configurations in such systems. The most important performance metric for the quality of service is the distribution of the Shannon rate experienced by a typical user in the system, which is a function of signal-to-interference-plus-noise-ratio (SINR) at any time instant. Our results essentially deal with the SINR distribution at various “typical epochs” of interest, and their comparison.

To characterize the time evolution of Shannon rate, we define a dynamical communication network model with random locations of BSs on the Euclidean plane. This randomness is complemented by their random motion, and this makes the time evolution of the SINR and Shannon rate as a stochastic process. As a first step we derive our results only for the dynamical communication model on 2D, even though our motivation comes from 3D non-terrestrial networks with a spherical structure. The present paper is hence essentially a first step in the direction of the analysis of this higher dimension problem.

We use stochastic geometry as a central tool in our analysis, which plays an important role in developing suitable models to understand many statistical features of key performance metrics in such systems. As we will see, it allows one to derive ensemble averages of the system properties, including at such specific epochs.

In this article we focus on determining performance metrics of dynamical cellular communication systems at specific typical epochs of interest. For example, consider a time epoch when the power of the signal is maximum. This corresponds to the time when the serving BS is at its minimal distance from the user in its motion. The SINR at this typical epoch is measured using the distance to the serving BS and that of all others. To measure the performance of the system, we take an average over all possible configurations of the system at such epochs, instead of taking averages at all times.

We first define the dynamical cellular communication model on \(\mathbb{R}^2\). The BSs are distributed as a homogeneous Poisson point process and all move at constant speed \(v\) and with a uniformly chosen and independent random direction. A user located at the origin (\(o\)) connects to the nearest BS at all time instants and changes its connection to whichever is the closest. This phenomenon is called a handover. Due to the motion of every station, the user encounters changes in the distance to the serving station, along the spatio-temporal evolution of the system. These temporal variations result in a continuous change in the performance metrics in terms of signal and interference power. We are in particular interested in determining the behavior of the system performance at specific time epochs whenever the user sees an abrupt change, for example at a handover epoch, at a time when the signal power is locally maximum, or at a time when the nearest interferer is at its closest or farthest distance.

Our investigation is based on analyzing the fluctuations of the SINR with the additive total interference by considering interference as noise to the signal received from the serving station. As we will see, this is governed by the joint distribution of the distance to the nearest station, contributing to the signal power, as well as the distance to all other stations, contributing to the interference power. On the other hand, their comparison at different typical epochs is mainly governed by the comparison of the distribution of the distance to the serving BS or the closest interferer depending on the typical epochs. As a special case, we also use the maxitive (or tropical) interference, with a goal to capture the effect of the strongest interferer on the performance evaluation.

1.1 Motivation↩︎

Most of the literature about performance evaluation of wireless communication systems only deals with a snapshot analysis. For example, in a cellular system, under Poisson setting, it is is well known that the random variable for the distance to the nearest BS is distributed as Rayleigh. This is not the case at a handover epoch, as the system is conditioned on the existence of two BSs at equal distance from the typical user. As a result, the previous works fails to capture the performance of the system at such specific epochs, for example at a time when the signal or interference power is locally maximum or minimum. Our objective is to formally analyze a spatio-temporal model with the goal to capture the evolution of system performance at the typical epochs of interest listed before. As already mentioned, knowing a potentially extreme behavior at specific epochs can help the operator to prepare and dimension the system accordingly in all these best and perhaps more importantly worst case scenarios. From the system design perspective, it is also important to compare the performance at these time epochs.

1.2 Illustration for seasonal statistics↩︎

As the title of this article suggests, one can better visualize this interplay between the evolving strong or weak signal and interference powers, as the evolution of “seasons” over time, where there is a unique source of signal power, and a joint source of interference power. The entire analysis in based on the identification of typical events corresponding to max-signal, min-signal, min-interference or max-interference, and probabilistic features of these seasonal statistics. Our comparison results focus on the coverage probability and the Shannon rate. Even though the time evolution of the interplay between strong or weak signal and interference power resembles the dynamics of seasons, in contrast, our spatio-temporal model is highly non-periodic. For instance, it can be seen to possibly possess two max-interference seasons without even having a max-signal in between. Also, there can be two max-signal seasons and a max-interference season in between, and vice versa.

In our context, the temporal evolution or the time series of the Shannon rate, can be better visualized in terms of Figure 2. In the tropical case, the interference is considered to be from the maximum interferer only, and the effect of such a reduction can be seen in the entire analysis, from seasonal statistics to their comparison. Our analysis and results provide insight about the worst case scenarios at extreme seasons and can help the system designer to dimension the network based on this, rather on the typical time. This also prepares the system to be more proactive than reactive for the time around such typical epochs. This essentially allows one to derive recommendation to the operator to dimension the system based on the extreme behavior at those typical epochs, rather than the average one. Let us stress two important facts pertaining to the term “season" (a) throughout the paper, this term has to be understood as some extreme point in the season (e.g.”peak of winter", here a handover event, where by construction, signal power is locally minimal and interference power locally maximal) rather than a time interval (the whole winter, or some interval centered on the handover); (b) seasons are in now way periodic here, with periodicity replaced by a Markov structure here.

As we will see later, in our context, seasons mean specific epochs where an inflection, for instance a peak of the evolution of the Shannon rate is observed, rather than intervals where this rate is better or worse. Archetypes will be when signal power is locally maximal or when interference power is locally maximal. With this convention in mind, seasons can be refined further than what we do in the present paper, based e.g. on certain derivatives. For instance, we could distinguish a local maximum of signal power when interference power is decreasing, from a local maximum where interference power is increasing. The techniques developed here clearly allow one to analyze such refinements, but this is left for future research.

The Figure 1 depicts a realization of the location and trajectories of the BSs.

The evolution of the distance to the nearest and second nearest BSs, and the corresponding time series of the Shannon rates (or rates) are shown in Figure 2, for the simple case of tropical SIR with power law attenuation for different attenuation exponent \(\alpha\) and without fading. In the interference limited no-fading case, the Shannon rate at time \(t\) is defined as \[\mathcal{R}(t):=B\ln\left(1+ \frac{H(t)^{-\alpha}}{R(t)^{-\alpha}}\right)bps,\] where \(H(t)\) and \(R(t)\) are the distance to the nearest and second nearest BS, respectively, at time \(t\), \(\alpha\) is the exponent of the path-loss attenuation and \(B=b/{\ln 2}\), with \(b\) being the bandwidth. Depending on the time-evolution of the ratio of the distance to the serving BS and nearest interferer there are high or low “seasons” of Shannon rate along the evolution of the system. We are interested in the characteristics of it at the seasons alluded before and depicted with vertical lines in Figure 2. The non-periodicity of the seasons and their sequential but random appearance are also evident even in the simple tropical model without noise and fading, whereas handover epochs are the worst seasons in terms of Shannon rate experienced by the user. It can be seen as a one-to-one “tug of war” between the signal and tropical interference power over time. The local extreme nature of Shannon rate is very clear at handover and max-signal epochs, where as there is only a slight change of its slope at max-interference and min-interference epochs. The time series of the Shannon rate turns out to be even more complex and infinitely dependent, while moving beyond tropical scenario with or without fading and noise. As we will see later in Remark 1, in a “dual dynamical model” with a mobile user, which encounters the identical seasonality along its motion in a straight line, with BSs being fixed.

1.3 Contributions↩︎

Below, we list out the main contributions made in this article. Most of our results are based on the power law attenuation function \(\ell(r)= r^{-\alpha}\) and Rayleigh fading, where \(\alpha\geq 2\) is the path-loss exponent.

1. To set the stage for the performance evaluation, we identify and formalize the specific time epochs of interest within the spatio-temporal evolution of the system in Section [section:prelims]. This also includes the determination of the intensity of such epochs and the corresponding distance characteristics, which are established in Appendix [section:A-prelim].

2. To determine the distribution of the interference power, we derive the distribution of the point process of distance of the BSs, other than the serving one at these epochs, which in our case turns out to be conditionally Poisson, in Lemma 5.

3. This leads to a determination of the coverage probability and Shannon rate at each of the typical epochs for both non-tropical and tropical scenarios in Section [section:results], mainly in the form of Theorem 1- 5.

4. An analytic comparison of the performance metrics (coverage probability and Shannon rate) is also shown to hold between different typical epochs for both non-tropical and tropical scenarios in Subsection [subsection:comparison]. These comparisons are also validated by plotting these closed form or integral form expressions with respect to the user QoS parameter \(\tau\) in Section 6.

5. We also prove a scale-invariance property for the coverage probabilities and Shannon rates at these typical epochs in the interference limited case, in Subsection [subsection:SI].

6. We manage to establish some closed or integral form expressions for the coverage probabilities, as presented in Section [sec-closedforms]. A significant take away is that, in the interference limited regime with fading factor and with path-loss exponent \(\alpha=4\), the coverage probability at any of the typical epoch can be represented in terms of a non-linear function of the coverage probability at a typical time.

7. As we will see later in Subsection [subsection:example-AF], it can be adapted to a more general class of attenuation functions and fading factors. Some of the comparison results hold true for the SINR defined using attenuation function other than the power law \(\ell(r)=r^{-\alpha}\), for which we provide an example of bounded attenuation function \(\ell(r)=(1+r)^{-\alpha}\) and a counter example as well of step attenuation function with given radius.

1.4 Organization of the article↩︎

This article starts with the initial description of the underlying system model and a set of questions of interest in Section [section:system]. Section [section:prelims] is dedicated to mathematical preliminaries, including a complete description of the system variables of interest. The proofs of the main results for the last section are presented in Appendix [section:A-prelim]. Section [section:results] contains all the major results about the performance evaluation of the system at different typical events or equivalently seasons. This also includes the comparison among these QoS at typical events. Section [sec-closedforms] highlights some of the closed form or simple integral form expressions of the coverage probabilities in some of the regimes, due to specific values of the path-loss exponent \(\alpha\). We present the plots of the closed form or integral form expressions of coverage probabilities with respect to different threshold parameter \(\tau\) in Section 6. The proof of the main results about the performance evaluation and their expressions at different typical events are presented in Appendix [proofs-wf], for the case with fading. The proofs for the case without fading is presented in Appendix [section:PWFa]. Appendix [section:CPE] is dedicated to the proof of the comparison among the coverage probabilities at those typical epochs in cases with and without fading. The comparison of additive interference, as well as the tropical interference at different typical times is presented in Appendix [Int-order]. Lastly, the proof for scale invariance property in the interference limited regime is given in Appendix [section:SI].

1.5 Literature↩︎

The first closed form expressions for Shannon rate in cellular wireless communication systems using stochastic geometry were obtained in [1], when base stations are modelled as a Poisson point process. The analysis of cellular networks using tools from stochastic geometry gained significant traction following [1], and also [2]. Building on this foundation, performance evaluation of heterogeneous networks was explored in  [3], [4] and [5].

This was extended to the setting of satellite networks using stochastic geometry approach by Okati et. al. in [6] and [7] addressing the unique geometric and statistical challenges posed by non-terrestrial infrastructures. This line of work laid important groundwork for analyzing low Earth orbit (LEO) constellations, where the spatial distribution of satellites and their motion relative to ground users introduce non-trivial interference and coverage characteristics. The characterization of satellite constellations themselves has also received considerable attention. Choi et al. [8] proposed a constellation model grounded in stochastic geometry, enabling the analysis of coverage and connectivity for large-scale LEO deployments. This was further generalized to multi-altitude constellation architectures [9], where satellites operating at different heights create layered interference environments. LEO satellite based communication systems are also studied in [10]. In a recent work [11], the authors evaluated the coverage probability for NTN system with LEO satellites in a heterogeneous environment.

The dynamic nature of satellite on the orbits has motivated the development of dedicated mobility and motion models. Baccelli and Zuyev [12] introduced a dual dynamical model that jointly accounts for the motion of user and the stochastic geometry of static terrestrial network. The question about handover probability in both the dual dynamical models and their equivalence is studied by Banagar et al. in [13]. Correlation structure of SNR in such a mobility model was studied in [14]. Complementing this, the random direction model [15] provides a tractable abstraction for user or node mobility, and has found application in both terrestrial and non-terrestrial network analysis. Random geometric graph with similar mobility of random direction model is studied in [16].

The work [17] and [18] examines the impact of mobility on network performance, characterizing how node movement affects interference statistics and spatial reuse. Gong et al. [19] further incorporated fading in terms of mobility and channel randomness shapes coverage and outage probabilities. These contributions highlight that a complete performance analysis must account for both the spatial and temporal dimensions of network dynamics. A closely related thread concerns the temporal correlation of interference, which is of central importance when evaluating link-layer protocols and re-transmission schemes. Ganti and Haenggi [20] and, Krishnan and Dhillon [21] provided a rigorous treatment of this temporal correlation of interference and mitigation of the adversity provided by the later in Poisson networks. In a new paradigm, the effect mobility on the association policies in terms of Doppler shift and the performance evaluation is explored in [22]. Finally, questions of resource sharing and data rate allocation in spatially distributed networks have been addressed by Madadi et al. [23], who studied shared data rate models that account for the stochastic geometry of the network alongside fairness and efficiency constraints. In a two-tier cellular system handover management has been studied in [24]. The notion of maximal power association in angular setting is studied in [25]. The current research is based on the handover analysis for the mobility model in [26] and as an initial ingredient about distance characteristics and Palm distributions are borrowed from there. The distance characteristics of similar interest in Poisson Voronoi tessellations are also studied in [27].

2 system Model and problem formulation↩︎

 

2.1 System Model↩︎

In this article, we work on a dynamical wireless communication network model based on a Poisson point process on \(\mathbb{R}^2\), as introduced in [26]. Consider a collection of BSs located at points of a homogeneous Poisson point process \(\Phi_\lambda\equiv \Phi= \sum_{i}\delta_{X_i}\) on \(\mathbb{R}^2\) with intensity \(\lambda\). Suppose each of the BS moves at constant speed \(v\), in a uniformly and independently chosen random direction. The BSs do not change their direction along the rest of the course. Suppose the directions are given by a collection of i.i.d. random variables \(\{\Theta_i\}_i\). A user located at the origin stays connected to the nearest BS at any time, using a distance based association rule. During this connection period, the serving BS provides the downlink signal to the user and the rest of the BSs contribute to the total interference experienced by the user. The user changes its connection from one BS to next over time. Such a change of BS is called a handover. The time epochs of these handover events form a point process on the time axis.

For any BS starting at a location \(X\in\Phi\) and moving in the direction \(\Theta\), it is located at \(X^t:=X+Vt\), at time \(t\), where \(V=v(\cos \Theta,\sin \Theta)\) and \(v\) is the modulus of the speed. We denote the point process corresponding to the new locations as \(\Phi^t:=\sum_{i}\delta_{X^t_i}\). It is well known from the displacement theorem [28] that for all \(t\), \(\Phi^t\) is a homogeneous Poisson point process with intensity \(\lambda\). We always write the location of the nearest BS as \(X_0^t\). The time varying distance of a BS starting at \(X=(|X|\cos\Psi,|X|\sin\Psi)\) is given by the function \[d_X(t):=\vert|X^t\vert|=\left(||X||^2+2vt\cos\Xi+v^2t^2\right)^{{\frac{1}{2}}}, \label{eq:distancef}\tag{1}\] where \(\Xi:= (\Theta-\Psi) \;\,mod(2\pi)\). Note that \(\Xi\stackrel{d}{\sim} U[-\pi, \pi)\). The curve of the distance function will be used throughout the article. As we will see later, these curves are the branches of certain hyperbolas on the the upper half space \(\mathbb{H}^+:=\mathbb{R}\times \mathbb{R}^+\).

In the first model, in principle, at any time instant, the power of the signal emitted from each BS is a function of the distance from the user, using the power law attenuation function \(\ell(\vert|X\vert|)= \vert|X\vert|^{-\alpha}\) for a BS located at \(X\), where \(\alpha\geq 2\) is the path-loss exponent. In a more realistic scenario, one can incorporate a fading factor, given by a random variable with a given distribution. In the modelling of wireless communications, the performance of the serving BS is usually measured in terms of the SINR experienced by the user at time \(t\), which is \[SINR(t):= \frac{\rho \vert|X_0^t\vert|^{-\alpha}}{\sigma^2+\sum_{Y\in \Phi^t\setminus \{X_0^t\}} \rho_Y \vert|Y\vert|^{-\alpha}}, \label{eq:SINRt}\tag{2}\] where \(\sigma^2\) is the thermal noise power, \(Y\in \Phi^t\setminus \{X_0^t\}\) are the locations of all other BSs at time \(t\) and \(\rho, \rho_Y\) are the i.i.d. fading factors. These can be considered as Rayleigh fading, but more general classes of fading distributions can cover a large set of use cases. All other performance metrics, namely, signal-to-noise-ratio (SNR) and signal-to-interference-ratio (SIR) are defined similarly both in the fading and no-fading cases.

In addition, we also define a more restrictive version of the performance metrics using tropical interference, by this we mean that we replace the additive interference by maxitive interference over \(\Phi^t\), see [29] and [30]. In this case we are interested in the signal-to-tropical interference-plus-noise-ratio (STINR) defined as \[STINR(t):= \frac{\rho \vert|X_0^t\vert|^{-\alpha}}{\sigma^2+\max_{Y\in \Phi^t\setminus \{X_0^t\}} \rho_Y \vert|Y\vert|^{-\alpha}}, \label{eq:STINRt}\tag{3}\] and similarly for the no-fading case. We can define signal-to-tropical interference-ratio (STIR) similarly, both in the fading and no-fading cases. The Shannon rate is the metric of interest here, which is proportional to \(\ln(1+SINR)\).

2.2 Problem formulation: performance metrics↩︎

In this article, building upon the theory developed in the previous work [26], we focus on the performance evaluation of the system using various metrics of interest. Suppose the user QoS requirement is \(\tau>0\). At any time \(t\), we say that a typical user is \(\tau\)-covered by the system with coverage probability \[p^c(\tau,\mu,\lambda,\alpha)(t):=\mathbb{P}(\mathcal{S}(t)>\tau),\] at time \(t\), where \(\mathcal{S}\in \{SNR, SIR, SINR, STIR, STINR\}\). In all these cases, we define the Shannon rate as \[\begin{align} \mathcal{R}(\mu,\lambda,\alpha)(t):=\mathbb{E}[B\ln(1+\mathcal{S}(t))]bps, \label{eq:Rcal} \end{align}\tag{4}\] at time \(t\), where \(B=b/{\ln 2}\), with \(b\) being the bandwidth. We use natural log \(\ln\) in the definition of the Shannon rate in (4 ), instead of \(\log_2\) for convenience in later computational tractability.

The main results in the article are about the coverage probability and average data rate experienced by the user at the origin, at the different typical epochs alluded to above, namely, at a typical min signal & max interference (mS-MI) epoch, max signal (MS) epoch, max interference (MI) epoch, min interference (mI) epoch and at any typical time epoch. All these scenarios are described in Figure 3 using the collection of distance functions (1 ) described in [26]. We describe these scenarios in Table ¿tbl:tab:abb? in terms of proximity of the serving BS and all others, and also with in our seasonal analogy as follows:

1. Min signal & max interference (mS-MI) epoch: At a handover epoch the user experiences a comparable interference power component compared to the signal received from the serving station, which is a time of weakest signal and strongest interference.

2. Max signal (MS) epoch: At such an epoch, the power of the signal is maximal, as the serving BS is at its nearest position.

3. Max interference (MI) epoch: At such an epoch, the nearest interferer is at the closest position with respect to the user.

4. Min interference (MI) epoch: Such an epoch corresponds to the time when the nearest interferer gets swapped with another interferer. We will also call this event a tropical interference handover.

5. Typical time epoch (t): It correspond to the system at any other time epoch. At this time, there exists a BS at a Rayleigh distance. This leads to classical snapshot analysis of the system as in [1].

Table ¿tbl:tab:abb? summarizes the scenarios at different typical epochs, their correspondence with seasons, along with the notations for the variables for the distances used later in the signal and interference components.

This article has three major parts. In the first part, we determine the probability distribution of several distance characteristics at various typical epochs, using the analysis in [26] and especially the Palm distribution, Theorem 5.4, therein. Next, using these distributions, we establish the coverage probability using a threshold parameter \(\tau\) corresponding to the user QoS, under different metrics of interest. This also includes the average Shannon rate perceived by the user with respect to different metrics. The last part focuses on the comparison of coverage probabilities and Shannon rates for various regime at different typical epochs. Moreover, this also includes a scale invariance property of the performance metrics in the interference limited regime.

2.3 Strategy↩︎

The results in this article can be considered as a continuation of the theory developed in [26]. Along with the Palm probability distributions of quantities or distances useful for the distribution of SINR, in Section [section:prelims], we identify and develop characteristics of few more ingredients of interest with the help of the spatio-temporal evolution of the distances of serving BS and other BSs. The distribution of these distances allows us to determine the distribution of SINR and Shannon rate at those typical events. Leveraging the properties of the Poisson point process, the probability generating functional (PGFl) of max-shot noise in particular, we provide integral form expressions or even closed form expressions in specific cases. We manage to compare the key performance indicators in terms of Laplace order or stochastic domination, with the help of the same ordering among some of the distances at typical epochs.

3 Preliminaries↩︎

  In the evaluation of the performance metrics at different time epochs, we need the probability distribution of system characteristics, for example, the distance to the serving station, to the nearest interferer(s) and that of all others at these epochs. This invites us to make use of the spatio-temporal evolution of the distance functions of all network participants, provided by the radial bird particle process developed in [26]. In the following subsections, we recall some of the constructions from there and devise new tools for deriving the distribution of the distances of interest at these different typical time epochs. Without loss of generality, we assume that \(v=1\), since the distance distribution is invariant with respect to the speed parameter \(v\), as seen in [26], and also observed in (5 ) later in this article. We borrow some of the notation and names from the paper [26], or explicitly define some others otherwise. We consider this section as a preliminary to our main objective of this article.

3.1 Radial bird particle process↩︎

Consider the particle process \(\mathcal{H}_c:= \sum_i\delta_{C_i}\), on the upper half plane \(\mathbb{H}^+\) where \(C_i:=\{\left(t,d_{X_i}(t)\right): t\in \mathbb{R}\}\), and \(d_{X_i}(\cdot)\) is the distance function of the BS starting at \(X_i\), for a homogeneous Poisson point process \(\Phi_\lambda=\Phi= \sum_i\delta_{X_i}\). We call the infinite closed set \(C_i\) a radial bird and \[\mathcal{H}_c:= \sum_i\delta_{C_i}\] the radial bird particle process. For each \(X\in \Phi\), we define \[T_X:=\arg\inf_{t\in \mathbb{R}}\{d_X(t)\}andH_X:=d_X(T_X).\] It turns out that \[\begin{align} (T_X, H_X) &= \left(-|X| \cos\Xi,|X| \,\vert\sin\Xi\vert \right),\nonumber \end{align}\] which is the time-space location corresponding to global minimum of the curve of \(d_X(\cdot)\), where \(\Xi\) is the relative angle of motion of the BS starting at \(X\). In case of \(v\neq 1\), we have \[\begin{align} (T_X, H_X) &:=\left(-\frac{|X|}{v} \cos\Xi,|X| \,\vert\sin\Xi\vert \right). \label{eq:headb} \end{align}\tag{5}\] Using [26], \(\mathcal{H}:=\sum_i \delta_{(T_i,H_i)}\), forms a homogeneous Poisson point process on \(\mathbb{H}^+\) with intensity measure \(\nu\), where \((T_i,H_i)\equiv (T_{X_i},H_{X_i})\) and \({\rm d}\nu:={\rm d} t\otimes 2\lambda\, {\rm d} h\). We call \(\mathcal{H}\) the head point process.

We rewrite the radial bird particle process as \(\mathcal{H}_c:= \sum_i\delta_{(T_i,H_i),C_i}\) by associating the infinite set \(C_i\) to each point \((T_i,H_i)\) of \(\mathcal{H}\). The point process \(\mathcal{H}\) and the intricate properties of the distance function as a branch of a hyperbola work as the backbone of our analysis in the rest of the article. The following Figure 3 represents a local realization of the radial bird particle process \(\mathcal{H}_c\), in which the intersection points on the vertical line at typical epochs represent the distance of the serving BS and the interferers.

3.2 Lower envelope↩︎

The lower envelope is defined as the random closed subset \[\mathcal{L}_e:=\{(t, L(t)): t\in \mathbb{R}\},\] of \(\mathbb{H}\), where \(L(t):=\inf_{i\in \mathbb{N}}\{d_{X_i}(t)\}\) for \(t\in \mathbb{R}\). The lower envelope \(\mathcal{L}_e\) correspond to the lowest part of the radial bird particle process in Figure 3. Physically, \(\mathcal{L}_e\) represents the time evolution of the distance to the nearest BS from the user, which contributes to the signal component. It also encodes the information about the distance of the nearest interferer at handover events.

3.3 The k-th lower envelope and its characterization↩︎

Other than the lower envelope \(\mathcal{L}_e\), it will be important to look at the second layer or the \(k\)-th layer above the lower envelope. The second layer is defined as \[\mathcal{L}^{(2)}_e:=\left\{\left(t, L^{(2)}(t)\right):t\in \mathbb{R}\right\},\] where, \[L^{(2)}(t):=\inf\big\{\{d_{X_i}(t)\}_{i\in \mathbb{N}}\setminus \{L(t)\}\big\},\] and we call it the second lower envelope. This is obtained by peeling off the part of the curves which lies on \(\mathcal{L}_e\), see Figure 3. This object is interesting since the dominant contribution to the interference power is given by the distances measured with respect to \(\mathcal{L}^{(2)}_e\). For \(k\geq 2\) the \(k\)-th envelope is defined as \[\mathcal{L}^{(k)}_e :=\left\{\left(t, L^{(k)}(t)\right):t\in \mathbb{R}\right\},\] where, \[L^{(k)}(t):=\inf\left\{\{d_{X_i}(t)\}_{i\in \mathbb{N}}\setminus \left\{L^{(i)}(t)\right\}_{1\leq i\leq k-1}\right\},\] where \(L^{(1)}(t):=L(t)\) and \(\mathcal{L}^{(1)}_e:=\mathcal{L}_e\). For \((s,h)\in \mathbb{H}^+\), \(U_h^s\) denotes the upper half ball of radius \(h\) and center at \((s,h)\). Let us recall two main geometric properties from [26] which can be summarized as follows:

1. Suppose \((s,h)\in C_{(T,H)}\), for the set particle \(C_{(T,H)}\) located at a point \((T,H)\in \mathcal{H}\), corresponding to a BS starting at \(X \in \Phi\). Then \((T,H)\in \partial U_h^s\), see Figure 4.

   

2. Let \((s,h), (T,H)\in\mathbb{H}^+\). Let \(C_{(T,H)}\) be the radial bird with its head at \((T,H)\). Let \(\hat{h}\) be the height at which the radial bird \(C_{(T,H)}\) intersects the vertical line \(t=s\). Then \((T,H)\in U_{h}^{s}\) if and only if \(\hat{h}<h\), as depicted in Figure 5.

   

This leads to the following characterization of points lying on the \(k\)-th lower envelope \(\mathcal{L}_e^{(k)}\), and the proof of which is given in Subsection [subsection:Lk].

3.4 Characteristics of min signal & max interference (mS-MI)↩︎

The time epochs of local minimum signal and local maximum interference (mS-MI) correspond to the handover point process, as seen in [26]. Therein the handover point process is defined as \[\begin{align} \mathcal{V}&:=\sum_{(T_i,H_i)\in \mathcal{H}}\;\;\sum_{(T_j,H_j)\in \mathcal{H}\,:\, T_j < T_i} \delta_{\hat{S}}\, {\mathbb{1}}_{A(\hat{S}, \hat{H})}, \label{eq:Vcal} \end{align}\tag{6}\] where \(A(\hat{S}, \hat{H}):= \left\{\mathcal{H}(U^{\hat{S}}_{\hat{H}})=0\right\}\) and \((\hat{S},\hat{H})\) is the intersection of two radial birds \(C_i\) and \(C_j\) corresponding to \((T_i,H_i)\) and \((T_j,H_j)\), respectively. The intensity of the handover point process \(\mathcal{V}\) is calculated in [26] as \(\lambda_{\scriptscriptstyle\mathcal{V}}=\frac{4\sqrt{\lambda}}{\pi}\).

At a handover event, there exist two BSs at equal distance from the user. One of them corresponds to the serving BS and the other one to the nearest interferer. From [26], the typical handover distance, denoted by \(H_{\scriptscriptstyle\mathcal{V}}\), follows a Nakagami distribution with parameter \(\left(\frac{3}{2}, \frac{3}{2\lambda\pi}\right)\), with density \[f_{H_{\scriptscriptstyle\mathcal{V}}}(h)=4\pi\lambda^{3/2} h^2 e^{-\lambda\pi h^2},\text{ for } h\geq 0. \label{eq:pdfhatH0}\tag{7}\]

3.5 Characteristics of max signal (MS)↩︎

At a typical time of maximum signal power, the serving BS is at its closest distance to the user. We denote the point process of max-signal epoch on the time axis by \(\mathcal{V}_{\scriptscriptstyle S}\). Intensity of such a point process is \(\lambda_{\scriptscriptstyle S}=\sqrt{\lambda}\), as derived in [26]. We write the corresponding head point process \(\mathcal{H}_{\scriptscriptstyle I}\) as \[\mathcal{H}_{\scriptscriptstyle S}:=\sum_{j\in \mathbb{N}}\delta_{(T^{\scriptscriptstyle S}_j, H^{\scriptscriptstyle S}_j)} = \sum_{i\in \mathbb{N}}\delta_{(T_i, H_i)} {\mathbb{1}}_{\left\{(T_i, H_i)\in \mathcal{L}_e \right\}}.\nonumber\] Let \(H_{\scriptscriptstyle S}\) denote the distance to the nearest position of a BS. It is known from [26] that \(H_{\scriptscriptstyle S}\) has a density \[\begin{align} f_{H_{\scriptscriptstyle S}}(h)= 2 \lambda^{\frac{1}{2}}\; e^{-\lambda\pi h^2}, \text{ for } h\geq 0.\nonumber \end{align}\] Let \(R_{\scriptscriptstyle S}\) be the distance to the nearest interferer at a typical max-signal epoch. The following result is about the joint density of \(H^2_{\scriptscriptstyle S}\) and \(R^2_{\scriptscriptstyle S}\) and the proof of which is given in Appendix [subsection:JHRS].

The conditional probability density of \(R_{\scriptscriptstyle S}\), given \(H_{\scriptscriptstyle S}=h\) can be obtained using inverse Laplace transform.

3.6 Characteristics of a typical time↩︎

Suppose \(R\) denotes the distance to the serving BS at a typical time. Using Poissonianity, it is well known that \(R\) is distributed as Rayleigh with pdf \(f_R(r):=2\lambda\pi r \, e^{-\lambda\pi r^2}\).

3.7 Characteristics of max interference (MI)↩︎

Recall that the head point process is \(\mathcal{H}:=\sum_{i\in \mathbb{N}}\delta_{(T_i,H_i)}\). Suppose \(\mathcal{H}_{\scriptscriptstyle I}\leq \mathcal{H}\), i.e., \(Supp(\mathcal{H}_{\scriptscriptstyle I})\subset Supp(\mathcal{H})\), is the collection of points from \(\mathcal{H}\) restricted to the second lower envelope \(\mathcal{L}^{(2)}_e\), i.e., \(\mathcal{H}_{\scriptscriptstyle I}:=\mathcal{H}\vert_{\mathcal{L}^{(2)}_e}\). In this case, in short, we use \(\scriptscriptstyle I\) in the subscript or superscript in our notation, to mean max interference. We write the corresponding head point process \(\mathcal{H}_{\scriptscriptstyle I}\) as \[\mathcal{H}_{\scriptscriptstyle I}:=\sum_{j\in \mathbb{N}}\delta_{(T^{\scriptscriptstyle I}_j, H^{\scriptscriptstyle I}_j)} = \sum_{i\in \mathbb{N}}\delta_{(T_i, H_i)} {\mathbb{1}}{\left\{(T_i, H_i)\in \mathcal{L}^{(2)}_e \right\}}, \label{eq:Hcal95I}\tag{8}\] where we use the notation \((T^{\scriptscriptstyle I}_j, H^{\scriptscriptstyle I}_j)\) for a head point on \(\mathcal{L}_e^{(2)}\). Observe that from Proposition 1 with \(k=2\), \[\left\{(T_i, H_i)\in \mathcal{L}^{(2)}_e\right\}=\left\{\mathcal{H}(U^{T_i}_{H_i})=1\right\}, \label{eq:Hcal95I1}\tag{9}\] see Figure 6. Also notice that, for \((T_i, H_i)\in \mathcal{L}^{(2)}_e\), it is not necessary for the radial bird at \((T_i, H_i)\) to incur a handover. The sequence of points in \(\mathcal{H}_{\scriptscriptstyle I}\) gives rise to a stationary point process, say \(\mathcal{V}_{\scriptscriptstyle I}\) on \(\mathbb{R}\), consisting of the abscissas of the points in \(\mathcal{H}_{\scriptscriptstyle I}\).

Formally, \[\begin{align} \mathcal{V}_{\scriptscriptstyle I}&:= \sum_{j\in \mathbb{N}\,:\, (T^{\scriptscriptstyle I}_j, H^{\scriptscriptstyle I}_j)\in \mathcal{H}_{\scriptscriptstyle I}}\delta_{T^{\scriptscriptstyle I}_j},\nonumber \end{align}\] is the point process of the abscissas of the points from \(\mathcal{H}_{\scriptscriptstyle I}\) on the time axis, which we call the max interference point process. The time-stationarity of the point processes \(\mathcal{H}_{\scriptscriptstyle I}\) and \(\mathcal{V}_{\scriptscriptstyle I}\) is inherited from that of \(\mathcal{H}\). We can re-write \(\mathcal{V}_{\scriptscriptstyle I}\) using (8 ) and (9 ) as \[\begin{align} \mathcal{V}_{\scriptscriptstyle I}& = \!\!\!\!\!\!\!\sum_{(T_i, H_i)\in \mathcal{H}} \!\!\!\!\!\!\delta_{T_i} {\mathbb{1}}{\left\{(T_i, H_i){\in} \mathcal{L}^{(2)}_e\right\}} = \!\!\!\!\!\!\! \sum_{(T_i, H_i)\in \mathcal{H}} \!\!\!\!\!\! \delta_{T_i}{\mathbb{1}}{\left\{\mathcal{H}(U^{T_i}_{H_i}){=}1\right\}}. \label{eq:Acal95I2} \end{align}\tag{10}\] In the following result, we determine the intensity of the point process \(\mathcal{V}_{\scriptscriptstyle I}\), proof of which is provided in the Appendix ([subsection:L-IVH]).

This enables us to define the Palm distribution with respect to the point process \(\mathcal{V}_{\scriptscriptstyle I}\), equivalently with respect to a typical max interference (MI) event. Let us denote the distance of the nearest interferer at typical epoch of MI as \(H_{\scriptscriptstyle I}\). In the following, we derive the Palm probability distribution of \(H_{\scriptscriptstyle I}\) with respect to \(\mathcal{V}_{\scriptscriptstyle I}\). The proof of this is given in Appendix [subsection:T-IVHd].

Let \(R_{\scriptscriptstyle I}\) be the distance of the serving BS under the Palm probability of \(\mathcal{V}_{\scriptscriptstyle I}\), where \(R_{\scriptscriptstyle I}\) and \(H_{\scriptscriptstyle I}\) are dependent. In the following, we give the joint Laplace transform of \(H^2_{\scriptscriptstyle I}, R^2_{\scriptscriptstyle I}\) characterizing their joint distribution. The proof is given in Appendix [subsection:T-JHRI].

3.8 Characteristics of min interference (mI)↩︎

The tropical interference is the contribution from the second closest BS only. We are interested in analyzing the scenario when the tropical interference is minimum, and this happens at an epoch when the second closest BS is swapped. Two types of swaps can happen, namely, swaps given by pure handovers between nearest serving BS and nearest interferer, and swaps between second and third closest interferer. We call the first type a signal handover and second type a tropical interference handover. We denote the tropical interference handover point process by \(\mathcal{V}_{\scriptscriptstyle\mathcal{I}}\). In this case, in short, we use \(\scriptscriptstyle\mathcal{I}\) in the subscript or superscript in some of our notations as it corresponds to tropical interference.

We say that an intersection point \((s,h)\) of two radial birds is a tropical interference handover if \((s,h)\in \mathcal{L}^{(2)}_e\). Using the characterization from Proposition 1, the tropical interference handover point process is defined as \[\mathcal{V}_{\scriptscriptstyle\mathcal{I}}:=\sum_{(T_i,H_i)\in \mathcal{H}}\;\;\sum_{(T_j,H_j)\in \mathcal{H}\,:\, T_j < T_i} \delta_{\hat{S}}\, {\mathbb{1}}_{A_1(\hat{S}, \hat{H})}, \label{eq:VcalIHPP}\tag{11}\] where \(A_1(\hat{S}, \hat{H}):= \{\mathcal{H}(U^{\hat{S}}_{\hat{H}})=1\}\) similarly to (6 ). The following result is about the intensity of the point process \(\mathcal{V}_{\scriptscriptstyle\mathcal{I}}\), which is proved later in Appendix [subsection:L-VTI].

Let us denote the distance of a typical tropical interference handover as \(H_{\scriptscriptstyle\mathcal{I}}\). In the following, we give the Palm probability distribution of \(H_{\scriptscriptstyle\mathcal{I}}\) with respect to \(\mathcal{V}_{\scriptscriptstyle\mathcal{I}}\) and the proof is given in Appendix [subsection:T-TIH].

Let us denote the distance to the serving BS at this typical epoch by \(R_{\scriptscriptstyle\mathcal{I}}\), which is dependent on \(H_{\scriptscriptstyle\mathcal{I}}\). The following result is about the joint distribution of \(H^2_{\scriptscriptstyle\mathcal{I}}, R^2_{\scriptscriptstyle\mathcal{I}}\) and the proof is given in Appendix [subsection:T-JHRTI].

3.9 Comparison of distances: Laplace transform and MGF order↩︎

Similarly to the Laplace order \(H_{\scriptscriptstyle\mathcal{V}}\geq_{\textrm{L}} R \geq_{\textrm{L}}H_{\scriptscriptstyle S}\), form [26], we also have the following orderings.

The following lrd ordering will be useful in comparing the coverage probabilities in the tropical case, without fading.

3.10 Summary of densities of the typical distances↩︎

For ease of reading and navigation, we provide the list (Table ¿tbl:tab:tdensity?) for probability densities of the distances of interest and which will be essential in the performance evaluation later.

3.11 Distances of all other stations↩︎

Given the distance \(H \in \{H_{\scriptscriptstyle\mathcal{V}}, H_{\scriptscriptstyle S}, H_{\scriptscriptstyle I}, H_{\scriptscriptstyle\mathcal{I}}, R\}\) at different typical epochs, let us define the distances of all other BSs can be represented as a one dimensional point process \(\eta_H=\sum_i\delta_{D_i}\), on \((H,\infty)\). The following result can be shown to hold using arguments similar to those in [26]:

4 Performance evaluation & comparison↩︎

  For simplicity, let us assume that the both transmit power of each of the transmitter and the combined channel gain is \(1\).

4.1 System with fading↩︎

4.1.1 Performance metrics↩︎

The main quantities of interest are SINR and its tropical version, STINR, at the typical epochs, defined by \[SINR:=\frac{S}{\sigma^2+I},\; STINR:=\frac{S}{\sigma^2+\mathcal{T}},\] aligning with the definition (2 ) and (3 ), where \(\sigma^2\) is the thermal noise power, \(S\) is the received signal power to the user at the corresponding typical epoch \[S:=\begin{cases} \rho H_{\scriptscriptstyle\mathcal{V}}^{-\alpha} & \text{ at typical mS-MI epoch},\\ \rho H_{\scriptscriptstyle S}^{-\alpha}& \text{ at typical MS epoch},\\ \rho R_{\scriptscriptstyle I}^{-\alpha} & \text{ at typical MI epoch },\\ \rho R_{\scriptscriptstyle\mathcal{I}}^{-\alpha} & \text{ at typical mI epoch},\\ \rho R^{-\alpha} & \text{ at any typical epoch}, \end{cases} \label{eq:Int}\tag{12}\] using the random variable for the distance to the serving station, listed in Table ¿tbl:tab:abb?, and \(I\) and \(\mathcal{T}\) are the additive and maxitive (or tropical) interference power. Given \(H= H_{\scriptscriptstyle\mathcal{V}}, H_{\scriptscriptstyle S}, R_{\scriptscriptstyle I}, R_{\scriptscriptstyle\mathcal{I}}, R\), the interference \(I\) and the tropical interference \(\mathcal{T}\) are defined in terms of the distances \(H_{\scriptscriptstyle I}\) or \(H_{\scriptscriptstyle\mathcal{I}}\) to the nearest interferer(s), and distances \(\{D_i\}_{i\in \mathbb{N}}\) to all others, at the respective places in the proof of our results. The signal-to-interference-ratio (SIR), signal-to-tropical interference-ratio (STIR) and signal-to-noise-ratio (SNR) are defined naturally.

The factors \(\rho\) used for the signal and \(\{\rho_i\}_{i\in \mathbb{N}}\) that will be used for the interference correspond to the fading of the channels from the transmitters, which are assumed to be i. i. d. and distributed as Exp(\(\mu\)), for \(\mu>0\). The Table ¿tbl:table:CP95metrics? contains the formulas for the coverage probability metrics (SINR and STINR) in the scenario with fading.

At all the typical epochs, the coverage probability and Shannon rate or data rate are defined using \(\mathcal{S}\in \{SINR, STINR, SNR, SIR, STIR\}\) as \[p^c(\tau,\mu,\lambda,\alpha):= \mathbb{P}^0(\mathcal{S}\geq \tau),\] and \[\mathcal{R}(\mu,\lambda,\alpha):=\mathbb{E}^0[B\ln(1+\mathcal{S})],\] respectively, where \(B=b\log_2e\), with \(b\) being the bandwidth of the channel, which we assume to be a constant. Here \(\mathbb{P}^0\) is the corresponding Palm probability measure and \(\mathbb{E}^0\) is the Palm expectation with respect to \(\mathbb{P}^0\).

In the following, we determine the coverage probability and the average data rate using different performance metrics \(\mathcal{S}\in \{SNR, SIR, SINR, STIR, STINR\}\), at typical mS-MI, MS and typical time epoch. We present the proof in Appendix [proofs-wf].

In the case of typical max-interference (MI) and typical min-interference (mI), there exists an interferer at distance \(H \in \{H_{\scriptscriptstyle I}, H_{\scriptscriptstyle\mathcal{I}}\}\) and a serving BS is at distance \(R\in \{R_{\scriptscriptstyle I}, R_{\scriptscriptstyle\mathcal{I}}\}\) such that \(R\leq H\) a.s., and all other BSs are farther than \(H\). Subsequently the performance metrics are given by the following result, the proof of which is given in Section [proofs-wf].

4.2 System without fading↩︎

The main quantities of interest are the SINR and its tropical version STINR, at the typical epochs, are given by \[SINR:=\frac{S}{\sigma^2+I}, \; STINR:=\frac{S}{\sigma^2+\mathcal{T}},\] aligning with the definition (2 ) and (3 ), where the power of the received signal is defined as \[S:=\begin{cases} H_{\scriptscriptstyle\mathcal{V}}^{-\alpha} & \text{ at typical mS-MI epoch},\\ H_{\scriptscriptstyle S}^{-\alpha}& \text{ at typical MS epoch},\\ R_{\scriptscriptstyle I}^{-\alpha} & \text{ at typical MI epoch },\\ R_{\scriptscriptstyle\mathcal{I}}^{-\alpha} & \text{ at typical mI epoch},\\ R^{-\alpha} & \text{ at any typical epoch}, \end{cases} \label{eq:Int-nf}\tag{13}\] using the random variable for the distance to the serving station, listed in Table ¿tbl:tab:abb?, \(H_{\scriptscriptstyle\mathcal{V}}, H_{\scriptscriptstyle S}, R_{\scriptscriptstyle I}, R_{\scriptscriptstyle\mathcal{I}}, R\), for the corresponding cases, respectively. The additive interference \(I\) and the maxitive or tropical interference \(\mathcal{T}\) are introduced similarly to the case with fading, at the respective places in the proof of our results. In the following, we determine the coverage probability and the Shannon rate using different performance metrics \(\mathcal{S}\in \{SINR, STINR, SNR, SIR, STIR\}\), at different typical epochs of mS-MI, MS and typical time, as follows:

In the case of typical max-interference (MI) and typical min-interference (mI), there exists an interferer at distance \(H \in \{H_{\scriptscriptstyle I}, H_{\scriptscriptstyle\mathcal{I}}\}\) and a serving BS at a distance \(R\in \{R_{\scriptscriptstyle I}, R_{\scriptscriptstyle\mathcal{I}}\}\) such that \(R\leq H\) almost surely. All other BSs are farther than \(H\) and the locations of the latter are Poisson distributed. In this case, the interference \(I_H\) (or \(\mathcal{T}_H\) for tropical interference) is computed slightly differently with respect to the distance \(H\) to the nearest interferer. The performance metrics are given in:

In all our computations, we derive the distribution \(F_{I_h}\) or \(F_{\mathcal{T}_h}\) of the additive interference or tropical interference, respectively, given the distance \(H=h\) in different typical epochs. Due to the absence of fading factor, the proofs are much simpler.

For ease of reading, we present a table of notation for the coverage probabilities and the data rates with respect to different time epochs in Table ¿tbl:tab:tnotation?, as continuation of Table ¿tbl:tab:abb? and Table ¿tbl:tab:tdensity?.

4.3 Comparison of performance metrics↩︎

 

We now state another important result of this work that compares the coverage probability and the average data rates among different typical epochs, the proof of which is given in Appendix [section:CPE]

Let us now write the performance metrics as \(\mathcal{S}_{*}\) for SINR and \(\mathcal{S}_{\scriptscriptstyle*;\mathcal{T}}\) for STINR, where \(*\in \{\scriptscriptstyle{\mathcal{V}, S, I, \mathcal{I}, t}\) \(\}\). A natural stochastic domination \(\leq_{\textrm{st}}\) among these metrics follows from Theorem 6, when their distributions are compared under their respective Palm probability spaces.

Note that for any of the performance metrics \(\mathcal{S}\), the Laplace transform of \(\mathcal{S}\) can be written for any \(\gamma\geq 0\) as \[\begin{align} \mathbb{E}^0_{\#}\left[e^{-\gamma\mathcal{S}}\right] & = \gamma\int e^{-\gamma x} F_{\mathcal{S}}(x){\rm d}x \nonumber\\ &= \gamma\int e^{-\gamma x} \left(1-p^c(x)\right){\rm d}x, \label{eq:laplace-S} \end{align}\tag{14}\] where \(\#\in \{\scriptscriptstyle{\mathcal{V}, S, I, \mathcal{I}, t}\) \(\}\), \(p^c(x)=p^c_*(x,\mu,\lambda,\alpha)\) or \(p^c_{\scriptscriptstyle*;\mathcal{T}}(x,\mu,\lambda,\alpha)\) for the system with fading and \(p^c(x)=p^c_*(x,\lambda,\alpha)\) or \(p^c_{\scriptscriptstyle*;\mathcal{T}}(x,\lambda,\alpha)\) for the system without fading and \(*\in \{\scriptscriptstyle{\mathcal{V}, S, I, \mathcal{I}, t}\) \(\}\). Laplace order comparison of the performance metrics \(\mathcal{S}_{*}\) for SINR and \(\mathcal{S}_{\scriptscriptstyle*;\mathcal{T}}\) for STINR follows from (14 ), Theorem 6, and also from the stochastic domination in Corollary 8.

4.4 Comparison of interference power: Stochastic domination & Laplace transform order↩︎

Let us denote the additive interference power as \(I_{*}\) and maxitive interference power as \(\mathcal{T}_{*}\), for \(*\in \{\scriptscriptstyle{\mathcal{V}, S, I, \mathcal{I}, t}\) \(\}\). Then we have the following ordering, the proof of which is given in Appendix [Int-order].

4.5 Scale invariance in the interference limited regime↩︎

  We have the following scale invariance property of the coverage probability and data rate with respect to the intensity \(\lambda\) of the BSs and the fading parameter \(\mu\) (in an environment with fading), in the interference limited regime, i.e., \(\sigma=0\). In all these cases, the ratio \(\frac{H^{-\alpha}}{I_{H}}\) is scale invariant with respect to \(\lambda\), where \(H\) is the distance to the nearest BS and \(I_H\) is the additive or maxitive interference. As a result we have the scale invariance property of the coverage probabilities at different typical epochs, in the interference limited regime. The proof is given in Appendix [section:SI].

4.6 Attenuation functions: example and counter example↩︎

 

4.6.1 Bounded path-loss function↩︎

  All our comparison results in Theorem 6, Corollary 7, Corollary 9 and Theorem 7 can be shown to hold true for the bounded path-loss function \(\ell(r)= \left(1+r\right)^{-\alpha}\), as in [34], where \(\alpha\geq 2\) is the path-loss exponent. Our results can be verified along the same type of analysis, see Appendix [Example] for an instance. The type of attenuation function is very useful to eliminate the question of singularity of infinite signal power from a BS infinitesimally close to the user, in case of power law attenuation. In general, the bounded path-loss model naturally lacks the scale invariance property of the performance metrics in the interference limited regime, due to the extra additive term inside the attenuation function.

4.6.2 Step attenuation function↩︎

  On the other hand, the comparison results do not hold true for other attenuation functions, for example \(\ell(r)= p\, {\mathbb{1}}_{\{r\leq d\}}\), where \(p\) is a constant power and \(d\) is a fixed cut-off distance. For example, a disordered behavior, depending on the user QoS \(\tau\), among \(p_{\scriptscriptstyle\mathcal{V}}^c, p_{t}^c, p_{\scriptscriptstyle S}^c\) and \(p^c_{\scriptscriptstyle\mathcal{I}}, p^c_{\scriptscriptstyle I}\) with additive interference, and also for the tropical interference case, is shown in Appendix [Example]. As we will see therein, we also loose the scale invariance property, in this case.

5 Closed forms for coverage probabilities↩︎

  Much of our computations are focused on the determination of closed forms or simple integral forms for the coverage probabilities at various typical epochs. We list out all such expressions in different regimes and typical epochs, with or without fading, and having specific values of the path-loss exponent \(\alpha\). Nevertheless, the coverage probabilities in all these cases can be simulated numerically.

5.1 With fading↩︎

These closed forms are mainly for SINR, SIR and SNR regimes in the presence of fading.

5.1.1 SINR regime↩︎

In the SINR regime, for the specific case of \(\alpha=2\) we have \(p^c_*=0\), in any of the typical epochs, i.e., \(*\in \{\scriptscriptstyle{\mathcal{V}, S, I, \mathcal{I}, t}\) \(\}\). For \(\alpha=4\) we have \[\begin{align} p^c_* &=\frac{1}{(1+\tau)^{{\mathbb{1}}_{\{*=\mathcal{V}\}}}}\mathbb{E}^0_{H_*}\left[e^{-\mu\tau H_*^4\sigma^2 -\pi\lambda H_*^2 \kappa(\tau,4)} \right],\nonumber \end{align}\] where \(\kappa(\tau,4)= \tau^{{\frac{1}{2}}} \arctan(\tau^{{\frac{1}{2}}})\) and \(*\in \{\scriptscriptstyle{\mathcal{V}, S, t}\) \(\}\). In the other two typical epochs, \[\begin{align} p^c_*&{=} \frac{\tau^{-1/4}}{4}\int_0^\tau \!\!\!\frac{z^{-3/4}}{(1{+}z)^{1+{\mathbb{1}}_{\{*=\scriptscriptstyle\mathcal{I}\}}}}\mathbb{E}^0_{H_*}\!\! \left[e^{-\mu z H_*^4 \sigma^2-\pi\lambda H_*^2 \kappa(z, 4)}\right]{\rm d}z,\nonumber \end{align}\] where \(\kappa(z,4)= z^{{\frac{1}{2}}} \arctan(z^{{\frac{1}{2}}})\) and \(*\in \{\scriptscriptstyle{I, \mathcal{I}}\) \(\}\).

5.1.2 SIR regime↩︎

For \(\alpha{=}2\), we have \(p^c_{*}{=}0\), at any typical epoch, i.e., \(*\in \{\scriptscriptstyle{\mathcal{V}, S, I, \mathcal{I}, t}\) \(\}\). For \(\alpha=4\), we have \[p^c_{\scriptscriptstyle\mathcal{V}} = (1{+}\tau)^{-1}\left(1{+}\tau^{{\frac{1}{2}}} \arctan(\tau^{{\frac{1}{2}}})\right)^{-3/2},\] \[p_{t}^c =\left(1{+}\tau^{{\frac{1}{2}}} \arctan(\tau^{{\frac{1}{2}}})\right)^{-1}\!\!,\; p_{\scriptscriptstyle S}^c =\left(1{+}\tau^{{\frac{1}{2}}} \arctan(\tau^{{\frac{1}{2}}})\right)^{-1/2}\!\!,\] \[p^c_{\scriptscriptstyle I}=\frac{\tau^{-1/2}}{2}\!\!\!\int_0^{\tau} \!\! \frac{z^{-1/2}}{1+z} \left(1{+}z^{\frac{1}{2}}\arctan(z^{\frac{1}{2}})\right)^{-3/2}{\rm d}z,\] \[p^c_{\scriptscriptstyle\mathcal{I}}=\frac{\tau^{-1/2}}{2}\!\! \int_0^{\tau}\!\!\! \frac{z^{-1/2}}{(1{+}z)^2} \left(1{+}z^{\frac{1}{2}}\arctan(z^{\frac{1}{2}})\right)^{-5/2}{\rm d}z.\] The formula of \(p^c_t\) naturally matches that in [1]. Moreover, we have the following relations among the coverage probabilities in terms of \(p^c_{\scriptscriptstyle t}(\tau)\), in the interference limited regime, as \[p^c_{\scriptscriptstyle\mathcal{V}}(\tau)= \frac{(p^c_{\scriptscriptstyle t}(\tau))^{3/2}}{1+\tau},\;\; p^c_{\scriptscriptstyle S}(\tau)= (p^c_{\scriptscriptstyle t}(\tau))^{1/2},\] \[p^c_{\scriptscriptstyle I}(\tau)=\frac{\tau^{-1/2}}{2}\!\!\!\int_0^{\tau} \!\! \frac{z^{-1/2}}{1+z} (p^c_{\scriptscriptstyle t}(z))^{3/2}{\rm d}z,\] \[p^c_{\scriptscriptstyle\mathcal{I}}(\tau)=\frac{\tau^{-1/2}}{2}\!\!\!\int_0^{\tau} \!\! \frac{z^{-1/2}}{(1+z)^2} (p^c_{\scriptscriptstyle t}(z))^{5/2}{\rm d}z,\] by writing them as a function of \(\tau\) only.

5.1.3 SNR regime↩︎

In particular, for the coverage probability in the SNR regime with \(\alpha=2\), we have from (?? ) that \[p^c_*= \mathbb{E}^0_H\left[e^{-\mu\tau H^2\sigma^2}\right] = \left(1+\frac{\mu\tau\sigma^2}{\lambda\pi}\right)^{-\zeta},\] where \(*\in \{\scriptscriptstyle{\mathcal{V}, S, t}\) \(\}\), \(\zeta=3/2, 1/2, 1\) for \(H= H_{\scriptscriptstyle\mathcal{V}}, H_{\scriptscriptstyle S}, R\), using the Laplace transform of \(H_{\scriptscriptstyle\mathcal{V}}^2, H_{\scriptscriptstyle S}^2, R^2\), under their respective Palm probability spaces. Also from (?? ), Proposition 4 and Proposition 6, we have \[p^c_* {=}\mathbb{E}^0_H\left[e^{-\mu\tau H^2\sigma^2}\right]{=} \frac{2\lambda\pi}{\mu\tau\sigma^2}\!\!\left[1{-}\left(1{+}\frac{\mu\tau\sigma^2}{\lambda\pi}\right)^{-\zeta}\right],\] where \(*\in \{\scriptscriptstyle{I, \mathcal{I}}\) \(\}\), \(\zeta=1/2\) and \(3/2\), for \(H=R_{\scriptscriptstyle I}\) and \(R_{\scriptscriptstyle\mathcal{I}}\), respectively.

5.2 Without fading↩︎

In no-fading scenario, the closed form or integral form expressions we have are for any value of \(\alpha\), and these are mainly for the STINR, STIR and SNR regimes.

5.2.1 STINR regime↩︎

\[\begin{align} p^c_{\scriptscriptstyle\mathcal{V};\mathcal{T}}&= \begin{cases} F^0_{H_{\scriptscriptstyle\mathcal{V}}}\left(\left(\frac{1-\tau}{\tau\sigma^2}\right)^{\frac{1}{\alpha}}\right) & \text{ if } \tau<1\\ 0 & \text{ otherwise.} \end{cases}\nonumber \end{align}\] At typical max-interference and min-interference we have, for \(*\in \{\scriptscriptstyle{I, \mathcal{I}}\) \(\}\) \[p^c_{*;\scriptscriptstyle\mathcal{T}}= \tau^{-\frac{2}{\alpha}}\mathbb{E}^0_{H_{*}}\left[\left(1+\sigma^2 H_{*}^{\alpha}\right)^{-\frac{2}{\alpha}}\right].\]

5.2.2 STIR regime↩︎

For any \(\tau\geq 0\) \[p^c_{\scriptscriptstyle\mathcal{V};\mathcal{T}} =\begin{cases} 1if\tau < 1,\\ 0if\tau\geq 1, \end{cases} p^c_{\scriptscriptstyle t;\mathcal{T}}= \begin{cases} \tau^{-2/\alpha} & \text{ if }\tau>1\\ 1 &\text{ if } \tau\leq 1 \end{cases},\] \[p^c_{\scriptscriptstyle S;\mathcal{T}} = \begin{cases} \frac{1}{2\sqrt{\pi}}\mathbb{E}^0_{H_{\scriptscriptstyle S}}\left[\Gamma\left(0, \lambda\pi (\tau^{2/\alpha}-1)H^2_{\scriptscriptstyle S}\right)\right] & \text{ if }\tau>1\\ 1 &\text{ if } \tau\leq 1. \end{cases}\nonumber\] For any \(\tau\geq 1\), \(p^c_{\scriptscriptstyle I;\mathcal{T}}= \tau^{-2/\alpha}= p^c_{\scriptscriptstyle\mathcal{I};\mathcal{T}}\).

5.2.3 SNR regime↩︎

The coverage probability, in general written as \(p^c_*\), is given by \[p^c_* = \mathbb{P}^0\left(H\leq (\tau \sigma^2)^{-\frac{1}{\alpha}}\right),\] for \(H= H_{\scriptscriptstyle\mathcal{V}}, H_{\scriptscriptstyle S}, R_{\scriptscriptstyle I}, R_{\scriptscriptstyle\mathcal{I}}, R\) and \(*\in \{\scriptscriptstyle{\mathcal{V}, S, I, \mathcal{I}, t}\) \(\}\), under their respective Palm probability measures \(\mathbb{P}^0\).

6 Numerical results and discussions↩︎

In this part we present the numerical illustrations of our results on the performance evaluation in terms of coverage probabilities and their comparison at different typical epochs, in both cases with and without fading. The closed form or semi explicit integral form derived by leveraging the Poissonianity, enable us to create plots of these performance metrics quite conveniently. The coverage probabilities are plotted against the user QoS requirement variable \(\tau\) in dB scale. The system parameters are the intensity of BSs \(\lambda\), the exponent \(\alpha\) of the path-loss attenuation function, \(\mu\) the fading parameter and \(\sigma\) the parameter for the thermal noise.

The plots and their comparisons are made for different \(\lambda\) and \(\alpha\) values. Figure 7 plots coverage probabilities against the SINR threshold \(\tau\) in dB scale, at typical handover, typical time and typical max-signal, when \(\alpha=3,4\), \(\mu=2\), \(\sigma^2=10^{-6}\), and \(\lambda=10^{-4}\) (or \(5\times 10^{-4}\) in some cases) corresponding to LEO constellation scale and \(\lambda=1\) (or \(10^{-1}\) in some cases) corresponding to cellular scale, respectively. Figure 8 is similar for the cases of typical max-interference and min-interference. A similar comparison of plots is also provided in Figure 9 and Figure 10 with the same values of the parameters.

In the tropical cases, we have similar comparison plots of coverage probabilities in Figure 11 and Figure 12 in the case with fading, Figure [fig:T-HotMSNF] and Figure 14 in the case without fading. All these plots validate our comparison of coverage probabilities stated in Theorem 6, except for the comparison of coverage probabilities \(p_{\scriptscriptstyle\mathcal{V};\mathcal{T}}^c, p_{\scriptscriptstyle t;\mathcal{T}}^c\) and \(p_{\scriptscriptstyle S;\mathcal{T}}^c\) at typical handover, typical time and typical max-signal, respectively, for the case of tropical SINR without fading, as shown at the end of Appendix [section:CPE] and also validated in Figure [fig:T-HotMSNF]. Our plot shows a different comparison among \(p_{\scriptscriptstyle\mathcal{V};\mathcal{T}}^c, p_{\scriptscriptstyle S;\mathcal{T}}^c\) and \(p_{\scriptscriptstyle t;\mathcal{T}}^c\) holds for the threshold parameter \(\tau\geq 0\) [dB]. Performance is better at the typical time epoch for high density \((\lambda)\) of the BSs, which is not the case in low density regimes.

For brevity we have presented the numerical comparison just for the ordering among the coverage probabilities, presented in Theorem 6, but not for the Shannon rates (Corollary 7) and the interferences (Theorem 7).

7 Conclusions and future research↩︎

This work addresses the question of performance evaluation in a Poisson based dynamical cellular communication system at various typical epochs, at which a UE tend to suffer or benefit from the presence of interferer(s) or server close or far from it. Building on top of the mobility model described in [26] using stochastic geometry, we have derived the coverage probabilities at these epochs, depending on the system parameter, and made comparison between them. Our approach based on the determination of the characteristics of several typical distances that are essential for the performance evaluation. The ergodic data rate among different typical epochs correspond to different seasons of Shannon rate with respect to our seasonal analogy of the system performance. This article contains a complete study of system performance in a Poisson BSs setting in the absence or presence of fading and noise, along with simple assumption of Rayleigh fading and classical use case of power law path-loss attenuation function. The in-built tractability of the Poisson analysis allows one to analyze many other fading distribution and path-loss attenuation function.

As a future direction of research, we plan to characterize various typical epochs of interest, investigate the performance evaluation at those typical epochs and their comparisons in other dynamical wireless system models, mentioned in [26], for example 3D Poisson setting, 2D & 3D Cox setting, Poisson and Cox setting under the spherical set up, with single and multiple speed and altitude scenarios, with or without visibility constraint. The Markov chain construction of the successive handover process developed in [26], allows one to investigate the evolution of the seasons as a Markov chain, which we keep as a future work.

8 Proofs: Preliminaries↩︎

 

8.1 Proof of Proposition 1↩︎

  The result is a corollary of Property [in-out]. Indeed, if there are more than \((k-1)\) point in \(U^{s}_{h}\), then the radial bird at those points intersects the vertical line \(t=s\) at a level below \(h\), contradicting the fact that \((s,h)\in \mathcal{L}^{(k)}_e\). For part [b], if \((s,h)\) is an intersection point of two radial birds, then \(\overline{U^s_h}\) will contain two extra points corresponding to the two radial birds. \(\square\)

8.2 Proof of Proposition 2↩︎

  Here we use the intensity of the point process \(\mathcal{V}_{\scriptscriptstyle S}\), \(\lambda_{\scriptscriptstyle S}= \sqrt{\lambda}\) from [26]. The joint Laplace transform of \(H_{\scriptscriptstyle S}^2, R_{\scriptscriptstyle S}^2\) with parameter \(\gamma, \beta\) under the Palm probability measure of \(\mathcal{V}_{\scriptscriptstyle S}\) is \[\begin{align} \nonumber\\ &= \frac{1}{\lambda_{\scriptscriptstyle S}}\mathbb{E}\left[\sum_{(T^{\scriptscriptstyle S}_i, H^{\scriptscriptstyle S}_i)\in \mathcal{H}_{\scriptscriptstyle S}: T^{\scriptscriptstyle S}_i\in [0,1]}e^{-\gamma \left(H^{\scriptscriptstyle S}_i\right)^2-\beta(R_i^{\scriptscriptstyle S})^2}\right]\nonumber\\ &= \frac{1}{\sqrt{\lambda}}\mathbb{E}\left[\sum_{(T_i,H_i)\in \mathcal{H}: T_i\in [0,1]}e^{-\gamma H_i^2-\beta(R_i^{\scriptscriptstyle S})^2}{\mathbb{1}}{\left\{\mathcal{H}(U^{T_i}_{H_i})=0\right\}}\right.\nonumber\\ & \times {\mathbb{1}}{\left\{\exists (T_j,H_j)\in \mathcal{H}: (T_i,H_i)\in U^{T_i}_{R_i^{\scriptscriptstyle S}}\right\}}\Bigg]\nonumber\\ &= \frac{1}{\sqrt{\lambda}}\mathbb{E}\left[\sum_{(T_i,H_i)\in \mathcal{H}: T_i\in [0,1]}e^{-\gamma H_i^2}{\mathbb{1}}_{\left\{\mathcal{H}(U^{T_i}_{H_i})=0\right\}}\sum_{(T_j,H_j)\in \mathcal{H}\vert_{(U^{T_i}_{H_i})^c}} e^{-\beta R_{i,j}^2}\right.\nonumber\\ & \times {\mathbb{1}}{\left\{\mathcal{H}\left(U^{T_i}_{R_{i,j}}\setminus U^{T_i}_{H_i}\right)=1\right\}}\Bigg]\nonumber\\ &= \frac{1}{\sqrt{\lambda}} \mathbb{E}\left[\sum_{\substack{(T_i,H_i)\in \mathcal{H}\\ T_i\in [0,1]}}e^{-\gamma H_i^2}\sum_{(T_j,H_j)\in \mathcal{H}\vert_{(U^{T_i}_{H_i})^c}} e^{-\beta R_{i,j}^2} {\mathbb{1}}_{\mathcal{H}(U^{T_i}_{R_{i,j}})=1}\right],\nonumber \end{align}\] where \((R_i^{S})^2 \equiv R_{i,j}^2= (T_i-T_j)^2+H_j^2\). By applying the multivariate Campbell-Mecke formula for the factorial power of order 2 of \(\mathcal{H}\), i.e., \(\mathcal{H}^{2,\neq}\), we get \[\begin{align} \nonumber\\ &=\!\!\frac{4\lambda^2}{\sqrt{\lambda}} \int_0^1\!\!\int_0^\infty \!\!\!\int_{(U^{t_i}_{h_i})^c} e^{-\gamma h_i^2} e^{-(\beta+\lambda\pi) ((t_i-t_j)^2+h_j^2)} {\rm d}h_j {\rm d}t_j {\rm d}h_i {\rm d}t_i\nonumber\\ &=4\lambda^{\frac{3}{2}} \int_0^\infty \int_{(U^0_{h_i})^c}e^{-\gamma h_i^2} e^{-(\beta+\lambda\pi) (t^2+h_j^2)} {\rm d}h_j {\rm d}t {\rm d}h_i\nonumber\\ &=2\lambda^{3/2} \pi\int_0^\infty e^{-\gamma h_i^2} K_{\beta+\lambda\pi}(h_i) {\rm d}h_i, \label{eq:Jbeta0} \end{align}\tag{15}\] where \[\begin{align} K_{\beta+\lambda\pi}(h_i)&:=\frac{2}{\pi}\int_{(U^0_{h_i})^c} e^{-(\beta+\lambda\pi) (t^2+h_j^2)} {\rm d}h_j {\rm d}t\nonumber\\ &=\frac{2}{\pi}\int_0^\pi\int_{h_i}^\infty e^{-(\beta+\lambda\pi) r^2} r{\rm d}r{\rm d}\theta \nonumber\\ &= \frac{1}{\beta+\lambda\pi}e^{-(\beta+\lambda\pi) h_i^2}. \label{eq:Jbeta1} \end{align}\tag{16}\] Substituting from (16 ) to (15 ), we get \[\begin{align} \mathbb{E}^0_{\mathcal{V}_{\scriptscriptstyle S}}\left[e^{-\gamma H_{\scriptscriptstyle S}^2-\beta R_{\scriptscriptstyle S}^2}\right] &=\frac{2\lambda^{3/2} \pi}{\beta+\lambda\pi} \int_0^\infty e^{-(\gamma+\beta+\lambda\pi) h_i^2} {\rm d}h_i\nonumber\\ &=\frac{\lambda^{3/2} \pi^{3/2}}{\beta+\lambda\pi} (\gamma+\beta+\lambda\pi)^{-{\frac{1}{2}}}\nonumber\\ &{=} \left(1{+}\frac{\beta}{\lambda\pi}\right)^{-1}\!\! \left(1{+}\frac{\gamma+\beta}{\lambda\pi}\right)^{-{\frac{1}{2}}}.\nonumber \end{align}\] Note that for \(\gamma=0\), we have \(\mathbb{E}^0_{\mathcal{V}_{\scriptscriptstyle S}}\left[e^{-\beta R_{\scriptscriptstyle S}^2}\right]=\left(1+\frac{\beta}{\lambda\pi}\right)^{-3/2}\) and hence \(R_{\scriptscriptstyle S}\) has a Nakagami distribution with parameters \(\left(\frac{3}{2}, \frac{3}{2\lambda\pi}\right)\). \(\square\)

8.3 Proof of Lemma 1↩︎

  Using the alternative definition of \(\mathcal{V}_{\scriptscriptstyle I}\) in (10 ) the intensity of the point process \(\mathcal{V}_{\scriptscriptstyle I}\) is \[\begin{align} \lambda_{\scriptscriptstyle I}&=\mathbb{E}\left[\sum_{T^{\scriptscriptstyle I}_j\in \mathcal{V}_{\scriptscriptstyle I}\,:\, 0\leq T^{\scriptscriptstyle I}_j\leq 1} \!\!\!\! 1\right] = \mathbb{E}\left[\sum_{\substack{(T_i,H_i)\in \mathcal{H}\\0\leq T_i\leq 1}} \!\!\!\!{\mathbb{1}}{\left\{\mathcal{H}(U^{T_i}_{H_i})=1\right\}} \right].\nonumber \end{align}\] Applying Campbell-Mecke formula, we have \[\begin{align} \lambda_{\scriptscriptstyle I}&=2\lambda\int_0^1\int_0^\infty \lambda\pi h^2 e^{-\lambda\pi h^2} {\rm d}h {\rm d}t\nonumber\\ &= 2\lambda\int_0^\infty \lambda\pi h^2 e^{-\lambda\pi h^2} {\rm d}h\nonumber\\ &= \lambda\int_0^\infty \!\!\!\!\!\!x e^{-x} \frac{1}{\sqrt{\lambda\pi x}}{\rm d}x = \sqrt{\frac{\lambda}{\pi}} \int_0^\infty \!\!\!x^{\frac{3}{2}-1} e^{-x}{\rm d}x = \frac{\sqrt \lambda}{2}.\nonumber\hfill \square \end{align}\]

8.4 Proof of Proposition 3↩︎

  The Laplace transform of \(H_{\scriptscriptstyle I}^2\) under the Palm probability measure of \(\mathcal{V}_{\scriptscriptstyle I}\) is \[\begin{align} \mathbb{E}^0_{\mathcal{V}_{\scriptscriptstyle I}}\left[e^{-\gamma H_{\scriptscriptstyle I}^2}\right] &= \frac{1}{\lambda_{\scriptscriptstyle I}}\mathbb{E}\left[\sum_{(T^{\scriptscriptstyle I}_j, H^{\scriptscriptstyle I}_j)\in \mathcal{H}_{\scriptscriptstyle I}: T^{\scriptscriptstyle I}_j\in [0,1] }e^{-\gamma \left(H^{\scriptscriptstyle I}_j\right)^2}\right]\nonumber\\ &{=} \frac{1}{\lambda_{\scriptscriptstyle I}}\mathbb{E}\!\!\left[\sum_{\substack{(T_i,H_i)\in \mathcal{H}\\ T_i\in [0,1]}}e^{-\gamma H_i^2}{\mathbb{1}}{\left\{\mathcal{H}(U^{T_i}_{H_i})=1\right\}}\right]. \label{eq:H2I} \end{align}\tag{17}\] Applying the Campbell-Mecke formula and using \(\lambda_{\scriptscriptstyle I}={\frac{1}{2}}\sqrt{\lambda}\) from Lemma 1, the last term in (17 ) equals \[\begin{align} \nonumber\\ &=4\sqrt{\lambda} \int_0^\infty \lambda\pi h^2 e^{-(\gamma+\lambda\pi) h^2} {\rm d}h {\rm d}t\nonumber\\ &= 4\sqrt{\lambda} \frac{\lambda\pi}{\gamma+\lambda\pi} \int_0^\infty x e^{-x} \frac{1}{2\sqrt{(\gamma+\lambda\pi)x}}{\rm d}x\nonumber\\ &= 2\pi \left(\frac{\lambda}{\gamma+\lambda\pi}\right)^{\frac{3}{2}} \Gamma\left(\frac{3}{2}\right)= \left(1+\frac{\gamma}{\lambda\pi}\right)^{-\frac{3}{2}}.\nonumber \end{align}\] This shows that \(H_{\scriptscriptstyle I}\) follows a Nakagami distribution with parameters \(\left(\frac{3}{2}, \frac{3}{2\lambda\pi}\right)\). \(\square\)

8.5 Proof of Proposition 4↩︎

  Given \((T^{\scriptscriptstyle I}_i,H^{\scriptscriptstyle I}_i)\in \mathcal{H}_{\scriptscriptstyle I}\), if \((T_j,H_j)\in \mathcal{H}\vert_{U^{T_i^{\scriptscriptstyle I}}_{H_i^{\scriptscriptstyle I}}}\), then \(R^{\scriptscriptstyle I}_i:=\left((T_i^{\scriptscriptstyle I}-T_j)^2+ H_j^2\right)^{\frac{1}{2}}\leq H_i^{\scriptscriptstyle I}\). The joint Laplace transform of \(H_{\scriptscriptstyle I}^2, R_{\scriptscriptstyle I}^2\) is given by \[\begin{align} \nonumber\\ &= \frac{1}{\lambda_{\scriptscriptstyle I}}\mathbb{E}\left[\sum_{(T^{\scriptscriptstyle I}_i, H^{\scriptscriptstyle I}_i)\in \mathcal{H}_{\scriptscriptstyle I}: T^{\scriptscriptstyle I}_i\in [0,1]}e^{-\gamma \left(H^{\scriptscriptstyle I}_i\right)^2-\beta(R_i^{\scriptscriptstyle I})^2}\right]\nonumber\\ &= \frac{1}{\lambda_{\scriptscriptstyle I}}\mathbb{E}\left[\sum_{\substack{(T_i,H_i)\in \mathcal{H}\\ T_i\in [0,1]}}e^{-\gamma H_i^2-\beta(R_i^{\scriptscriptstyle I})^2}{\mathbb{1}}_{\left\{\mathcal{H}(U^{T_i}_{H_i})=1\right\}}\right.\nonumber\\ & \times {\mathbb{1}}{\left\{\exists (T_j,H_j)\in \mathcal{H}: (T_j,H_j)\in U^{T_i}_{H_i}\right\}}\Bigg]\nonumber\\ &= \frac{1}{\lambda_{\scriptscriptstyle I}}\mathbb{E}\left[\sum_{\substack{(T_i,H_i)\in \mathcal{H}\\ T_i\in [0,1]}}e^{-\gamma H_i^2} {\mathbb{1}}_{\left\{\mathcal{H}(U^{T_i}_{H_i})=1\right\}}\sum_{(T_j,H_j)\in \mathcal{H}\vert_{U^{T_i}_{H_i}}} e^{-\beta R_{i,j}^2}\right],\nonumber \end{align}\] where, \(R_{i,j}^2= (T_i-T_j)^2+H_j^2\). By applying the multivariate Campbell-Mecke formula for the factorial power of order 2 of \(\mathcal{H}\), i.e., \(\mathcal{H}^{2,\neq}\) similar to the Palm distribution in [26], we obtain that \[\begin{align} \nonumber\\ &=\frac{4\lambda^2}{\lambda_{\scriptscriptstyle I}} \int_0^1\int_0^\infty \!\!\!\int_{U^{t_i}_{h_i}} e^{-(\gamma+\lambda\pi) h_i^2} e^{-\beta((t_i-t_j)^2+h_j^2)} {\rm d}h_j {\rm d}t_j {\rm d}h_i {\rm d}t_i\nonumber\\ &=8\lambda^{\frac{3}{2}} \int_0^\infty \int_{U^0_{h_i}}e^{-(\gamma+\lambda\pi) h_i^2} e^{-\beta(t^2+h_j^2)} {\rm d}h_j {\rm d}t {\rm d}h_i\nonumber\\ &=4\lambda^{3/2} \pi\int_0^\infty e^{-(\gamma+\lambda\pi) h_i^2} J_{\beta}(h_i) {\rm d}h_i, \label{eq:HIR-2} \end{align}\tag{18}\] where \[\begin{align} J_{\beta}(h_i)&:=\frac{2}{\pi}\int_{ U^0_{h_i}} e^{-\beta(t^2+h_j^2)} {\rm d}h_j {\rm d}t\nonumber\\ &=\frac{2}{\pi}\int_0^\pi\int_0^{h_i} e^{-\beta r^2} r{\rm d}r{\rm d}\theta = \frac{1}{\beta}\left(1-e^{-\beta h_i^2}\right). \label{eq:Jbeta} \end{align}\tag{19}\] This implies that, conditioned on \(H_{\scriptscriptstyle I}= h_{\scriptscriptstyle I}\), \(R^2_{\scriptscriptstyle I}\) is uniformly distributed on \([0,h^2_{\scriptscriptstyle I}]\). Hence, from (18 ) and (19 ), we have \[\begin{align} \nonumber\\ &=\frac{4\lambda^{3/2}\pi}{\beta} \int_0^\infty e^{-(\gamma+\lambda\pi) h_{\scriptscriptstyle I}^2} \left(1-e^{-\beta h_{\scriptscriptstyle I}^2}\right) {\rm d}h_{\scriptscriptstyle I}\nonumber\\ &=\frac{4\lambda^{3/2}\pi}{\beta}\left[ \int_0^\infty e^{-(\gamma+\lambda\pi) h_{\scriptscriptstyle I}^2} {\rm d}h_{\scriptscriptstyle I} {-}\int_0^\infty e^{-(\gamma+\lambda\pi+\beta) h_{\scriptscriptstyle I}^2} {\rm d}h_{\scriptscriptstyle I} \right]\nonumber\\ &=\frac{2\lambda^{3/2}\pi^{3/2}}{\beta}\left[(\gamma+\lambda\pi)^{-{\frac{1}{2}}}-(\gamma+\lambda\pi+\beta)^{-{\frac{1}{2}}}\right]\nonumber\\ &=\frac{2\lambda\pi}{\beta}\left[\left(1+\frac{\gamma}{\lambda\pi}\right)^{-{\frac{1}{2}}}-\left(1+\frac{\gamma+\beta}{\lambda\pi}\right)^{-{\frac{1}{2}}}\right],\nonumber \end{align}\] as required in this result. \(\square\)

8.6 Proof of Lemma 2↩︎

  For any two points \((t_1,h_1), (t_2,h_2)\) with \(t_2<t_1\), we have that the square of the distance to the intersection of two radial birds at \((t_1,h_1), (t_2,h_2)\) is given by \[\begin{align} \hat{h}^2 &= \frac{1}{4}\left[(t_1-t_2)^2+2(h_1^2+h_2^2)+\frac{(h_2^2-h_1^2)^2}{(t_1-t_2)^2}\right]. \label{eq:hat-h} \end{align}\tag{20}\] Applying the multivariate Campbell-Mecke formula for the factorial power of order 2 of \(\mathcal{H}\), i.e., \(\mathcal{H}^{2,\neq}\), the intensity of \(\mathcal{V}_{\scriptscriptstyle\mathcal{I}}\) is \[\begin{align} \lambda_{\scriptscriptstyle\mathcal{I}} &= 4\lambda^2\int_0^1\int_{0}^{\infty}\int_{0}^{\infty}\int_{-\infty}^{t_1} \lambda\pi \hat{h}^2 e^{-\lambda\pi \hat{h}^2} {\rm d} t_2\, {\rm d} h_2\, {\rm d} h_1 \, {\rm d} t_1 \nonumber\\ &= 4\lambda^2\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty} \lambda\pi \hat{h}^2 e^{-\lambda\pi \hat{h}^2} {\rm d} t\, {\rm d} h_1\, {\rm d} h_2, \label{eq:h2} \end{align}\tag{21}\] where \(\hat{h}\) is a function of \(t=t_1-t_2\), \(h_1\) and \(h_2\). We write the inner integral as \[\begin{align} \nonumber\\ &= e^{-\frac{\lambda\pi}{2}(h_1^2+h_2^2)} \left(I(a,b) + \frac{\lambda\pi}{2} (h_1^2+h_2^2) J(a,b) \right), \label{eq:IJ} \end{align}\tag{22}\] where \(a=\frac{\lambda\pi}{4}\) and \(b=\frac{\lambda\pi}{4} (h_2^2-h_1^2)^2\) and \[\begin{align} I(a,b)=\int_0^\infty \left(a t^2+\frac{b}{t^2}\right) e^{-\left(a t^2+\frac{b}{t^2}\right)}{\rm d}t, \label{eq:Iab} \end{align}\tag{23}\] \[\begin{align} J(a,b)=\int_0^\infty e^{-\left(a t^2+\frac{b}{t^2}\right)}{\rm d}t = \frac{\sqrt{\pi}}{2\sqrt{a}}e^{-2\sqrt{ab}}. \label{eq:Jab} \end{align}\tag{24}\] The integral \(J(a,b)\) in (24 ) is computed using Cauchy-Schlömilch transformation [35]. We first compute an integral of the form (23 ). Using Leibniz’s rule for differentiation under the integral sign for two parameter \(a\) and \(b\), we have \[\begin{align} I(a,b)&= -a\frac{\partial}{\partial a} J(a,b)- b\frac{\partial}{\partial b} J(a,b)\nonumber\\ & {=}{-}\frac{\sqrt{\pi}}{2} e^{-2\sqrt{ab}}\left[a\left({-}{\frac{1}{2}}a^{-3/2}{-} \sqrt{b}/a\right)+b \left({-}\frac{1}{\sqrt{b}}\right) \right]\nonumber\\ & =\frac{\sqrt{\pi}}{2} e^{-2\sqrt{ab}}\left[{\frac{1}{2}}a^{-1/2}+ 2\sqrt{b} \right] \nonumber\\ & =\frac{\sqrt{\pi}}{4\sqrt{a}} e^{-2\sqrt{ab}}\left(1+ 4\sqrt{ab} \right). \nonumber \end{align}\] Then using \(a=\frac{\lambda\pi}{4}\) and \(b=\frac{\lambda\pi}{4} (h_2^2-h_1^2)^2\) in (22 ), we have \[\begin{align} \nonumber\\ &= e^{-\frac{\lambda\pi}{2}(h_1^2+h_2^2)} e^{-2\sqrt{ab}} \frac{\sqrt{\pi}}{4\sqrt{a}} \left(1+ 4\sqrt{ab} + \lambda\pi (h_1^2+h_2^2) \right)\nonumber\\ &{=} e^{-\frac{\lambda\pi}{2}\left(h_1^2+h_2^2+|h_1^2-h_2^2|\right)} \frac{1}{2\sqrt{\lambda}} \left[1+ \lambda\pi(|h_1^2-h_2^2|{+}h_1^2+h_2^2) \right]\nonumber\\ &= \frac{1}{2\sqrt{\lambda}} e^{-\lambda\pi(h_1^2\vee h_2^2)} \left(1+ 2\lambda\pi(h_1^2\vee h_2^2)\right). \label{eq:IJ1} \end{align}\tag{25}\] Using the value of the inner integral from (25 ) in (21 ) we have \[\begin{align} \lambda_{\scriptscriptstyle\mathcal{I}}&{=} 2\lambda^{3/2}\!\!\int_{0}^{\infty}\!\!\!\!\int_{0}^{\infty}\!\!\!\! e^{-\lambda\pi(h_1^2\vee h_2^2)} \left(1+ 2\lambda\pi(h_1^2\vee h_2^2)\right) {\rm d} h_1\, {\rm d} h_2\nonumber\\ &= \frac{2\sqrt{\lambda}}{\pi} \int_{0}^{\infty}\!\!\!\! \int_{0}^{\infty}\!\!\!\! e^{-h_1^2\vee h_2^2} \left(1{+} 2(h_1^2\vee h_2^2)\right){\rm d} h_1 {\rm d} h_2. \label{eq:h3} \end{align}\tag{26}\] Since the integrand is a symmetric function of \(h_1,h_2\), it is enough to compute the integral for \(h_2\in [0,\infty)\) and \(h_1\in [0,h_2]\). Using this, it can be proved that \[\int_{(\mathbb{R}^+)^2}\!\!\!\!\! \!\!\! e^{-(h_1^2\vee h_2^2)} {\rm d} h_1\, {\rm d} h_2 = 1 = \int_{(\mathbb{R}^+)^2} \!\!\!\!\! (h_1^2\vee h_2^2) e^{-(h_1^2\vee h_2^2)} {\rm d} h_1\, {\rm d} h_2.\] This shows from (26 ) that \(\lambda_{\scriptscriptstyle\mathcal{I}} = \frac{6\sqrt{\lambda}}{\pi}\). \(\square\)

8.7 Proof of Proposition 5↩︎

  The Laplace transform of \(H_{\scriptscriptstyle\mathcal{I}}^2\) is obtained under the Palm probability measure of \(\mathcal{V}_{\scriptscriptstyle\mathcal{I}}\) similarly to the Palm distribution in [26] as follows: \[\begin{align} \nonumber\\ &= \frac{1}{\lambda_{\scriptscriptstyle\mathcal{I}}}\mathbb{E}\left[\sum_{\substack{(T_i,H_i)\in \mathcal{H}\\ T_i\in [0,1]}} \sum_{\substack{(T_j,H_j)\in \mathcal{H}\\ T_j<T_i}}e^{-\gamma \hat{H}^2}{\mathbb{1}}_{\left\{\mathcal{H}(U^{\hat{S}}_{\hat{H}})=1\right\}}\right]\nonumber\\ &{=}\frac{4\lambda^2}{\lambda_{\scriptscriptstyle\mathcal{I}}}\int_0^1 \!\! \!\int_{0}^{\infty} \!\!\! \int_{0}^{\infty} \!\! \!\int_{-\infty}^{t_1} \!\!\!\!\! \lambda\pi \hat{h}^2 e^{-(\gamma+\lambda\pi) \hat{h}^2} {\rm d} t_2\, {\rm d} h_2\, {\rm d} h_1 \, {\rm d} t_1 \nonumber\\ &=\frac{2\lambda^{3/2}\pi}{3}\int_{0}^{\infty} \!\! \int_{0}^{\infty} \!\!\int_0^{\infty} \!\! \lambda\pi \hat{h}^2 e^{-(\gamma+\lambda\pi) \hat{h}^2} {\rm d} t\, {\rm d} h_2\, {\rm d} h_1, \label{eq:HTI1} \end{align}\tag{27}\] by applying the multivariate Campbell-Mecke formula for the factorial power of order 2 of \(\mathcal{H}\), where \(\lambda_{\scriptscriptstyle\mathcal{I}} = \frac{6\sqrt{\lambda}}{\pi}\), from Lemma 2, and \(\hat{h}\) is function of \(t=t_1-t_2, h_1\) and \(h_2\) as derived in (20 ). We evaluate the inner integral in (27 ) as \[\begin{align} \nonumber\\ &= \frac{\sqrt{\pi}}{2\sqrt{\gamma+\lambda\pi}} e^{-(\gamma+\lambda\pi)(h_1^2\vee h_2^2)} \left(1+ 2(\gamma+\lambda\pi)(h_1^2\vee h_2^2)\right)\nonumber\\ &{=}\frac{1}{2\sqrt{\lambda}} \!\!\left(\frac{\lambda\pi}{\gamma{+}\lambda\pi}\right)^{1/2} \!\!\! e^{-(\gamma{+}\lambda\pi)(h_1^2\vee h_2^2)}\! \left(1{+}2(\gamma{+}\lambda\pi)(h_1^2\vee h_2^2)\right), \label{eq:HTI2} \end{align}\tag{28}\] using similar computation up to (25 ). Substituting from (28 ) to (27 ), we have \[\begin{align} \mathbb{E}^0_{\mathcal{V}_{\scriptscriptstyle\mathcal{I}}}\left[e^{-\gamma H_{\scriptscriptstyle\mathcal{I}}^2}\right] &= \frac{\lambda\pi}{3} \left(\frac{\lambda\pi}{\gamma+\lambda\pi}\right)^{3/2}\int_{0}^{\infty}\!\!\!\int_{0}^{\infty} \!\!\! e^{-(\gamma+\lambda\pi)(h_1^2\vee h_2^2)} \nonumber\\ &\times \left(1+ 2(\gamma+\lambda\pi)(h_1^2\vee h_2^2)\right)\,{\rm d} h_2\, {\rm d} h_1\nonumber\\ &{=}\frac{\lambda\pi}{3}\!\! \left(\!\frac{\lambda\pi}{\gamma{+}\lambda\pi}\!\right)^{3/2}\!\!\!\!\!\!\frac{3}{\gamma{+}\lambda\pi}{=} \left(1{+}\frac{\gamma}{\lambda\pi}\right)^{-5/2}, \label{eq:LIcal} \end{align}\tag{29}\] since the double integral in the first step in (29 ) equals to \(\frac{3}{\gamma+\lambda\pi}\) similarly to (26 ). \(\square\)

8.8 Proof of Proposition 6↩︎

  The joint distribution of \(H_{\scriptscriptstyle\mathcal{I}}^2, R_{\scriptscriptstyle\mathcal{I}}^2\) is given by the joint Laplace transform obtained similarly to the Palm distribution in [26], as follows: \[\begin{align} \nonumber\\ &= \frac{1}{\lambda_{\scriptscriptstyle\mathcal{I}}}\mathbb{E}\left[\sum_{\substack{(T_i,H_i)\in \mathcal{H}\\ T_i\in [0,1]}} \sum_{\substack{(T_j,H_j)\in \mathcal{H}\\ T_j<T_i}}e^{-\gamma H_{\scriptscriptstyle\mathcal{I}}^2-\beta R_{\scriptscriptstyle\mathcal{I}}^2}{\mathbb{1}}{\left\{\mathcal{H}(U^{\hat{S}}_{\hat{H}})=1\right\}}\right.\nonumber\\ &\times {\mathbb{1}}{\left\{\exists (T_k,H_k)\in \mathcal{H}\vert_{U^{\hat{S}}_{\hat{H}}}\right\}}\Bigg]\nonumber\\ &= \frac{1}{\lambda_{\scriptscriptstyle\mathcal{I}}}\mathbb{E}\left[\sum_{\substack{(T_i,H_i)\in \mathcal{H}\\ T_i\in [0,1]}} \sum_{\substack{(T_j,H_j)\in \mathcal{H}\\ T_j<T_i}}e^{-\gamma H_{\scriptscriptstyle\mathcal{I}}^2}{\mathbb{1}}{\left\{\mathcal{H}(U^{\hat{S}}_{\hat{H}})=1\right\}} \right.\nonumber\\ &\times \left.\sum_{(T_k,H_k)\in \mathcal{H}\vert_{U^{\hat{S}}_{\hat{H}}}} e^{-\beta R_{\hat{S},k}^2} \right]\nonumber\\ &=\frac{4\lambda^2}{\lambda_{\scriptscriptstyle\mathcal{I}}}\!\!\int_0^1\!\!\int_{0}^{\infty}\!\!\!\int_{0}^{\infty} \!\!\! \int_{-\infty}^{t_1} \!\!\!\! e^{-(\gamma+\lambda\pi) \hat{h}^2} \lambda\pi J_{\beta}(\hat{s},\hat{h}) {\rm d} t_2\, {\rm d} h_2\, {\rm d} h_1 \, {\rm d} t_1, \label{eq:HTIRI} \end{align}\tag{30}\] where \(R^2_{\scriptscriptstyle\mathcal{I}}=R_{\hat{S},k}^2:= (\hat{S}-T_k)^2+H_k^2\), \(J_{\beta}(\hat{s},\hat{h})=\frac{2}{\pi}\int_{U^{\hat{s}}_{\hat{h}}} e^{-\beta \hat{r}^2} {\rm d} t_3\, {\rm d} h_3\), \(\hat{r}^2=(\hat{s}-t_3)^2+h_3^2\) is a realisation of \(R_{\scriptscriptstyle\mathcal{I}}\) and \(\hat{s}=\frac{t_1^2+h_1^2- t_2^2-h_2^2}{2(t_1-t_2)}\).

Let \(\hat{s}-t_3= u\cos\theta\) and \(h_3=u\sin \theta\). Then similarly to (19 ) we can compute the integral \[\begin{align} J_{\beta}(\hat{s},\hat{h})&=\frac{2}{\pi}\int_{U^{\hat{s}}_{\hat{h}}} e^{-\beta \hat{r}^2} {\rm d} t_3\, {\rm d} h_3\nonumber\\ &{=}\frac{2}{\pi} \int_0^\pi\!\!\! \int_0^{\hat{h}} \!\!e^{-\beta u^2} u{\rm d}u\,{\rm d}\theta {=} \frac{1}{\beta}\left(1{-}e^{-\beta \hat{h}^2}\right). \end{align}\] Conditioned on \(H_{\scriptscriptstyle\mathcal{I}}=\hat{h}\), we have \(R^2_{\scriptscriptstyle\mathcal{I}}\) is uniform on \([0,\hat{h}^{2}]\). Then the integral in (30 ) becomes \[\begin{align} \nonumber\\ &=\frac{4\lambda^3\pi}{\lambda_{\scriptscriptstyle\mathcal{I}}\beta}\int_{0}^{\infty}\int_{0}^{\infty}\int_0^{\infty} e^{-(\gamma+\lambda\pi) \hat{h}^2} \left(1-e^{-\beta \hat{h}^2}\right) {\rm d} t\, {\rm d} h_2\, {\rm d} h_1\nonumber\\ &=\frac{2\lambda\pi}{3\beta} \left[\left(1+\frac{\gamma}{\lambda\pi}\right)^{-3/2}-\left(1+\frac{\gamma+\beta}{\lambda\pi}\right)^{-3/2}\right],\nonumber \end{align}\] using the fact that \(\lambda_{\scriptscriptstyle\mathcal{I}}=\frac{6\sqrt{\lambda}}{\pi}\) from Lemma 2 and for any \(\eta\geq 0\) \[\pi\lambda^{3/2}\!\! \int_{0}^{\infty}\!\! \int_{0}^{\infty}\!\!\int_{0}^{\infty} \!\! e^{-(\lambda\pi+\eta) \hat{h}^2} {\rm d} t\, {\rm d} h_2\, {\rm d} h_1= \left(1+\frac{\eta}{\lambda\pi}\right)^{-3/2}\!\!\!. \hfill \square\]

9 Proofs: case with fading↩︎

  This section contains the proof of Theorem 1 and the derivation of these performance metrics for individual typical epochs with SINR and STINR, with respect to the corresponding Palm probability measure.

9.1 Proof of Theorem 1↩︎

The coverage probability with respect to the Palm probability distribution of the epoch of interest is \[\begin{align} p^c(\tau,\mu,\lambda,\alpha)&= \mathbb{E}^0_{H}[\mathbb{P}(\mathcal{S}>\tau\vert H)]. \label{eq:OP1} \end{align}\tag{31}\] Conditioned on \(H=h\) the inner term (31 ) is \[\begin{align} \mathbb{P}(\mathcal{S}>\tau\vert H=h) &= \mathbb{E}_{I_{h}}\left[\mathbb{P}(\mathcal{S}>\tau\vert I_h)\right]\nonumber\\ &=\mathbb{E}_{I_{h}}\left[\mathbb{P}\left(\rho >\tau h^{\alpha}\left(\sigma^2+I_{h}\right)\right)\right]\nonumber\\ &=\mathbb{E}_{I_{h}}\left[e^{-\mu\tau h^{\alpha}\left(\sigma^2+I_{ h}\right)}\right]\nonumber\\ &= e^{-\mu\tau h^{\alpha}\sigma^2}\mathcal{L}_{I_{h}}(\mu\tau h^{\alpha}).\nonumber \end{align}\] Then the coverage probability is given by the formula \[\begin{align} p^c(\tau,\mu,\lambda,\alpha)&= \mathbb{E}^0_{H}\left[e^{-\mu\tau H^{\alpha}\sigma^2} \mathcal{L}_{I_{H}}(\mu\tau H^{\alpha})\right]. \nonumber\;\;\;\qquad\quad \hfill \square \end{align}\]

9.2 Proof of Theorem 2↩︎

The proof follows the same steps as in Theorem 1, together with taking into account the randomness due to the distance \(H\) to the nearest interferer as well, jointly with the distance \(R\) to the serving BS. Given \(H=h\) and \(R=r\), the coverage probability is \[\begin{align} \mathbb{P}(\mathcal{S}>\tau\vert H=h, R=r) &= \mathbb{E}_{I_{h}}\left[\mathbb{P}(\mathcal{S}>\tau\vert I_h)\right]\nonumber\\ &=\mathbb{E}_{I_{h}}\left[\mathbb{P}\left(\rho >\tau r^{\alpha}\left(\sigma^2+I_{h}\right)\right)\right]\nonumber\\ &=\mathbb{E}_{I_{h}}\left[e^{-\mu\tau r^{\alpha}\left(\sigma^2+I_{ h}\right)}\right]\nonumber\\ &= e^{-\mu\tau r^{\alpha}\sigma^2}\mathcal{L}_{I_{h}}(\mu\tau r^{\alpha}).\nonumber \end{align}\] Jointly averaging of \(H\) and \(R\), we have \[\begin{align} p^c(\tau,\mu,\lambda,\alpha)&= \mathbb{E}^0_{H,R}\left[e^{-\mu\tau R^{\alpha}\sigma^2} \mathcal{L}_{I_{H}}(\mu\tau R^{\alpha})\right]. \nonumber\qquad\hfill \square \end{align}\] For each of the typical epoch of interest, ms-MI, MS, MI, mI and t, in each of the subsections, we just state the expression of the metrics before proving them using the formula obtained in Theorem 1 and Theorem 2, without formally stating them as a theorem or corollary. From now on, for simplicity we write \(p^c_{*}\equiv p^c_{*}(\tau,\mu,\lambda,\alpha)\), \(p^c_{\scriptscriptstyle*;\mathcal{T}}\equiv p^c_{\scriptscriptstyle*;\mathcal{T}}(\tau,\mu,\lambda,\alpha)\) and \(\mathcal{R}_{*}\equiv \mathcal{R}_{*}(\mu,\lambda,\alpha)\), \(\mathcal{R}_{\scriptscriptstyle*;\mathcal{T}}\equiv \mathcal{R}_{\scriptscriptstyle*;\mathcal{T}}(\mu,\lambda,\alpha)\), which are understood to a function of the set of parameters \(\{\tau,\mu,\lambda,\alpha\}\) and \(\{\mu,\lambda,\alpha\}\), respectively, where \(*\in \{\scriptscriptstyle{\mathcal{V}, S, I, \mathcal{I}, t}\) \(\}\).

9.3 Min-signal-max-interference (mS-MI): typical handover↩︎

In this case the distance to the serving BS is given by the handover distance \(H_{\scriptscriptstyle\mathcal{V}}\). The Palm probability distribution of a typical handover distance \(H_{\scriptscriptstyle\mathcal{V}}\) is given by a Nakagami distribution with parameter \(\left(\frac{3}{2}, \frac{3}{2\lambda\pi}\right)\), having density (7 ) established in [26].

Conditioned on the typical handover distance \(H_{\scriptscriptstyle\mathcal{V}}=h\), the distance of all other BSs is given by a Poisson point process \(\eta_h=\sum_i \delta_{D_i}\) on \([h, \infty)\) with intensity measure having density \(2\pi\lambda r {\rm d}r\), as seen in Lemma 5 (see also [26]). Here is a result about the probability of coverage. In the following, we use the subscript \(\mathcal{V}\) to denote the typical handover.

9.3.1 Theorem 1 (for SINR at typical mS-MI)↩︎

The probability of coverage in the system with threshold \(\tau\) at a typical handover is given by \[\begin{align} p_{\scriptscriptstyle\mathcal{V}}^c &{=} \frac{1}{1{+}\tau}\mathbb{E}^0_{H_{\scriptscriptstyle\mathcal{V}}}\left[ e^{-\mu\tau H_{\scriptscriptstyle\mathcal{V}}^{\alpha}\sigma^2-\pi\lambda H_{\scriptscriptstyle\mathcal{V}}^2 \kappa(\tau,\alpha)}\right], \label{eq:hocp1} \end{align}\tag{32}\] where \(\kappa(\tau,\alpha):=\tau^{2/\alpha} \int_{\tau^{-2/\alpha}}^\infty\frac{1}{1+z^{\alpha/2}} {\rm d}z\) as in (83 ).

Using the formula (?? ) we have \[\begin{align} p_{\scriptscriptstyle\mathcal{V}}^c &=\int_{0}^\infty e^{-\mu\tau h^{\alpha}\sigma^2}\mathcal{L}_{I_{h}}(\mu\tau h^{\alpha})f_{H_{\scriptscriptstyle\mathcal{V}}}(h){\rm d}h, \label{eq:cp1} \end{align}\tag{33}\] where \(\mathcal{L}_{I_{h}}(\gamma)\) is the Laplace transform of the total interference \[I_{h}:=\rho h^{-\alpha}+\sum_i \rho_i D_i^{-\alpha},\] given the handover distance \(H_{\scriptscriptstyle\mathcal{V}}=h\). Using the Poisson point process \(\eta_{h}\) of distances of all other stations, we compute the Laplace transform of \(I_{h}\) as follows: \[\begin{align} \mathbb{E}_{I_{h}}\left[e^{-\gamma I_{h}}\right] &= \mathbb{E}_{\eta_h, \rho, \{\rho_i\}}\left[\mathbb{E}\left[e^{-\gamma I_{h}} \vert \eta_h, \rho, \{\rho_i\}\right]\right]\nonumber\\ &=\mathbb{E}_{\rho}\left[e^{-\gamma\rho h^{-\alpha}}\right]\mathbb{E}_{\eta_h, \{\rho_i\}}\left[e^{-\gamma\sum_{i\in \mathbb{N}} \rho_i D_i^{-\alpha}}\right]\nonumber\\ &=\frac{\mu}{\mu+\gamma h^{-\alpha}}\mathbb{E}_{\eta_h, \{\rho_i\}}\left[\prod_{i\in\mathbb{N}}e^{-\gamma\rho_i D_i^{-\alpha} }\right]\nonumber\\ &= \frac{\mu}{\mu+\gamma h^{-\alpha}} \mathbb{E}_{\eta_h}\left[\prod_{i\in\mathbb{N}}\mathbb{E}_{\rho_i}\left[e^{-\gamma\rho_i D_i^{-\alpha} }\right]\right]. \label{eq:cp3} \end{align}\tag{34}\] Computing the PGFl with respect to the point process \(\eta_h\), we have \[\begin{align} \nonumber\\ &=\exp\left(-2\pi\lambda\int_{h}^\infty\left(1-\mathbb{E}_\rho[e^{-\gamma\rho r^{-\alpha}}]\right)r{\rm d}r\right)\nonumber\\ &= \exp\left(-2\pi\lambda\int_{h}^\infty\frac{\gamma r^{-\alpha}}{\mu+\gamma r^{-\alpha}}r{\rm d}r\right). \label{eq:cp3a} \end{align}\tag{35}\] Substituting the Laplace transform of \(I_h\) with \(\gamma=\mu\tau h^{\alpha}\) from (34 ) to (33 ), we get \[\begin{align} p_{\scriptscriptstyle\mathcal{V}}^c &=\int_{0}^\infty e^{-\mu\tau h^{\alpha}\sigma^2}\mathcal{L}_{I_{h}}(\mu\tau h^{\alpha})f_{H_{\scriptscriptstyle\mathcal{V}}}(h){\rm d}h\nonumber\\ &{=}\frac{1}{1{+}\tau}\int_{0}^\infty \!\!\! e^{-\mu\tau h^{\alpha}\sigma^2 -2\pi\lambda\int_{h}^\infty\frac{\tau h^{\alpha}}{r^{\alpha}+\tau h^{\alpha}}r{\rm d}r} f_{H_{\scriptscriptstyle\mathcal{V}}}(h){\rm d}h\nonumber\\ &{=}\frac{1}{1{+}\tau}\int_{0}^\infty \!\!\!\! e^{-\mu\tau h^{\alpha}\sigma^2 -\pi\lambda h^2 \kappa(\tau,\alpha)} f_{H_{\scriptscriptstyle\mathcal{V}}}(h){\rm d}h,\nonumber \end{align}\] where we have computed the integral \[\int_{h}^\infty\frac{\tau h^{\alpha}}{r^{\alpha}+\tau h^{\alpha}}r{\rm d}r={\frac{1}{2}}h^2 \kappa(\tau,\alpha), \label{eq:kappa-a}\tag{36}\] where \(\kappa(\tau,\alpha):=\tau^{2/\alpha} \int_{\tau^{-2/\alpha}}^\infty\frac{1}{1+z^{\alpha/2}} {\rm d}z\) as in (83 ).  \(\square\)

9.3.2 Theorem 1 (for tropical SINR at typical mS-MI)↩︎

Under the tropical regime of SINR, the coverage probability at a threshold \(\tau\) at a typical handover are \[\begin{align} p_{\scriptscriptstyle\mathcal{V};\mathcal{T}}^c &{=} \mathbb{E}^0_{H_{\scriptscriptstyle\mathcal{V}}}\!\left[e^{-\mu\tau H_{\scriptscriptstyle\mathcal{V}}^{\alpha}\sigma^2} \!\!\mu\tau H_{\scriptscriptstyle\mathcal{V}}^{\alpha} \!\! \int \!\! e^{- \mu\tau H_{\scriptscriptstyle\mathcal{V}}^{\alpha} x} \left(1{-}e^{-\mu H_{\scriptscriptstyle\mathcal{V}}^{\alpha}x}\right)\right.\nonumber\\ & \times \left.e^{-2\lambda\pi K_\alpha(\mu,x, H_{\scriptscriptstyle\mathcal{V}})} {\rm d}x\right].\nonumber \end{align}\] Proof. In this case the tropical interference, given \(H_{\scriptscriptstyle\mathcal{V}}=h\), is \(\mathcal{T}_{h}:=\rho h^{-\alpha}\vee\max_i \rho_i D_i^{-\alpha}\). Using the formula (?? ) in Theorem 1, we have \[\begin{align} p_{\scriptscriptstyle{\mathcal{V};\mathcal{T}}}^c &=\int_{0}^\infty e^{-\mu\tau h^{\alpha}\sigma^2} \mathcal{L}_{\mathcal{T}_{h}}(\mu\tau h^{\alpha})f_{H_{\scriptscriptstyle\mathcal{V}}}(h){\rm d}h.\nonumber \end{align}\] Given a handover at a distance \(h\), the probability distribution of the tropical interference is \[\begin{align} \nonumber\\ &=\mathbb{E}_{\eta_{h},\rho, \{\rho_i\}}\left[\mathbb{P}(I_{h}\leq x)\vert \eta_{h},\rho, \{\rho_i\}\right]\nonumber\\ &=\mathbb{E}_{\eta_{h}}\left[\mathbb{E}_{\rho, \{\rho_i\}}\left( {\mathbb{1}}_{\rho h^{-\alpha}\leq x}\prod_{i\in \mathbb{N}}{\mathbb{1}}_{\rho_iD_i^{-\alpha}\leq x}\right)\vert \eta_{h}\right]\nonumber\\ &=\mathbb{P}_{\rho}\left(\rho h^{-\alpha}\leq x\right)\mathbb{E}_{\eta_h}\!\!\left[\prod_{i\in \mathbb{N}}\exp\left(\log\mathbb{E}_{\rho_i}\left[{\mathbb{1}}_{\rho_iD_i^{-\alpha}\leq x}\right]\right)\vert \eta_{h}\right]\nonumber\\ &=\mathbb{P}_{\rho}\left(\rho \leq h^{\alpha}x\right) \mathbb{E}_{\eta_h}\!\!\left[\exp\left( \sum_{i\in \mathbb{N}}\log\mathbb{E}_{\rho_i}\left[{\mathbb{1}}_{\rho_iD_i^{-\alpha}\leq x}\right]\right)\vert \eta_{h}\right]\nonumber\\ &=\left(1{-}e^{-\mu h^{\alpha}x}\right)\exp\left(-2\lambda\pi\int_{h}^{\infty}\!\!\!\left(1-\mathbb{P}_\rho(\rho r^{-\alpha}\leq x)\right)r{\rm d}r\right)\nonumber\\ &=\left(1{-}e^{-\mu h^{\alpha}x}\right) \exp\left(-2\lambda\pi\int_{h}^{\infty}e^{-\mu r^\alpha x}r{\rm d}r\right)\nonumber\\ &=\left(1{-}e^{-\mu h^{\alpha}x}\right) e^{-2\lambda\pi K_\alpha(\mu,x, h)}, \label{eq:STINR-h} \end{align}\tag{37}\] where \(K_\alpha(\mu,x, h)\) is as defined in (87 ). Thus the Laplace transform of the interference \(\mathcal{T}_h\) is \[\begin{align} \mathcal{L}_{\mathcal{T}_{h}}(\gamma)&= \gamma\int e^{- \gamma x} F_{\mathcal{T}_{h}}(x){\rm d}x. \label{eq:LapIh} \end{align}\tag{38}\] Then from (38 ) with \(\gamma=\mu\tau h^{\alpha}\) and (?? ), the coverage probability is \[\begin{align} p_{\scriptscriptstyle\mathcal{V}; \mathcal{T}}^c &=\int_{0}^\infty \!\!e^{-\mu\tau h^{\alpha}\sigma^2}\!\! \mu\tau h^{\alpha} \int e^{- \mu\tau h^{\alpha} x} \left(1{-}e^{-\mu h^{\alpha}x}\right)\nonumber\\ & \times e^{-2\lambda\pi K_\alpha(\mu,x, h)} {\rm d}x f_{H_{\scriptscriptstyle\mathcal{V}}}(h){\rm d}h.\nonumber \hfill \square \end{align}\]

9.4 Typical time↩︎

At any typical time, let \(R\) be the random variable for the nearest BS from the user. It is well known in literature that \(R\) follows a Rayleigh distribution with parameter \(1/{\sqrt{2\pi\lambda}}\), i.e., it has density \[f_R(r):=2\lambda\pi r \, e^{-\lambda\pi r^2}.\]

9.4.1 Theorem 1 (for SINR at a typical time)↩︎

  In this case the coverage probability with user QoS \(\tau\) is \[\begin{align} p_{t}^c &= \mathbb{E}^0_R\left[e^{-\mu\tau R^{\alpha}\sigma^2-\pi\lambda R^2 \kappa(\tau,\alpha)}\right],\nonumber \end{align}\] where the function \(\kappa\) is as defined in (83 ).

Proof. To apply the formula for the \(\tau\)-coverage probability (?? ) we determine the Laplace transform \(\mathcal{L}_{I_{r}}(\gamma)\) with \(\gamma=\mu\tau r^{\alpha}\) as \[\begin{align} \mathcal{L}_{I_{r}}(\mu\tau r^{\alpha})&=\exp\left(-2\pi\lambda\int_{r}^\infty\frac{\mu\tau r^{\alpha} v^{-\alpha}}{\mu+\mu\tau r^{\alpha} v^{-\alpha}}v{\rm d}v\right)\nonumber\\ &=\exp\left(-2\pi\lambda\int_{r}^\infty\frac{\tau r^{\alpha}}{v^{\alpha}{+}\tau r^{\alpha}}v{\rm d}v\right)= e^{-\pi\lambda r^2 \kappa(\tau,\alpha)},\nonumber \end{align}\] where \(\kappa\) is as in (83 ). Then, from (?? ), the coverage probability is \[\begin{align} p_{t}^c &= \int_{0}^\infty e^{-\mu\tau r^{\alpha}\sigma^2} e^{-\pi\lambda r^2 \kappa(\tau,\alpha)}f_{R}(r){\rm d}r.\nonumber \hfill \square \end{align}\]

9.4.2 Theorem 1 (for STINR at a typical time)↩︎

  At a typical time, the probability of coverage with user QoS \(\tau\) is given by \[\begin{align} p_{t;\scriptscriptstyle{\mathcal{T}}}^c {=} \mu\tau\mathbb{E}_{R}^0\!\!\left[ R^\alpha\!\!\int \!\! e^{-\mu\tau R^\alpha(\sigma^2+x)-2\lambda\pi K_\alpha(\mu,x,R)}{\rm d}x\right]. \label{eq:} \end{align}\tag{39}\] We find the coverage probability using the formula (?? ) we first determine the Laplace transform of \(\mathcal{T}_r\) given \(R=r\). For this, the probability distribution of \(\mathcal{T}_r\) is \[\begin{align} \mathbb{P}(\mathcal{T}_{r}\leq x) &=\mathbb{E}_{\eta_{r}, \{\rho_i\}}\left[\mathbb{P}(\mathcal{T}_{r}\leq x)\vert \eta_{r}, \{\rho_i\}\right]\nonumber\\ &=\mathbb{E}_{\eta_{r}}\left[\mathbb{E}_{\{\rho_i\}}\left(\prod_{i\in \mathbb{N}}{\mathbb{1}}_{\rho_i D_i^{-\alpha}\leq x}\right)\vert \eta_{r}\right]\nonumber\\ &=\mathbb{E}_{\eta_{r}}\left[\prod_{i\in \mathbb{N}}\exp\left(\log\mathbb{E}_{\rho_i}\left[{\mathbb{1}}_{\rho_iD_i^{-\alpha}\leq x}\right]\right)\vert \eta_{r}\right]\nonumber\\ &=\mathbb{E}\left[\exp\left( \sum_{i\in \mathbb{N}}\log\mathbb{E}_{\rho_i}\left[{\mathbb{1}}_{\rho_iD_i^{-\alpha}\leq x}\right]\right)\vert \eta_{r}\right].\nonumber \end{align}\] Using the PGFl of the Poisson point process \(\eta_r\), we have \[\begin{align} \mathbb{P}(\mathcal{T}_{r}\leq x) &=\exp\left(-2\lambda\pi\int_{r}^{\infty}\left(1-\mathbb{P}_\rho(\rho v^{-\alpha}\leq x)\right)v{\rm d}v\right)\nonumber\\ &=\exp\left(-2\lambda\pi\int_{r}^{\infty}\mathbb{P}_\rho(\rho> v^\alpha x)v{\rm d}v\right)\nonumber\\ &=e^{-2\lambda\pi\int_{r}^{\infty}e^{-\mu v^\alpha x}v{\rm d}v} =e^{-2\lambda\pi K_\alpha(\mu,x, h)}, \label{eq:STINR-R} \end{align}\tag{40}\] where the function \(K_\alpha(\mu,x, h)\) is defined in (87 ). The Laplace transform of the tropical interference is \[\begin{align} \mathbb{E}_{\mathcal{T}_{r}}\left[e^{-\gamma\mathcal{T}_{r}}\right] &= \gamma\int e^{-\gamma x} \mathbb{P}(\mathcal{T}_r\leq x){\rm d}x\nonumber\\ &= \gamma\int e^{-\gamma x} e^{-2\lambda\pi K_\alpha(\mu,x, r)}{\rm d}x, \label{eq:LT-t-T} \end{align}\tag{41}\] using \(\gamma=\mu\tau r^\alpha\), the coverage probability turns out to be \[\begin{align} p_{t;\scriptscriptstyle{\mathcal{T}}}^c &{=}\mu\tau\!\!\int_{0}^\infty \!\! r^\alpha e^{-\mu\tau r^\alpha\sigma^2}\int \!\! e^{-\mu\tau r^\alpha x-2\lambda\pi K_\alpha(\mu,x,r)}{\rm d}x f_{R}(r){\rm d}r.\nonumber \end{align}\]  \(\square\)

9.5 Typical max-signal (MS)↩︎

Let \(H_{\scriptscriptstyle S}\) be the distance to the typical visible head point. It follows from [26] that the distance \(H_{\scriptscriptstyle S}\) to the typical visible head follows a Nakagami distribution with parameter \(\left({\frac{1}{2}}, \frac{1}{2\lambda\pi}\right)\), i.e., \(f_{H_{\scriptscriptstyle S}}(h)= 2\lambda^{\frac{1}{2}}e^{-\lambda\pi h^2}\).

9.5.1 Theorem 1 (for SINR at a typical MS)↩︎

  The \(\tau\)-coverage probability is \[\begin{align} p_{\scriptscriptstyle S}^c &= \mathbb{E}^0_{H_{\scriptscriptstyle S}}\left[e^{-\mu\tau H_{\scriptscriptstyle S}^{\alpha}\sigma^2-\pi\lambda H_{\scriptscriptstyle S}^2\kappa(\tau,\alpha)}\right]. \label{eq:cp-V} \end{align}\tag{42}\] To find the \(\tau\)-coverage probability using (?? ) we first compute the Laplace transform of the interference \(I_h\) given \(H_{\scriptscriptstyle S}=h\) as \[\begin{align} \mathbb{E}_{I_{h}}[e^{-\gamma I_{h}}] &=\mathbb{E}_{\eta_h,\{\rho_i\}}\left[e^{-\gamma\sum_{i\in \mathbb{N}} \rho_i D_i^{-\alpha}} \vert \eta_{h}, \{\rho_i\}\right]\nonumber\\ &= \mathbb{E}_{\eta_{h}}\left[\prod_{i\in\mathbb{N}}\mathbb{E}_{\rho_i}[e^{-\gamma\rho_i D_i^{-\alpha} }]\vert \eta_{h} \right]. \label{eq:pgfl} \end{align}\tag{43}\] Applying the PGFl with respect to the point process \(\eta_h\) in (43 ), we have \[\begin{align} \mathbb{E}_{I_{h}}[e^{-\gamma I_{h}}] &=\exp\left(-2\pi\lambda\int_{h}^\infty\left(1-\mathbb{E}_\rho[e^{-\gamma\rho r^{-\alpha}}]\right)r{\rm d}r\right)\nonumber\\ &=\exp\left(-2\pi\lambda\int_{h}^\infty\frac{\gamma r^{-\alpha}}{\mu+\gamma r^{-\alpha}}r{\rm d}r\right) \tag{44} \\ &\stackrel{(\gamma=\mu\tau h^\alpha)}{=}\exp\left(-2\pi\lambda\int_{h}^\infty \frac{\tau h^{\alpha} r^{-\alpha}}{1+\tau h^{\alpha} r^{-\alpha}}r{\rm d}r\right)\nonumber\\ &=e^{-2\pi\lambda\int_{h}^\infty\frac{\tau h^{\alpha}}{r^{\alpha}+\tau h^{\alpha}}r{\rm d}r} = e^{-\pi\lambda h^2 \kappa(\tau,\alpha)}, \tag{45} \end{align}\] where \(\kappa\) as in (83 ). Then using (?? ), the \(\tau\)-coverage probability is \[\begin{align} p_{\scriptscriptstyle S}^c &=\int_{0}^\infty e^{-\mu\tau h^{\alpha}\sigma^2-\pi\lambda h^2\kappa(\tau,\alpha)}f_{H_{\scriptscriptstyle S}}(h){\rm d}h.\nonumber \end{align}\]  \(\square\)

9.5.2 Theorem 1 (for STINR at a typical MS)↩︎

  In this case the \(\tau\)-coverage probability is \[\begin{align} p_{{\scriptscriptstyle S};\scriptscriptstyle{\mathcal{T}}}^c &{=} \mathbb{E}^0_{H_{\scriptscriptstyle S}}\!\!\left[e^{-\mu\tau H_{\scriptscriptstyle S}^{\alpha}\sigma^2} \!\! \mu\tau H_{\scriptscriptstyle S}^{\alpha} \int e^{-\mu\tau H_{\scriptscriptstyle S}^{\alpha} x-2\lambda\pi K_\alpha(\mu,x,H_{\scriptscriptstyle S})}{\rm d}x\right].\nonumber \end{align}\] Given \(H_{\scriptscriptstyle S}=h\), the tropical interference is \(\mathcal{T}_h= \max_{i} \rho_i D_i^{-\alpha}\) and its distribution is \[\begin{align} \mathbb{P}(\mathcal{T}_h\leq x) &=\mathbb{E}_{\eta_{h}, \{\rho_i\}}\left[\mathbb{P}(\mathcal{T}_h \leq x)\vert \eta_{h}, \{\rho_i\}\right]\nonumber\\ &=\mathbb{E}_{\eta_{h}}\left[\mathbb{E}_{\{\rho_i\}}\left(\prod_{i\in \mathbb{N}}{\mathbb{1}}_{ \rho_i D_i^{-\alpha} \leq x}\right)\vert \eta_{h_{\scriptscriptstyle I}}\right]\nonumber\\ &= \mathbb{E}_{\eta_{h}}\left[\prod_{i\in \mathbb{N}}\mathbb{E}_{\rho_i}\left[{\mathbb{1}}_{ \rho_i D_i^{-\alpha} \leq x}\right]\vert \eta_{h}\right]\nonumber\\ &= \exp\left(-2\lambda\pi\int_{h}^\infty \mathbb{P}_{\rho}(\rho>v^\alpha x) v{\rm d}v\right)\nonumber\\ &= \exp\left(-2\lambda\pi\int_{h}^\infty e^{-\mu v^\alpha x} v{\rm d}v\right)\nonumber\\ &= e^{-2\lambda\pi K_\alpha(\mu,x, h)}. \label{eq:dnst-trop-sinr-V} \end{align}\tag{46}\] where \(K_\alpha(\mu,x, h)\) is defined in (87 ). Using the distribution of \(\mathcal{T}_h\) (46 ), we obtain the Laplace transform of \(\mathcal{T}_h\) as \[\begin{align} \mathcal{L}_{I_h}(\mu\tau h^{\alpha}) &= \mu\tau h^{\alpha} \int e^{-\mu\tau h^{\alpha} x-2\lambda\pi K_\alpha(\mu,x, h)}{\rm d}x.\nonumber \end{align}\] We obtain the coverage probability by using (?? ).  \(\square\)

9.6 Typical max-interference (MI)↩︎

Recall that the distance to the nearest interferer and the serving BS is denoted by \(H_{\scriptscriptstyle I}\) and \(R_{\scriptscriptstyle I}\), where \(H_{\scriptscriptstyle I}\stackrel{d}{\sim} Na\left(\frac{3}{2}, \frac{3}{2\lambda\pi}\right)\) from Proposition 3 and, from Proposition 4, \(R^2_{\scriptscriptstyle I}\stackrel{d}{\sim} U[0,h^2]\), given \(H_{\scriptscriptstyle I}=h\).

9.6.1 Theorem 2 (for SINR at a typical MI)↩︎

  The coverage probability for user QoS \(\tau\) is \[\begin{align} p^c_{\scriptscriptstyle I} &{=} \frac{2\tau^{-2/\alpha}}{\alpha}\int_0^\tau \!\frac{z^{2/\alpha-1}}{1{+}z}\mathbb{E}^0_{H_{\scriptscriptstyle I}}\!\! \left[e^{-\mu z H_{\scriptscriptstyle I}^\alpha\sigma^2-\pi\lambda H_{\scriptscriptstyle I}^2 \kappa(z, \alpha)}\right]{\rm d}z.\nonumber \end{align}\] In this case, the total interference is \[I_{H_{\scriptscriptstyle I}}:=\rho H_{\scriptscriptstyle I}^{-\alpha}+ \sum_{i\in \mathbb{N}} \rho_i D_i^{-\alpha},\] where \(H_{\scriptscriptstyle I}\) is the distance to the nearest interferer. Given \(H_{\scriptscriptstyle I}=h\) and distance to the serving BS \(R_{\scriptscriptstyle I}=r\), the Laplace transform of the interference is \[\begin{align} \mathbb{E}_{I_h}\left[e^{-\gamma I_h}\right] &=\mathbb{E}_{\eta_{h}, \rho, \{\rho_i\}}\left[e^{-\gamma(\rho h^{-\alpha}+\sum_{i\in \mathbb{N}} \rho_i D_i^{-\alpha})} \vert \eta_{h}, \rho, \{\rho_i\}\right]\nonumber\\ &=\mathbb{E}_{\rho}\left[e^{-\gamma\rho h^{-\alpha}}\right] \mathbb{E}_{\eta_{h}, \{\rho_i\}}\left[\prod_{i\in\mathbb{N}}e^{-\gamma\rho_i D_i^{-\alpha} }\vert \eta_{h}, \{\rho_i\}\right]\nonumber\\ &= \frac{\mu}{\mu+\gamma h^{-\alpha}} \mathbb{E}_{\eta_{h}}\left[\prod_{i\in\mathbb{N}}\mathbb{E}_{\rho_i}[e^{-\gamma\rho_i D_i^{-\alpha} }]\vert \eta_{h} \right]\nonumber\\ &= \frac{\mu}{\mu+\gamma h^{-\alpha}} e^{-2\pi\lambda\int_{h}^\infty\frac{\gamma u^{-\alpha}}{\mu+\gamma u^{-\alpha}}u{\rm d}u}\tag{47}\\ &\stackrel{\gamma=\mu\tau r^{\alpha}}{=} \frac{1}{1+\tau r^{\alpha} h^{-\alpha}} e^{-2\pi\lambda\int_{h}^\infty\frac{\tau r^{\alpha} u^{-\alpha}}{1+\tau r^{\alpha} u^{-\alpha}}u{\rm d}u}\nonumber\\ &= \frac{h^{\alpha}}{h^{\alpha}+\tau r^{\alpha}} e^{-2\pi\lambda\int_{h}^\infty\frac{\tau r^{\alpha}}{u^{\alpha}+\tau r^{\alpha}}u{\rm d}u}.\tag{48} \end{align}\] Hence the coverage probability is \[\begin{align} \!\! p_{\scriptscriptstyle I}^c &{=} E^0_{H_{\scriptscriptstyle I}, R_{\scriptscriptstyle I}}\!\! \left[ \!\frac{H_{\scriptscriptstyle I}^\alpha}{H_{\scriptscriptstyle I}^\alpha{+}\tau R_{\scriptscriptstyle I}^\alpha} e^{-\mu\tau R_{\scriptscriptstyle I}^{\alpha}\sigma^2-2\pi\lambda\int_{H_{\scriptscriptstyle I}}^\infty\frac{\tau R_{\scriptscriptstyle I}^{\alpha}}{u^\alpha{+} \tau R_{\scriptscriptstyle I}^{\alpha}}u{\rm d}u}\right]. \end{align}\] Considering the random variable \(R_{\scriptscriptstyle I}^2\), the integral in the exponent in (48 ) equals \[\begin{align} \int_h^\infty\!\!\! \frac{\tau r^{\alpha/2}}{u^{\alpha}+\tau r^{\alpha/2}}u{\rm d}u &= \frac{h^2}{2}\kappa\left(\tau (r/h^2)^{\alpha/2}, \alpha\right), \label{eq:hralpha} \end{align}\tag{49}\] using (86 ). Hence the coverage probability turns out to be \[\begin{align} p^c_{\scriptscriptstyle I} & {=}\int_0^\infty\!\!\! \int_0^{h^2} \!\!\! \frac{e^{-\mu\tau r^{\alpha/2}\sigma^2}}{1+\tau (r/{h^2})^{\alpha/2}} e^{-\pi\lambda h^2 \kappa(\tau (r/h^2)^{\alpha/2}, \alpha)}\nonumber\\ & \times \frac{1}{h^2}{\rm d}r f_{H_{\scriptscriptstyle I}}(h){\rm d}h\nonumber\\ &\stackrel{\frac{r}{h^2}=w}{=} \!\!\!\int_0^1\!\!\! \int_0^\infty\!\! \frac{e^{-\mu\tau h^\alpha w^{\alpha/2}\sigma^2}}{1+\tau w^{\alpha/2}} e^{-\pi\lambda h^2 \kappa(\tau w^{\alpha/2}, \alpha)} f_{H_{\scriptscriptstyle I}}(h){\rm d}h{\rm d}w \nonumber\\ &{=} \int_0^1 \!\!\!\frac{1}{1{+}\tau w^{\alpha/2}}\mathbb{E}^0_{H_{\scriptscriptstyle I}}\!\! \left[e^{-\mu\tau w^{\alpha/2} H_{\scriptscriptstyle I}^\alpha\sigma^2-\pi\lambda H_{\scriptscriptstyle I}^2 \kappa(\tau w^{\alpha/2}, \alpha)}\right]{\rm d}w\nonumber\\ &{=} \frac{2\tau^{-2/\alpha}}{\alpha} \!\!\! \int_0^\tau \!\!\frac{z^{2/\alpha-1}}{1{+}z}\mathbb{E}^0_{H_{\scriptscriptstyle I}}\!\! \left[e^{-\mu z H_{\scriptscriptstyle I}^\alpha\sigma^2-\pi\lambda H_{\scriptscriptstyle I}^2 \kappa(z, \alpha)}\right]{\rm d}z,\label{eq:OPI-4} \end{align}\tag{50}\] by taking \(\tau w^{\alpha/2}=z\).  \(\square\)

9.6.2 Theorem 2 (for STINR at a typical MI)↩︎

  In this case the \(\tau\)-coverage probability is \[\begin{align} p^c_{\scriptscriptstyle I;\mathcal{T}} &{=} \mathbb{E}^0_{H_{\scriptscriptstyle I}, R_{\scriptscriptstyle I}}\left[ e^{-\mu\tau R_{\scriptscriptstyle I}^{\alpha}\sigma^2} \mu\tau R_{\scriptscriptstyle I}^{\alpha} \int e^{- \mu\tau R_{\scriptscriptstyle I}^{\alpha} x} \left(1-e^{-\mu H_{\scriptscriptstyle I}^\alpha x}\right) \right.\nonumber\\ &\left.\times e^{-2\lambda\pi K_\alpha(\mu,x,H_{\scriptscriptstyle I})}{\rm d}x \right].\nonumber \end{align}\] Given the distance to the nearest interferer \(H_{\scriptscriptstyle I}=h\), the tropical interference is \(\mathcal{T}_h= \rho h^{-\alpha}\vee \max_{i} \rho_i D_i^{-\alpha}\). The probability distribution of \(\mathcal{T}_h\) is \[\begin{align} \mathbb{P}(\mathcal{T}_h\leq x) &=\mathbb{E}_{\eta_{h},\rho, \{\rho_i\}}\left[\mathbb{P}(\mathcal{T}_h\leq x)\vert \eta_{h}, \{\rho_i\}\right]\nonumber\\ &=\mathbb{E}_{\eta_{h}}\left[\mathbb{E}_{\rho, \{\rho_i\}}\left(\rho h^{-\alpha}\prod_{i\in \mathbb{N}}{\mathbb{1}}_{ \rho_i D_i^{-\alpha} \leq x}\right)\vert \eta_{h}\right]\nonumber\\ &= \mathbb{P}_{\rho}(\rho h^{-\alpha}\leq x) \mathbb{E}_{\eta_{h}}\left[\prod_{i\in \mathbb{N}}\mathbb{E}_{\rho_i}\left[{\mathbb{1}}_{ \rho_i D_i^{-\alpha} \leq x}\right]\vert \eta_{h}\right]\nonumber\\ &=\left(1-e^{-\mu h^\alpha x}\right) e^{-2\lambda\pi\int_{h}^\infty \mathbb{P}_{\rho}(\rho>v^\alpha x) v{\rm d}v}\nonumber\\ &=\left(1-e^{-\mu h^\alpha x}\right) e^{-2\lambda\pi\int_{h}^\infty e^{-\mu v^\alpha x} v{\rm d}v}\nonumber\\ &=\left(1-e^{-\mu h^\alpha x}\right) e^{-2\lambda\pi K_\alpha(\mu,x, h)}, \label{eq:dnst-trop-sinr-T} \end{align}\tag{51}\] where \(K_\alpha\) is as defined in (87 ). We compute the Laplace transform \(\mathcal{L}_{\mathcal{T}_h}(\gamma)\), for \(\gamma=\mu\tau r^{\alpha}\) given \(R_{\scriptscriptstyle I}=r\), similarly to (38 ) as \[\begin{align} \mathbb{E}\left[e^{-\gamma\mathcal{T}_{h}}\right]&{=} \mu\tau r^{\alpha}\!\! \int\!\! e^{- \mu\tau r^{\alpha} x} \left(1{-}e^{-\mu h^\alpha x}\right) e^{-2\lambda\pi K_\alpha(\mu,x, h)}{\rm d}x. \label{eq:LT-TI} \end{align}\tag{52}\] We obtain the coverage probability using the Laplace transform (52 ) in the formula (?? ) as \[\begin{align} p^c_{\scriptscriptstyle I;\mathcal{T}} &{=} \mathbb{E}^0_{H_{\scriptscriptstyle I}, R_{\scriptscriptstyle I}}\left[ e^{-\mu\tau R_{\scriptscriptstyle I}^{\alpha}\sigma^2} \mu\tau R_{\scriptscriptstyle I}^{\alpha} \int e^{- \mu\tau R_{\scriptscriptstyle I}^{\alpha} x} \left(1-e^{-\mu H_{\scriptscriptstyle I}^\alpha x}\right) \right.\nonumber\\ &\left.\times e^{-2\lambda\pi K_\alpha(\mu,x,H_{\scriptscriptstyle I})}{\rm d}x \right]\nonumber\\ &{=} \int_0^1 \mathbb{E}_{H_{\scriptscriptstyle I}}\left[ e^{-\mu\tau H_{\scriptscriptstyle I}^{\alpha} w^{\alpha/2}\sigma^2} \mu\tau H_{\scriptscriptstyle I}^{\alpha} w^{\alpha/2} \int e^{- \mu\tau H_{\scriptscriptstyle I}^{\alpha} w^{\alpha/2} x} \right.\nonumber\\ &\left.\times \left(1-e^{-\mu H_{\scriptscriptstyle I}^\alpha x}\right) e^{-2\lambda\pi K_\alpha(\mu,x,H_{\scriptscriptstyle I})}{\rm d}x \right] {\rm d}w,\nonumber \end{align}\] by taking \(R^2_{\scriptscriptstyle I}/h^2\) as a uniform random variable on \([0,1]\), given \(H_{\scriptscriptstyle I}=h\).  \(\square\)

9.7 Typical min-interference (mI): typical tropical interference handover↩︎

Recall that the distance to the nearest interferer and the serving BS is denoted by \(H_{\scriptscriptstyle\mathcal{I}}\) and \(R_{\scriptscriptstyle\mathcal{I}}\), where \(H_{\scriptscriptstyle\mathcal{I}}\stackrel{d}{\sim} Na\left(\frac{3}{2}, \frac{3}{2\lambda\pi}\right)\) from Proposition 5 and, from Proposition 6, \(R^2_{\scriptscriptstyle\mathcal{I}}\stackrel{d}{\sim} U[0,h^2]\), given \(H_{\scriptscriptstyle\mathcal{I}}=h\).

9.7.1 Theorem 2 (for SINR at a typical mI)↩︎

  In this case, the \(\tau\)-coverage probability is \[\begin{align} \!\! p_{\scriptscriptstyle\mathcal{I}}^c &{=} \frac{2\tau^{-2/\alpha}}{\alpha}\int_0^\tau \!\frac{z^{2/\alpha-1}}{(1{+}z)^2}\mathbb{E}^0_{H_{\scriptscriptstyle\mathcal{I}}}\!\! \left[e^{-\mu z H_{\scriptscriptstyle\mathcal{I}}^\alpha\sigma^2-\pi\lambda H_{\scriptscriptstyle\mathcal{I}}^2 \kappa(z,\alpha)}\right]{\rm d}z.\nonumber \end{align}\] Suppose the distance to the typical handover is \(H_{\scriptscriptstyle\mathcal{I}}\). Given \(H_{\scriptscriptstyle\mathcal{I}}=h\), let \(R_{\scriptscriptstyle\mathcal{I}}\) be the distance to serving station. In this case the total interference is \[I_{h}:=(\rho^++\rho^-) h^{-\alpha} + \sum_{i\in \mathbb{N}} \rho_i D_i^{-\alpha}.\] Given \(H_{\scriptscriptstyle\mathcal{I}}=h\) and \(R_{\scriptscriptstyle\mathcal{I}}=r\), the Laplace transform of the interference is \[\begin{align} \mathbb{E}\left[e^{-\gamma I_{h}}\right] &= \left(\frac{\mu}{\mu+\gamma h^{-\alpha}}\right)^2 e^{-2\pi\lambda\int_{h}^\infty\frac{\gamma v^{-\alpha}}{\mu+\gamma v^{-\alpha}}v{\rm d}v}\label{eq:LTmI1}\\ &=\left(\frac{h^\alpha}{h^\alpha+\tau r^\alpha}\right)^2 e^{-2\pi\lambda\int_{h}^\infty\frac{\tau r^{\alpha}}{v^\alpha+ \tau r^{\alpha}}v{\rm d}v},\nonumber \end{align}\tag{53}\] by substituting \(\gamma=\mu\tau r^\alpha\). Using the formula (?? ), the coverage probability is \[\begin{align} p_{\scriptscriptstyle\mathcal{I}}^c &{=}E^0_{H_{\scriptscriptstyle\mathcal{I}}, R_{\scriptscriptstyle\mathcal{I}}}\!\! \left[ \!\left(\frac{H_{\scriptscriptstyle\mathcal{I}}^\alpha}{H_{\scriptscriptstyle\mathcal{I}}^\alpha{+}\tau R_{\scriptscriptstyle\mathcal{I}}^\alpha}\right)^2 \!\! e^{-\mu\tau R_{\scriptscriptstyle\mathcal{I}}^{\alpha}\sigma^2-2\pi\lambda\int_{H_{\scriptscriptstyle\mathcal{I}}}^\infty\frac{\tau R_{\scriptscriptstyle\mathcal{I}}^{\alpha}}{v^\alpha{+} \tau R_{\scriptscriptstyle\mathcal{I}}^{\alpha}}v{\rm d}v}\right]. \label{eq:cpTTIH2} \end{align}\tag{54}\] By considering the random variable \(R^2_{\scriptscriptstyle\mathcal{I}}\) and performing an analysis similarly to (50 ), we have \[\begin{align} p_{\scriptscriptstyle\mathcal{I}}^c &{=} \int_0^1 \!\!\!\frac{1}{(1{+}\tau w^{\alpha/2})^2}\mathbb{E}^0_{H_{\scriptscriptstyle\mathcal{I}}}\!\! \left[e^{-\mu\tau w^{\alpha/2} H_{\scriptscriptstyle\mathcal{I}}^\alpha\sigma^2-\pi\lambda H_{\scriptscriptstyle\mathcal{I}}^2 \kappa(\tau w^{\alpha/2}, \alpha)}\right]{\rm d}w \nonumber\\ &= \frac{2\tau^{-2/\alpha}}{\alpha}\int_0^\tau \!\frac{z^{2/\alpha-1}}{(1{+}z)^2}\mathbb{E}^0_{H_{\scriptscriptstyle\mathcal{I}}}\!\! \left[e^{-\mu z H_{\scriptscriptstyle\mathcal{I}}^\alpha\sigma^2-\pi\lambda H_{\scriptscriptstyle\mathcal{I}}^2 \kappa(z,\alpha)}\right]{\rm d}z,\nonumber \end{align}\] where \(\tau w^{\alpha/2}=z\) and \(\kappa(z,\alpha)\) is defined in using (86 ).  \(\square\)

9.7.2 Theorem 2 (for STINR at a typical mI)↩︎

  The \(\tau\)-coverage probability is \[\begin{align} p^c_{\scriptscriptstyle\mathcal{I};\mathcal{T}} &{=} \mathbb{E}^0_{H_{\scriptscriptstyle\mathcal{I}},R_{\scriptscriptstyle\mathcal{I}}}\left[e^{-\mu\tau R_{\scriptscriptstyle\mathcal{I}}^{\alpha}\sigma^2} \mu\tau R_{\scriptscriptstyle\mathcal{I}}^{\alpha} \int e^{- \mu\tau R_{\scriptscriptstyle\mathcal{I}}^{\alpha} x} \left(1{-}e^{-\mu H_{\scriptscriptstyle\mathcal{I}}^{\alpha} x}\right)^2\right.\nonumber\\ &\left.\times e^{-2\lambda\pi K_\alpha(\mu,x,H_{\scriptscriptstyle\mathcal{I}})}{\rm d}x\right].\nonumber \end{align}\] Given \(H_{\scriptscriptstyle\mathcal{I}}=h\), let \(R_{\scriptscriptstyle\mathcal{I}}\) be the distance to serving station. In this case the tropical interference is \[\mathcal{T}_h:= \rho^+ h^{-\alpha}\vee \rho^- h^{-\alpha} \vee \max_{i\in \mathbb{N}} \rho_i D_i^{-\alpha}.\] We compute the distribution of \(\mathcal{T}_h\), given \(H_{\scriptscriptstyle\mathcal{I}}=h\) similarly to (51 ), as \[\begin{align} \mathbb{P}(\mathcal{T}_h\leq x)&=\left(1-e^{-\mu h^\alpha x}\right)^2 e^{-2\lambda\pi K_\alpha(\mu,x, h)}. \nonumber \end{align}\] Hence, similarly to (52 ), the Laplace transform of \(\mathcal{T}_h\) is \[\begin{align} \mathbb{E}\left[e^{-\gamma\mathcal{T}_{h}}\right] &{=} \mu\tau r^{\alpha} \!\!\int\!\! e^{- \mu\tau r^{\alpha} x} \left(1{-}e^{-\mu h^\alpha x}\right)^2 e^{-2\lambda\pi K_\alpha(\mu,x, h)}{\rm d}x.\nonumber \end{align}\] We obtain the coverage probability using the Laplace transform of the interference \(\mathcal{T}_h\) in the formula (?? ), with \(\gamma=\mu\tau r^\alpha\), given \((H_{\scriptscriptstyle\mathcal{I}}, R_{\scriptscriptstyle\mathcal{I}})= (h,r)\), as \[\begin{align} p^c_{\scriptscriptstyle\mathcal{I};\mathcal{T}} &{=} \mathbb{E}^0_{H_{\scriptscriptstyle\mathcal{I}},R_{\scriptscriptstyle\mathcal{I}}}\left[e^{-\mu\tau R_{\scriptscriptstyle\mathcal{I}}^{\alpha}\sigma^2} \mu\tau R_{\scriptscriptstyle\mathcal{I}}^{\alpha} \int e^{- \mu\tau R_{\scriptscriptstyle\mathcal{I}}^{\alpha} x} \left(1{-}e^{-\mu H_{\scriptscriptstyle\mathcal{I}}^{\alpha} x}\right)^2\right.\nonumber\\ &\left.\times e^{-2\lambda\pi K_\alpha(\mu,x,H_{\scriptscriptstyle\mathcal{I}})}{\rm d}x\right].\nonumber \end{align}\]  \(\square\)

10 Proofs: case without fading↩︎

  As an opening remark for this section, we use the same set of notations for the distance variables, coverage probabilities and data rates, as in the case with fading. Recall from Lemma 5 that at any of the typical epochs of interest, the point process \(\eta_h\) of the distance of all other BS is Poisson on \((h, \infty)\) with intensity measure \(\hat{\mu}\) with density \({\rm d}\hat{\mu}:=2\pi\lambda r\, {\rm d}r\), given \(H=h\), for \(H \in \{H_{\scriptscriptstyle\mathcal{V}}, H_{\scriptscriptstyle S}, R_{\scriptscriptstyle I}, R_{\scriptscriptstyle\mathcal{I}}, R\}\).

10.1 Proof of Theorem 3↩︎

The coverage probability is given by \[\begin{align} p^c(\tau,\lambda,\alpha)&= \mathbb{E}^0_{H}[\mathbb{P}(\mathcal{S}>\tau\vert H)], \label{eq:WFP1} \end{align}\tag{55}\] where, given \(H=h\), the inner term in (55 ) is \[\begin{align} \mathbb{P}(\mathcal{S}>\tau\vert H=h) &= \mathbb{E}_{I_{h}}\left[\mathbb{P}(\mathcal{S}>\tau\vert I_h)\right]\nonumber\\ &=\mathbb{P}_{I_{h}}\left(\tau \left(\sigma^2+I_{h}\right)< h^{-\alpha}\right)\nonumber\\ &=\mathbb{P}_{I_{h}}\left(I_{h}<h^{-\alpha}/{\tau}-\sigma^2\right)\nonumber\\ &=F_{I_{h}}\left(0\vee (h^{-\alpha}/{\tau}-\sigma^2)\right). \nonumber \end{align}\] Then the baseline formula for the coverage probability is \[\begin{align} p^c(\tau,\lambda,\alpha)&= \mathbb{E}^0_{H}\left[F_{I_{H}}\left(0\vee (H^{-\alpha}/{\tau}-\sigma^2)\right)\right].\nonumber\qquad \hfill \square \end{align}\]

10.2 Proof of Theorem 5↩︎

The proof follows the same steps as those in Theorem 3 by considering the distance \(R\) to the serving BS and the distance \(H\) to the nearest interferer as well. Given \(H=h\) and \(R=r\), the coverage probability is \[\begin{align} \mathbb{P}(\mathcal{S}>\tau\vert H=h, R=r) &= \mathbb{E}_{I_{h}}\left[\mathbb{P}(\mathcal{S}>\tau\vert h, I_h)\right]\nonumber\\ &=\mathbb{P}_{I_{h}}\left(\tau \left(\sigma^2+I_{h}\right)< r^{-\alpha}\right)\nonumber\\ &=\mathbb{P}_{I_{h}}\left(I_{h}<r^{-\alpha}/{\tau}-\sigma^2\right)\nonumber\\ &=F_{I_{h}}\left(0\vee (r^{-\alpha}/{\tau}-\sigma^2)\right). \label{eq:NFCP} \end{align}\tag{56}\] By jointly averaging over the distances \(R\) and \(H\), from (56 ), we have \[\begin{align} p^c(\tau,\lambda,\alpha)&= \mathbb{E}^0_{H,R}\left[F_{I_{H}}\left(0\vee (R^{-\alpha}/{\tau}-\sigma^2)\right)\right].\nonumber\qquad \hfill\square \end{align}\]

10.3 Proof of Theorem 4↩︎

The result can be proved similarly to the last one. \(\square\)

For simplicity we write \(p^c_{*}\equiv p^c_{*}(\tau,\lambda,\alpha)\), \(p^c_{\scriptscriptstyle*;\mathcal{T}}\equiv p^c_{\scriptscriptstyle*;\mathcal{T}}(\tau,\lambda,\alpha)\) and \(\mathcal{R}_{*}\equiv \mathcal{R}_{*}(\lambda,\alpha)\), \(\mathcal{R}_{\scriptscriptstyle*;\mathcal{T}}\equiv \mathcal{R}_{\scriptscriptstyle*;\mathcal{T}}(\lambda,\alpha)\), which are understood to a function of the set of parameters \(\{\tau,\lambda,\alpha\}\) and \(\{\lambda,\alpha\}\), respectively, where \(*\in \{\scriptscriptstyle{\mathcal{V}, S, I, \mathcal{I}, t}\) \(\}\).

10.4 Min-signal-max-interference (mS-MI): typical handover↩︎

The typical handover distance \(H_{\scriptscriptstyle\mathcal{V}}\) is given by a Nakagami distribution with parameter \(\left(\frac{3}{2}, \frac{3}{2\lambda\pi}\right)\) and with density (7 ).

10.4.1 Theorem 3 (for SINR at typical mS-MI)↩︎

In this case the interference is \(I_{h}:=h^{-\alpha}+\sum_i D_i^{-\alpha}\), given \(H_{\scriptscriptstyle\mathcal{V}}=h\). Suppose \(F_{I_h}\) is the distribution of the interference \(I_h\). By applying the formula (?? ) the coverage probability is \[\begin{align} p^c_{\scriptscriptstyle\mathcal{V}} &= \mathbb{E}^0_{H_{\scriptscriptstyle\mathcal{V}}}\left[F_{I_{H_{\scriptscriptstyle\mathcal{V}}}}\left(0\vee (H_{\scriptscriptstyle\mathcal{V}}^{-\alpha}/{\tau}-\sigma^2)\right)\right], \label{eq:cp-hp-nf} \end{align}\tag{57}\] where the distribution of \(I_h=h^{-\alpha}+I'_h\), given \(H_{\scriptscriptstyle\mathcal{V}}=h\), can be found using the Laplace transform \[\begin{align} \mathcal{L}_{I_h}(\nu) & = e^{-\nu h^{-\alpha}}\mathcal{L}_{I'_h}(\nu) \stackrel{(\ref{eq:LI1})}{=} e^{-\nu h^{-\alpha} -\pi\lambda L_\nu(h, \alpha)}. \end{align}\]

10.4.2 Theorem 3 (for STINR at typical mS-MI)↩︎

Given a handover at a distance \(H_{\scriptscriptstyle\mathcal{V}}=h\), the tropical interference in this case is given by the other BS at a distance \(h\), i.e., \(\mathcal{T}_h=h^{-\alpha}\). The coverage probability is \[\begin{align} p^c_{\scriptscriptstyle\mathcal{V};\mathcal{T}} &{=} \int_0^\infty {\mathbb{1}}_{h^{-\alpha}>\tau(\sigma^2+h^{-\alpha})} f_{H_{\scriptscriptstyle\mathcal{V}}}(h){\rm d}h\nonumber\\ &{=} \!\!\int_0^\infty\!\!\!\!\! {\mathbb{1}}_{h<\left(\frac{1-\tau}{\tau\sigma^2}\right)^\frac{1}{\alpha}} f_{H_{\scriptscriptstyle\mathcal{V}}}(h){\rm d}h {=} F_{H_{\scriptscriptstyle\mathcal{V}}}\left(\left(\frac{1{-}\tau}{\tau\sigma^2}\right)^\frac{1}{\alpha}\right). \label{eq:WFTP2} \end{align}\tag{58}\] which is computed using the density of \(H_{\scriptscriptstyle\mathcal{V}}\) from (7 ).

10.5 Typical time↩︎

The distance to the nearest BS follows a Rayleigh density \[f_R(r):=2\lambda\pi r \, e^{-\lambda\pi r^2}.\]

10.5.1 Theorem 3 (for SINR at typical time)↩︎

In this case the distance \(R\) to the nearest point is Rayleigh distributed and the interference is \[I_{r}:=\sum_i D_i^{-\alpha},\] given \(R=r\). The coverage probability is given by the formula (?? ) as \[\begin{align} p^c_{t} &= \int_{0}^\infty F_{I_{r}}\left(\left(\frac{ r^{-\alpha}}{\tau}-\sigma^2\right)\vee 0\right) f_{R}(r){\rm d}r. \label{eq:WFTtime1} \end{align}\tag{59}\] where \(F_{I_r}\) is the distribution of \(I_r\) when \(R=r\), is given by the Laplace transform as in (?? ) \[\mathcal{L}_{I_r}(\nu) = e^{-\pi\lambda L_\nu(r,\alpha)}.\]

10.5.2 Theorem 4 (for STINR at typical time)↩︎

In the tropical case the interference is given by the second closest station. Let \(R_1\) be the distance to the second closest station. In this case the tropical interference is given by \(\mathcal{T}= R_1^{-\alpha}\). From (?? ) the coverage probability turns out to be \[\begin{align} p^c_{t;\scriptscriptstyle{\mathcal{T}}} &= \int_{0}^\infty F_{\scriptscriptstyle\mathcal{T}}\left(\left(r^{-\alpha}/{\tau}-\sigma^2\right)\vee 0\right) f_{R}(r){\rm d}r~\tag{60}\\ &= \int_{0}^\infty \!\!\mathbb{P}_{R_1\vert r}\left(R_1{>}\left( \left(r^{-\alpha}/{\tau}{-}\sigma^2\right)\vee 0\right)^{-\frac{1}{\alpha}}\vee r\right) f_{R}(r){\rm d}r\nonumber\\ &= \int_{0}^{(\tau \sigma^2)^{-1/\alpha}} \mathbb{P}_{R_1\vert r}\left(R_1>\left(r^{-\alpha}/{\tau}-\sigma^2\right)^{-1/\alpha}\vee r\right) f_{R}(r){\rm d}r\nonumber\\ &{=} \int_{0}^{(\tau \sigma^2)^{-1/\alpha}} \!\!\!\! \!\! \int_{\left(h^{-\alpha}/{\tau}-\sigma^2\right)^{-1/\alpha}\vee h}^\infty \!\!\! f_{R_1\vert h}(r){\rm d}r f_{R}(h){\rm d}h.\tag{61} \end{align}\] Note that \(h^{-\alpha}/{\tau}-\sigma^2< h^{-\alpha}\) for all \(\tau\geq 1\) and under this case we have \[\begin{align} p^c_{{\scriptscriptstyle t;\mathcal{T}}} &= \int_{0}^{(\tau \sigma^2)^{-1/\alpha}} \!\!\!\! \! \int_{\left(h^{-\alpha}/{\tau}-\sigma^2\right)^{-1/\alpha}}^\infty f_{R_1\vert h}(r){\rm d}r f_{R}(h){\rm d}h.\nonumber \end{align}\] For \(\tau< 1\), \(h^{-\alpha}/{\tau}-\sigma^2\geq h^{-\alpha}\) if and only if \(h\leq \left(\frac{\sigma^2 \tau}{1-\tau}\right)^{-1/\alpha}\) and hence \[\begin{align} p^c_{{\scriptscriptstyle t;\mathcal{T}}} &= \int_{0}^{\left(\frac{\sigma^2 \tau}{1-\tau}\right)^{-1/\alpha}} \!\!\!\! \! \int_{h}^\infty f_{R_1\vert h}(r){\rm d}r f_{R}(h){\rm d}h\nonumber\\ & \;\; + \int_{\left(\frac{\sigma^2 \tau}{1-\tau}\right)^{-1/\alpha}}^{(\tau \sigma^2)^{-1/\alpha}} \int_{\left(h^{-\alpha}/{\tau}-\sigma^2\right)^{-1/\alpha}}^\infty f_{R_1\vert h}(r){\rm d}r f_{R}(h){\rm d}h\nonumber\\ &= \mathbb{P}\left(R\leq (1-\tau)^{1/\alpha}\left(\sigma^2 \tau\right)^{-1/\alpha}\right)\nonumber\\ & \;\; + \int_{\left(\frac{\sigma^2 \tau}{1-\tau}\right)^{-1/\alpha}}^{(\tau \sigma^2)^{-1/\alpha}} \int_{\left(h^{-\alpha}/{\tau}-\sigma^2\right)^{-1/\alpha}}^\infty f_{R_1\vert h}(r){\rm d}r f_{R}(h){\rm d}h,\nonumber \end{align}\] since \(\left(\frac{\sigma^2 \tau}{1-\tau}\right)^{-1/\alpha}\leq (\sigma^2 \tau)^{-1/\alpha}\) for \(\tau<1\).

10.6 Typical max-signal (MS)↩︎

Let \(H_{\scriptscriptstyle S}\) be the distance to the typical visible head point. It follows from [26] that the distance \(H_{\scriptscriptstyle S}\) to the typical visible head follows a Nakagami distribution with parameter \(\left({\frac{1}{2}}, \frac{1}{2\lambda\pi}\right)\), i.e., \[f_{H_{\scriptscriptstyle S}}(h)= 2\lambda^{\frac{1}{2}}e^{-\lambda\pi h^2}.\]

10.6.1 Theorem 3 (for SINR at typical MS)↩︎

Given the distance to the serving BS \(H_{\scriptscriptstyle S}=h\), the total interference is \(I_h=\sum_i D_i^{-\alpha}\), and suppose its distribution is \(F_{I_h}\) given by the Laplace transform \[\mathcal{L}_{I_h}(\nu) = e^{-\pi\lambda L_\nu(h,\alpha)}.\] We determine the coverage probability using (?? ) as \[\begin{align} p^c_{\scriptscriptstyle S} &= \int_{0}^\infty F_{I_{h}}\left(0\vee \left(\frac{ h^{-\alpha}}{\tau}-\sigma^2\right)\right) f_{H_{\scriptscriptstyle S}}(h){\rm d}h. \label{eq:WFV1} \end{align}\tag{62}\]

10.6.2 Theorem 4 (for STINR at typical MS)↩︎

The tropical interference is given by the second closest station, distance of which is \(R_{\scriptscriptstyle S}\). Given \(H_{\scriptscriptstyle S}=h\), we have \(\mathcal{T}_h= R_{\scriptscriptstyle S}^{-\alpha}\), where \(R_{\scriptscriptstyle S}\geq h\) is the distance to the nearest interferer. The distribution of interference is \[\begin{align} \mathbb{P}(\mathcal{T}_h\leq x)=\mathbb{P}(R_{\scriptscriptstyle S}^{-\alpha}\leq x)=\mathbb{P}(R_{\scriptscriptstyle S}>x^{-1/\alpha}).\nonumber \end{align}\] From (?? ) the coverage probability \(p^c_{{\scriptscriptstyle S;\mathcal{T}}}\) turns out to be \[\begin{align} \nonumber\\ &= \int_{0}^\infty \!\!\!\! \!\!\mathbb{P}_{R_{\scriptscriptstyle S}\vert h}\left(R_{\scriptscriptstyle S}{>}\left((h^{-\alpha}/{\tau}{-}\sigma^2)\vee 0\right)^{-1/\alpha} \vee h\right)\!\! f_{H_{\scriptscriptstyle S}}(h){\rm d}h\nonumber\\ &= \int_{0}^{(\tau \sigma^2)^{-1/\alpha}} \!\!\!\! \!\!\!\!\!\!\!\! \!\!\!\! \mathbb{P}_{R_{\scriptscriptstyle S}\vert h}\left(R_{\scriptscriptstyle S}>\left(h^{-\alpha}/{\tau}{-}\sigma^2\right)^{-1/\alpha}\vee h\right) \!\!f_{H_{\scriptscriptstyle S}}(h){\rm d}h\nonumber\\ &= \int_{0}^{(\tau \sigma^2)^{-1/\alpha}} \!\!\!\! \! \! \int_{\left(h^{-\alpha}/{\tau}-\sigma^2\right)^{-1/\alpha}\vee h}^\infty \!\!\! f_{R_{\scriptscriptstyle S}\vert h}(r){\rm d}r f_{H_{\scriptscriptstyle S}}(h){\rm d}h. \label{eq:TNFS} \end{align}\tag{63}\] Note that \(h^{-\alpha}/{\tau}-\sigma^2< h^{-\alpha}\) for all \(\tau\geq 1\) and in this case, we have \[\begin{align} p^c_{{\scriptscriptstyle S;\mathcal{T}}} &= \int_{0}^{(\tau \sigma^2)^{-1/\alpha}} \!\!\!\! \! \int_{\left(h^{-\alpha}/{\tau}-\sigma^2\right)^{-1/\alpha}}^\infty f_{R_{\scriptscriptstyle S}\vert h}(r){\rm d}r f_{H_{\scriptscriptstyle S}}(h){\rm d}h.\nonumber \end{align}\] For \(\tau< 1\), \(h^{-\alpha}/{\tau}-\sigma^2\geq h^{-\alpha}\) if \(h\leq \left(\frac{\sigma^2 \tau}{1-\tau}\right)^{-1/\alpha}\) and hence \[\begin{align} p^c_{{\scriptscriptstyle S;\mathcal{T}}} &= \int_{0}^{\left(\frac{\sigma^2 \tau}{1-\tau}\right)^{-1/\alpha}} \!\!\!\! \! \int_{h}^\infty f_{R_{\scriptscriptstyle S}\vert h}(r){\rm d}r f_{H_{\scriptscriptstyle S}}(h){\rm d}h\nonumber\\ & \;\; + \int_{\left(\frac{\sigma^2 \tau}{1-\tau}\right)^{-1/\alpha}}^{(\tau \sigma^2)^{-1/\alpha}} \int_{\left(h^{-\alpha}/{\tau}-\sigma^2\right)^{-1/\alpha}}^\infty f_{R_{\scriptscriptstyle S}\vert h}(r){\rm d}r f_{H_{\scriptscriptstyle S}}(h){\rm d}h\nonumber\\ &= \mathbb{P}\left( H_{\scriptscriptstyle S}\leq (1-\tau)^{1/\alpha}\left(\sigma^2 \tau\right)^{-1/\alpha}\right)\nonumber\\ & \;\; + \int_{\left(\frac{\sigma^2 \tau}{1-\tau}\right)^{-1/\alpha}}^{(\tau \sigma^2)^{-1/\alpha}} \int_{\left(h^{-\alpha}/{\tau}-\sigma^2\right)^{-1/\alpha}}^\infty f_{R_{\scriptscriptstyle S}\vert h}(r){\rm d}r f_{H_{\scriptscriptstyle S}}(h){\rm d}h,\nonumber \end{align}\] since \(\left(\frac{\sigma^2 \tau}{1-\tau}\right)^{-1/\alpha}\leq (\sigma^2 \tau)^{-1/\alpha}\) for \(\tau<1\).

10.7 Typical max-interference (MI)↩︎

Suppose the nearest BS and the nearest interferer are at distance \(R_{\scriptscriptstyle I}\) and \(H_{\scriptscriptstyle I}\), respectively.

10.7.1 Theorem 5 (for SINR at typical MI)↩︎

In this case given \(H_{\scriptscriptstyle I}\), the distance to the closest interferer at its nearest position, the total interference is \[I_{H_{\scriptscriptstyle I}}:= H_{\scriptscriptstyle I}^{-\alpha}+ \sum_{i\in \mathbb{N}} D_i^{-\alpha},\] and given \(R_{\scriptscriptstyle I} =r\), we have \(D_i\geq H_{\scriptscriptstyle I}\geq r\), for all \(D_i\in \eta_{h}\). Given the typical invisible head at distance \(H_{\scriptscriptstyle I}=h\) the Laplace transform of \(I_h\) is \[\begin{align} \mathcal{L}_{I_h}(\nu) &\stackrel{(\ref{eq:LI1})}{=}e^{-\nu h^{-\alpha}-\pi\lambda L_\nu(h, \alpha)}. \label{eq:Iint-sinr} \end{align}\tag{64}\] Let \(F_{I_{h}}\) be the distribution of the interference \(I_{h}\). Using that and the formula (?? ) we find the coverage probability as \[\begin{align} p^c_{\scriptscriptstyle I} &= \mathbb{E}^0_{H_{\scriptscriptstyle I},R_{\scriptscriptstyle I}}\left[F_{I_{H_{\scriptscriptstyle I}}}\left(\left(\frac{ R_{\scriptscriptstyle I}^{-\alpha}}{\tau}-\sigma^2\right)\vee 0\right) \right]. \label{eq:cpTIH} \end{align}\tag{65}\]

10.7.2 Theorem 5 (for STINR at typical MI)↩︎

In this case, we can only consider the case where \(\tau\geq 1\). The tropical interference is given by the distance to the second closest BS \(H_{\scriptscriptstyle I}\), \(\mathcal{T}= H_{\scriptscriptstyle I}^{-\alpha}\). The distribution of interference is \[\begin{align} F_{\mathcal{T}}(x)&= \mathbb{P}\left(H_{\scriptscriptstyle I}^{-\alpha}\leq x\right)=\mathbb{P}\left(H_{\scriptscriptstyle I}\geq x^{-1/\alpha}\right).\nonumber \end{align}\] From formula (?? ), we have \[\begin{align} p^c_{{\scriptscriptstyle I ;\mathcal{T}}} &= \mathbb{E}^0_{H_{\scriptscriptstyle I},R_{\scriptscriptstyle I}}\left[F_{\mathcal{T}}\left(\left(\frac{ R_{\scriptscriptstyle I}^{-\alpha}}{\tau}-\sigma^2\right)\vee 0\right) \right]\nonumber\\ &= \mathbb{P}^0\left(H_{\scriptscriptstyle I}\geq \left(\left(\frac{ R_{\scriptscriptstyle I}^{-\alpha}}{\tau}-\sigma^2\right)\vee 0\right)^{-1/\alpha}\right)\nonumber\\ &= \mathbb{P}^0\left(H_{\scriptscriptstyle I}\geq \left(\left(\frac{ R_{\scriptscriptstyle I}^{-\alpha}}{\tau}-\sigma^2\right)\vee 0\right)^{-1/\alpha}\!\!\! ; R^{-\alpha}_{\scriptscriptstyle I}\leq \sigma^2\tau \right)\nonumber\\ &\; \; +\mathbb{P}^0\left(H_{\scriptscriptstyle I}\geq \left(\left(\frac{ R_{\scriptscriptstyle I}^{-\alpha}}{\tau}-\sigma^2\right)\vee 0\right)^{-1/\alpha}\!\!\! ; R_{\scriptscriptstyle I}^{-\alpha}\geq\sigma^2\tau \right)\nonumber\\ &=\mathbb{P}^0(H_{\scriptscriptstyle I}=\infty; R^{-\alpha}_{\scriptscriptstyle I}\leq \sigma^2\tau)\nonumber\\ &\;\;\;\; +\mathbb{P}^0\left(R_{\scriptscriptstyle I}\leq \left(\tau(H_{\scriptscriptstyle I}^{-\alpha}+\sigma^2)\right)^{-1/\alpha}\!\!\! ; R_{\scriptscriptstyle I}^{-\alpha}\geq\sigma^2\tau \right)\nonumber\\ &=\mathbb{P}^0\left(R_{\scriptscriptstyle I}\leq \left(\tau(H_{\scriptscriptstyle I}^{-\alpha}+\sigma^2)\right)^{-1/\alpha} \right). \label{eq:tr-Ical-nf} \end{align}\tag{66}\] Note that \[\begin{align} \nonumber\\ &= \tau^{-2/\alpha}\mathbb{E}^0_{H_{\scriptscriptstyle I}}\left[H^{-2}_{\scriptscriptstyle I}\left(H_{\scriptscriptstyle I}^{-\alpha}+\sigma^2\right)^{-2/\alpha}\right]\nonumber\\ &= \tau^{-2/\alpha}\mathbb{E}^0_{H_{\scriptscriptstyle I}}\left[\left(1+\sigma^2 H_{\scriptscriptstyle I}^{\alpha}\right)^{-2/\alpha}\right]. \label{eq:WFtropI} \end{align}\tag{67}\]

10.8 Typical min-interference (mI): tropical interference handover↩︎

Suppose the nearest BS and nearest interferer are at distance \(R_{\scriptscriptstyle\mathcal{I}}\) and \(H_{\scriptscriptstyle\mathcal{I}}\).

10.8.1 Theorem 5 (for SINR at typical mI)↩︎

In this case, given \(H_{\scriptscriptstyle\mathcal{I}}\), the total interference is \[I_{H_{\scriptscriptstyle\mathcal{I}}}:=2H_{\scriptscriptstyle\mathcal{I}}^{-\alpha} + \sum_{i\in \mathbb{N}} D_i^{-\alpha}.\] Then using (?? ), the coverage probability is given by \[\begin{align} p^c_{\scriptscriptstyle\mathcal{I}} &= \mathbb{E}^0_{H_{\scriptscriptstyle\mathcal{I}},R_{\scriptscriptstyle\mathcal{I}}}\left[F_{I_{H_{\scriptscriptstyle\mathcal{I}}}}\left(\frac{ R_{\scriptscriptstyle\mathcal{I}}^{-\alpha}}{\tau}-\sigma^2\right) \right], \label{eq:TI-cpT} \end{align}\tag{68}\] where the distribution of the interference \(I_h\) given \(H_{\scriptscriptstyle\mathcal{I}}= h\), is computed similarly to (64 ) using the Laplace transform as \[\begin{align} \mathcal{L}_{I_h}(\nu) &\stackrel{(\ref{eq:LI1})}{=}e^{-2\nu h^{-\alpha}-\pi\lambda L_\nu(h, \alpha)}. \nonumber \end{align}\]

10.8.2 Theorem 5 (for STINR at typical mI)↩︎

This is only under the case where \(\tau\geq 1\). In this case, the tropical interference is \(\mathcal{T}:= H_{\scriptscriptstyle\mathcal{I}}^{-\alpha}\). The distribution of the interference is \[\begin{align} F_{\mathcal{T}}(x)&= \mathbb{P}\left(H_{\scriptscriptstyle\mathcal{I}}^{-\alpha}\leq x\right)=\mathbb{P}\left(H_{\scriptscriptstyle\mathcal{I}}\geq x^{-1/\alpha}\right). \nonumber \end{align}\] The coverage probability is computed similarly to (66 ) and (67 ) and using formula (?? ) as \[\begin{align} p^c_{{\scriptscriptstyle\mathcal{I};\mathcal{T}}} &= \mathbb{E}^0_{H_{\scriptscriptstyle\mathcal{I}},R_{\scriptscriptstyle\mathcal{I}}}\left[F_{\mathcal{T}}\left(\left(\frac{ R_{\scriptscriptstyle\mathcal{I}}^{-\alpha}}{\tau}-\sigma^2\right) \vee 0 \right)\right]\nonumber\\ &=\mathbb{P}^0\left(H_{\scriptscriptstyle\mathcal{I}}\geq \left(\left(\frac{ R_{\scriptscriptstyle\mathcal{I}}^{-\alpha}}{\tau}-\sigma^2\right)\vee 0\right)^{-\frac{1}{\alpha}}\right)\nonumber\\ &= \mathbb{P}^0\left(R_{\scriptscriptstyle\mathcal{I}}\leq \left(\tau( H_{\scriptscriptstyle\mathcal{I}}^{-\alpha}+\sigma^2)\right)^{-1/\alpha}\right)\nonumber\\ &= \tau^{-2/\alpha}\mathbb{E}^0_{H_{\scriptscriptstyle\mathcal{I}}}\left[H^{-2}_{\scriptscriptstyle\mathcal{I}}\left(H_{\scriptscriptstyle\mathcal{I}}^{-\alpha}+\sigma^2\right)^{-2/\alpha}\right]\nonumber\\ &= \tau^{-2/\alpha}\mathbb{E}^0_{H_{\scriptscriptstyle\mathcal{I}}}\left[\left(1+\sigma^2 H_{\scriptscriptstyle\mathcal{I}}^{\alpha}\right)^{-2/\alpha}\right]. \label{eq:cp-nftrop} \end{align}\tag{69}\]

11 Proofs: Comparison of performance metrics↩︎

 

11.1 Theorem 6 (with fading)↩︎

 

Part [CP-F1].  All these inequalities can be obtained by comparing the coverage probabilities \[p_{\scriptscriptstyle\mathcal{V}}^c = \frac{1}{1+\tau}\int_{0}^\infty e^{-\mu\tau h^{\alpha}\sigma^2-\pi\lambda h^2 \kappa(\tau,\alpha)} f_{H_{\scriptscriptstyle\mathcal{V}}}(h){\rm d}h,\] \[p_{t}^c = \int_{0}^\infty e^{-\mu\tau r^{\alpha}\sigma^2-\pi\lambda r^2 \kappa(\tau,\alpha)} f_{R}(r){\rm d}r,\] \[p_{\scriptscriptstyle S}^c =\int_{0}^\infty e^{-\mu\tau h^{\alpha}\sigma^2-\pi\lambda h^2\kappa(\tau,\alpha)}f_{H_{\scriptscriptstyle S}}(h){\rm d}h,\] and comparing the densities of \(H_{\scriptscriptstyle\mathcal{V}}, R, H_{\scriptscriptstyle S}\). Note that the integrand in all the integrals are same and we define \[\begin{align} G(h):=e^{-\mu\tau h^{\alpha}\sigma^2-\pi\lambda h^2\kappa(\tau,\alpha)}, \label{eq:Gh} \end{align}\tag{70}\] writing it in same variable \(h\). Recall the probability densities of the random variables \(H_{\scriptscriptstyle\mathcal{V}}\), \(R\) and \(H_{\scriptscriptstyle S}\), \(f_{H_{\scriptscriptstyle\mathcal{V}}}(h)=4\pi\lambda^{3/2} h^2 e^{-\lambda\pi h^2}, f_R(h)=2\lambda\pi h \, e^{-\lambda\pi h^2}\) and \(f_{H_{\scriptscriptstyle S}}(h)=2\lambda^{1/2} e^{-\lambda\pi h^2}\), for \(h\geq 0\), respectively. Observe that \(f_{H_{\scriptscriptstyle\mathcal{V}}}(h)\leq f_R(h)\) on \([0, 1/{2\sqrt{\lambda}}]\) and \(f_{H_{\scriptscriptstyle\mathcal{V}}}(h)>f_R(h)\) on \((1/{2\sqrt{\lambda}}, \infty)\). Hence \[\begin{align} p_t^c -(1+\tau) p_{\scriptscriptstyle\mathcal{V}}^c &= \int_0^{1/2\sqrt{\lambda}} G(h) \left[f_R(h)-f_{H_{\scriptscriptstyle\mathcal{V}}}(h)\right]{\rm d}h\nonumber\\ &- \int_{1/2\sqrt{\lambda}}^\infty G(h) \left[f_{H_{\scriptscriptstyle\mathcal{V}}}(h)- f_R(h)\right]{\rm d}h\nonumber\\ &\geq G(1/2\sqrt{\lambda}) \int_0^\infty \!\!\!\!\left[f_{H_{\scriptscriptstyle\mathcal{V}}}(h)- f_R(h)\right]{\rm d}h =0,\nonumber \end{align}\] which implies that \(p_t^c \geq (1+\tau) p_{\scriptscriptstyle\mathcal{V}}^c \geq p_{\scriptscriptstyle\mathcal{V}}^c\), since \(G(1/2\sqrt{\lambda})\geq 0\) and \(\tau\geq 0\).

For the other inequality, note that \(f_R(h)\leq f_{H_{\scriptscriptstyle S}}(h)\) on \([0, 1/{\pi\sqrt{\lambda}}]\) and \(f_R(h)>f_{H_{\scriptscriptstyle S}}(h)\) on \((1/{\pi\sqrt{\lambda}}, \infty)\). Then by similar comparison we have that \(p_{\scriptscriptstyle S}^c \geq p_{t}^c\).

For the second part, we compare the coverage probabilities \[\begin{align} p^c_{\scriptscriptstyle I}&{=} \mathbb{E}^0_{H_{\scriptscriptstyle I}, R_{\scriptscriptstyle I}}\left[ \frac{H_{\scriptscriptstyle I}^{\alpha}}{H_{\scriptscriptstyle I}^{\alpha}{+}\tau R_{\scriptscriptstyle I}^{\alpha}} e^{-\mu\tau R_{\scriptscriptstyle I}^{\alpha}\sigma^2-2\pi\lambda\int_{H_{\scriptscriptstyle I}}^\infty\frac{\tau R_{\scriptscriptstyle I}^{\alpha}}{r^{\alpha}{+}\tau R_{\scriptscriptstyle I}^{\alpha}}r{\rm d}r} \right], \nonumber \end{align}\] \[\begin{align} p_{\scriptscriptstyle\mathcal{I}}^c &{=}E^0_{H_{\scriptscriptstyle\mathcal{I}}, R_{\scriptscriptstyle\mathcal{I}}}\!\!\left[\left(\frac{H_{\scriptscriptstyle\mathcal{I}}^\alpha}{H_{\scriptscriptstyle\mathcal{I}}^\alpha{+}\tau R_{\scriptscriptstyle\mathcal{I}}^\alpha}\right)^2\!\! e^{-\mu\tau R_{\scriptscriptstyle\mathcal{I}}^{\alpha}\sigma^2-2\pi\lambda\int_{H_{\scriptscriptstyle\mathcal{I}}}^\infty\frac{\tau R_{\scriptscriptstyle\mathcal{I}}^{\alpha}}{v^\alpha{+} \tau R_{\scriptscriptstyle\mathcal{I}}^{\alpha}}v{\rm d}v}\right].\nonumber \end{align}\] The densities of \(H_{\scriptscriptstyle I}\) and \(H_{\scriptscriptstyle\mathcal{I}}\) are \[f_{H_{\scriptscriptstyle I}}(h)=4\pi\lambda^{3/2} h^2 e^{-\lambda\pi h^2}, f_{H_{\scriptscriptstyle\mathcal{I}}}(h)=\frac{8}{3}\pi^2\lambda^{5/2} h^4 e^{-\lambda\pi h^2},\] respectively for all \(h\geq 0\). Note that \(f_{H_{\scriptscriptstyle\mathcal{I}}}(h)\leq f_{H_{\scriptscriptstyle I}}(h)\) on \([0, {\sqrt{3/{2\lambda\pi}}}]\) and \(f_{H_{\scriptscriptstyle\mathcal{I}}}(h)>f_{H_{\scriptscriptstyle I}}(h)\) on \((\sqrt{3/{2\lambda\pi}}, \infty)\). Define \[G_1(h,r):= \frac{h^\alpha}{h^\alpha+\tau r^\alpha} e^{-\mu\tau r^{\alpha}\sigma^2- 2\pi\lambda\int_{h}^\infty\frac{\tau r^{\alpha}}{v^\alpha+ \tau r^{\alpha}}v{\rm d}v},\] \[G_2(h,r):=\left(\frac{h^\alpha}{h^\alpha+\tau r^\alpha}\right)^2e^{-\mu\tau r^{\alpha}\sigma^2- 2\pi\lambda\int_{h}^\infty\frac{\tau r^{\alpha}}{v^\alpha+ \tau r^{\alpha}}v{\rm d}v}.\] Observe that \(G_2(h,r)\leq G_1(h,r)\) for all pair \((h,r)\), because of the extra factor of the form \(\frac{h^\alpha}{h^\alpha+\tau r^\alpha}\leq 1\), in \(G_2(h,r)\), for all \(h,r,\tau\in \mathbb{R}^+\). Let us define \[\begin{align} G_1(h):=\mathbb{E}_{R_{\scriptscriptstyle I}\vert h}[G_1(h,R_{\scriptscriptstyle I}\vert h)], G_2(h):= \mathbb{E}_{R_{\scriptscriptstyle\mathcal{I}}\vert h}[G_2(h,R_{\scriptscriptstyle\mathcal{I}}\vert h)], \label{eq:G1G2-def} \end{align}\tag{71}\] where \(G_2(h)\leq G_1(h)\) for all \(h\geq 0\). Then \[\begin{align} p_{\scriptscriptstyle I}^c - p_{\scriptscriptstyle\mathcal{I}}^c &= \int_0^\infty \int_0^h G_1(h,r) f_{R_{\scriptscriptstyle I}\vert h}(r){\rm d}r f_{H_{\scriptscriptstyle I}}(h) {\rm d}h \nonumber\\ & -\int_0^\infty \int_0^h G_2(h,r)f_{R_{\scriptscriptstyle\mathcal{I}}\vert h}(r){\rm d}r f_{H_{\scriptscriptstyle\mathcal{I}}}(h) {\rm d}h\nonumber\\ &= \int_0^\infty \mathbb{E}_{R_{\scriptscriptstyle I}\vert h}[G_1(h,R_{\scriptscriptstyle I}\vert h)]f_{H_{\scriptscriptstyle I}}(h) {\rm d}h \nonumber\\ & -\int_0^\infty \mathbb{E}_{R_{\scriptscriptstyle\mathcal{I}}\vert h}[G_2(h,R_{\scriptscriptstyle\mathcal{I}}\vert h)] f_{H_{\scriptscriptstyle\mathcal{I}}}(h) {\rm d}h\nonumber\\ &\stackrel{(\ref{eq:G1G2-def})}{=} \int_0^\infty G_1(h) f_{H_{\scriptscriptstyle I}}(h) {\rm d}h -\int_0^\infty G_2(h) f_{H_{\scriptscriptstyle\mathcal{I}}}(h) {\rm d}h\nonumber\\ &\geq \int_0^\infty G_2(h) f_{H_{\scriptscriptstyle I}}(h) {\rm d}h -\int_0^\infty G_2(h) f_{H_{\scriptscriptstyle\mathcal{I}}}(h) {\rm d}h\nonumber\\ &= \int_0^{\sqrt{3/{2\lambda\pi}}} G_2(h) \left[f_{H_{\scriptscriptstyle I}}(h)- f_{H_{\scriptscriptstyle\mathcal{I}}}(h)\right] {\rm d}h \nonumber\\ & -\int_{\sqrt{3/{2\lambda\pi}}}^\infty G_2(h) \left[f_{H_{\scriptscriptstyle\mathcal{I}}}(h)- f_{H_{\scriptscriptstyle I}}(h)\right] {\rm d}h\nonumber\\ &\geq G_2(\sqrt{3/{2\lambda\pi}}) \int_0^\infty\!\! \left[f_{H_{\scriptscriptstyle I}}(h){-} f_{H_{\scriptscriptstyle\mathcal{I}}}(h)\right] {\rm d}h =0,\nonumber \end{align}\] since \(G_2(\sqrt{3/{2\lambda\pi}})\geq 0\). This proves the result for the non-tropical case. \(\square\)

Part [CP-F2]. The results \(p_{\scriptscriptstyle{\mathcal{V};\mathcal{T}}}^c \leq p_{t;\scriptscriptstyle{\mathcal{T}}}^c \leq p_{{\scriptscriptstyle S};\scriptscriptstyle{\mathcal{T}}}^c and p^c_{{\scriptscriptstyle\mathcal{I}; \mathcal{T}}} \leq p^c_{{\scriptscriptstyle I;\mathcal{T}}}\), in the tropical case can be proved similarly. \(\square\)

11.2 Theorem 6 (without fading)↩︎

 

Part [CP-F1]. The proof essentially follows by comparing the the random variables \(H_{\scriptscriptstyle\mathcal{V}}\), \(R\) and \(H_{\scriptscriptstyle S}\). Note that \(F_{I_{H_{\scriptscriptstyle\mathcal{V}}}}(x)\leq F_{I_R}(x)=F_{I_{\scriptscriptstyle S}}(x)\) for any \(x\geq 0\), since \[I_{H_{\scriptscriptstyle\mathcal{V}}=h}= h^{-\alpha} + \sum_i D_i^{-\alpha}, I_{R=h}= \sum_i D_i^{-\alpha} \stackrel{d}{=} I_{H_{\scriptscriptstyle S}=h}.\] We prove the result by comparing the coverage probabilities \[p^c_{\scriptscriptstyle\mathcal{V}} = \mathbb{E}^0_{H_{\scriptscriptstyle\mathcal{V}}}\left[F_{I_{H_{\scriptscriptstyle\mathcal{V}}}}\left(0\vee (H_{\scriptscriptstyle\mathcal{V}}^{-\alpha}/{\tau}-\sigma^2)\right)\right],\] \[p^c_t = \mathbb{E}^0_R\left[F_{I_R}\left(0\vee (R^{-\alpha}/{\tau}-\sigma^2)\right)\right],\] \[p^c_{\scriptscriptstyle S} = \mathbb{E}^0_{H_{\scriptscriptstyle S}}\left[F_{I_{H_{\scriptscriptstyle S}}}\left(0\vee (H_{\scriptscriptstyle S}^{-\alpha}/{\tau}-\sigma^2)\right)\right],\] and using the technique similar to the first part of Theorem 6, with \(G(h)= F_{I_{H=h}}\left(0\vee (h^{-\alpha}/{\tau}-\sigma^2)\right)\), for any \(H\in \{H_{\scriptscriptstyle\mathcal{V}}, R, H_{\scriptscriptstyle S}\}\).

For this part, we use a comparison argument similar to second part of Theorem 6 for \[p^c_{\scriptscriptstyle I} = \mathbb{E}^0_{H_{\scriptscriptstyle I},R_{\scriptscriptstyle I}}\left[F_{I_{H_{\scriptscriptstyle I}}}\left(0\vee\left( R_{\scriptscriptstyle I}^{-\alpha}/{\tau}-\sigma^2\right)\right) \right],\] \[p^c_{\scriptscriptstyle\mathcal{I}} = \mathbb{E}^0_{H_{\scriptscriptstyle\mathcal{I}},R_{\scriptscriptstyle\mathcal{I}}}\left[F_{I_{H_{\scriptscriptstyle\mathcal{I}}}}\left(0\vee\left( R_{\scriptscriptstyle\mathcal{I}}^{-\alpha}/{\tau}-\sigma^2\right)\right) \right],\] from (65 ) and (68 ), respectively. Define \[G_1(h,r):= F_{I_{H_{\scriptscriptstyle I}=h}}\left(0\vee\left(r^{-\alpha}/{\tau}-\sigma^2\right)\right),\] \[G_2(h,r)= F_{I_{H_{\scriptscriptstyle\mathcal{I}}=h}}\left(0\vee\left(r^{-\alpha}/{\tau}-\sigma^2\right)\right).\] Observe that by definition of the interference \(I_{H_{\scriptscriptstyle I}=h}\) and \(I_{H_{\scriptscriptstyle\mathcal{I}}=h}\) at a distance \(h\) and the expressions in (65 ) and (68 ) we have \(G_2(h,r)\leq G_1(h,r)\) for all \(h\geq 0\). Also \[\begin{align} G_2(h) &:= \mathbb{E}_{R_{\scriptscriptstyle\mathcal{I}}\vert h}[G_2(h,R_{\scriptscriptstyle\mathcal{I}})]\nonumber\\ &\leq \mathbb{E}_{R_{\scriptscriptstyle I}\vert h}[G_1(h,R_{\scriptscriptstyle I})]=: G_1(h). \label{eq:G1G2-I} \end{align}\tag{72}\] Then by comparing the two densities \[f_{H_{\scriptscriptstyle I}}(h)=4\pi\lambda^{3/2} h^2 e^{-\lambda\pi h^2}, f_{H_{\scriptscriptstyle\mathcal{I}}}(h)=\frac{8}{3}\pi^2\lambda^{5/2} h^4 e^{-\lambda\pi h^2},\] we obtain that \[\begin{align} p_{\scriptscriptstyle I}^c - p_{\scriptscriptstyle\mathcal{I}}^c &= \int_0^\infty \mathbb{E}_{R_{\scriptscriptstyle I}\vert h}[G_1(h,R_{\scriptscriptstyle I})]f_{H_{\scriptscriptstyle I}}(h) {\rm d}h \nonumber\\ & -\int_0^\infty \mathbb{E}_{R_{\scriptscriptstyle\mathcal{I}}\vert h}[G_2(h,R_{\scriptscriptstyle\mathcal{I}})] f_{H_{\scriptscriptstyle\mathcal{I}}}(h) {\rm d}h. \label{eq:G1G21-I} \end{align}\tag{73}\] Using the definition (72 ), the last expression in (73 ) is equal to \[\begin{align} \nonumber\\ &\geq \int_0^\infty G_2(h) f_{H_{\scriptscriptstyle I}}(h) {\rm d}h -\int_0^\infty G_2(h) f_{H_{\scriptscriptstyle\mathcal{I}}}(h) {\rm d}h\nonumber\\ &= \int_0^{\sqrt{3/{2\lambda\pi}}} G_2(h) \left[f_{H_{\scriptscriptstyle I}}(h)- f_{H_{\scriptscriptstyle\mathcal{I}}}(h)\right] {\rm d}h \nonumber\\ & -\int_{\sqrt{3/{2\lambda\pi}}}^\infty G_2(h) \left[f_{H_{\scriptscriptstyle\mathcal{I}}}(h)- f_{H_{\scriptscriptstyle I}}(h)\right] {\rm d}h\nonumber\\ &\geq G_2(\sqrt{3/{2\lambda\pi}}) \!\!\int_0^\infty \!\!\!\! (f_{H_{\scriptscriptstyle I}}(h)- f_{H_{\scriptscriptstyle\mathcal{I}}}(h)) {\rm d}h =0,\nonumber \end{align}\] where by using the definition of \(G_2\), it can be shown that \(G_2(\sqrt{3/{2\lambda\pi}})\geq 0\). Indeed, using the fact that for \(h=\sqrt{3/{2\lambda\pi}}\), \(R_{\scriptscriptstyle\mathcal{I}}^2\stackrel{d}{\sim} U\left[0, \sqrt{3/{2\lambda\pi}}\right]\), from (68 ) we have \[\begin{align} \nonumber\\ & = \sqrt{2\lambda\pi/3} \int_0^{\sqrt{3/{2\lambda\pi}}}F_{I_{H_{\scriptscriptstyle\mathcal{I}}=\sqrt{3/{2\lambda\pi}}}}\left(0\vee\left(\frac{r^{-\alpha/2}}{\tau}-\sigma^2\right)\right){\rm d}r\nonumber\\ & = \sqrt{2\lambda\pi/3} \int_0^{(\tau \sigma^2)^{-1/\alpha} \wedge \sqrt{3/{2\lambda\pi}}}F_{I_{H_{\scriptscriptstyle\mathcal{I}}=\sqrt{3/{2\lambda\pi}}}}\left(\frac{r^{-\alpha/2}}{\tau}-\sigma^2\right){\rm d}r,\nonumber \end{align}\] which is non-negative, and hence we have the result. \(\square\)

Part [CP-F3]. The result \(p^c_{{\scriptscriptstyle\mathcal{I}; \mathcal{T}}} \leq p^c_{{\scriptscriptstyle I;\mathcal{T}}}\) in the tropical case can be proved similarly. For this part, we use a comparison argument similar to second part of Theorem 6 for \[p^c_{\scriptscriptstyle I} = \mathbb{E}^0_{H_{\scriptscriptstyle I},R_{\scriptscriptstyle I}}\left[F_{I_{H_{\scriptscriptstyle I}}}\left(0\vee\left( R_{\scriptscriptstyle I}^{-\alpha}/{\tau}-\sigma^2\right)\right) \right],\] \[p^c_{\scriptscriptstyle\mathcal{I}} = \mathbb{E}^0_{H_{\scriptscriptstyle\mathcal{I}},R_{\scriptscriptstyle\mathcal{I}}}\left[F_{I_{H_{\scriptscriptstyle\mathcal{I}}}}\left(0\vee\left( R_{\scriptscriptstyle\mathcal{I}}^{-\alpha}/{\tau}-\sigma^2\right)\right) \right].\] Define \[G_1(h,R_{\scriptscriptstyle\mathcal{I}}):= F_{I_{H_{\scriptscriptstyle I}=h}}\left(0\vee\left(R_{\scriptscriptstyle I}^{-\alpha}/{\tau}-\sigma^2\right)\right),\] \[G_2(h,R_{\scriptscriptstyle\mathcal{I}})= F_{I_{H_{\scriptscriptstyle\mathcal{I}}=h}}\left(0\vee\left(R_{\scriptscriptstyle\mathcal{I}}^{-\alpha}/{\tau}-\sigma^2\right)\right).\] Observe that by definition of \(I_{H_{\scriptscriptstyle I}=h}\) and \(I_{H_{\scriptscriptstyle\mathcal{I}}=h}\) and the expressions in (67 ) and (69 ) we have \(G_2(h,R_{\scriptscriptstyle I})\leq G_1(h,R_{\scriptscriptstyle\mathcal{I}})\) for all \(h\). Moreover \[\begin{align} G_2(h) &:= \mathbb{E}_{R_{\scriptscriptstyle\mathcal{I}}\vert h}[G_2(h,R_{\scriptscriptstyle\mathcal{I}})]\nonumber\\ &= \tau^{-2/\alpha}\left[\left(1+\sigma^2 h^{\alpha}\right)^{-2/\alpha}\right]\nonumber\\ & = \mathbb{E}_{R_{\scriptscriptstyle I}\vert h}[G_1(h,R_{\scriptscriptstyle I})]=: G_1(h). \label{eq:G1G2} \end{align}\tag{74}\] Then by comparing the two densities \[f_{H_{\scriptscriptstyle I}}(h)=4\pi\lambda^{3/2} h^2 e^{-\lambda\pi h^2}, f_{H_{\scriptscriptstyle\mathcal{I}}}(h)=\frac{8}{3}\pi^2\lambda^{5/2} h^4 e^{-\lambda\pi h^2},\] we have \[\begin{align} p_{\scriptscriptstyle I}^c - p_{\scriptscriptstyle\mathcal{I}}^c &= \int_0^\infty \mathbb{E}_{R_{\scriptscriptstyle I}\vert h}[G_1(h,R_{\scriptscriptstyle I})]f_{H_{\scriptscriptstyle I}}(h) {\rm d}h \nonumber\\ & -\int_0^\infty \mathbb{E}_{R_{\scriptscriptstyle\mathcal{I}}\vert h}[G_2(h,R_{\scriptscriptstyle\mathcal{I}})] f_{H_{\scriptscriptstyle\mathcal{I}}}(h) {\rm d}h. \label{eq:G1G21} \end{align}\tag{75}\] Using the definition (74 ), the last term in (75 ) is equal to \[\begin{align} \nonumber\\ &= \int_0^\infty G_2(h) f_{H_{\scriptscriptstyle I}}(h) {\rm d}h -\int_0^\infty G_2(h) f_{H_{\scriptscriptstyle\mathcal{I}}}(h) {\rm d}h\nonumber\\ &= \int_0^{\sqrt{3/{2\lambda\pi}}} G_2(h) \left[f_{H_{\scriptscriptstyle I}}(h)- f_{H_{\scriptscriptstyle\mathcal{I}}}(h)\right] {\rm d}h \nonumber\\ & -\int_{\sqrt{3/{2\lambda\pi}}}^\infty G_2(h) \left[f_{H_{\scriptscriptstyle\mathcal{I}}}(h)- f_{H_{\scriptscriptstyle I}}(h)\right] {\rm d}h\nonumber\\ &\geq G_2(\sqrt{3/{2\lambda\pi}}) \!\!\int_0^\infty \!\!\!\! (f_{H_{\scriptscriptstyle I}}(h)- f_{H_{\scriptscriptstyle\mathcal{I}}}(h)) {\rm d}h =0,\nonumber \end{align}\] where, by definition of \(G_2\), it can be shown that \(G_2(\sqrt{3/{2\lambda\pi}})\geq 0\). Indeed for \(h=\sqrt{3/{2\lambda\pi}}\), from (69 ) we have \[\begin{align} G_2(\sqrt{3/{2\lambda\pi}}) &= \tau^{-2/\alpha}\left[\left(1+\sigma^2 h^{\alpha}\right)^{-2/\alpha}\right]\big\vert_{h=\sqrt{3/{2\lambda\pi}}}\nonumber\\ & = \tau^{-2/\alpha}\left[\left(1+\sigma^2\left({3}/{2\lambda\pi}\right)^{\alpha/2}\right)^{-2/\alpha}\right],\nonumber \end{align}\] which is non-negative for any \(\lambda, \tau, \sigma>0\) and \(\alpha\). Hence we have the result.

In contrast, the ordering among \(p_{\scriptscriptstyle\mathcal{V};\mathcal{T}}^c, p_{\scriptscriptstyle t;\mathcal{T}}^c\) and \(p_{\scriptscriptstyle S;\mathcal{T}}^c\) depends on the parameter values \(\tau, \lambda\) and \(\alpha\), which can be justified as follows and also validated in Figure [fig:T-HotMSNF]. From (61 ) and (63 ), \(p^c_{\scriptscriptstyle t;\mathcal{T}}- p^c_{\scriptscriptstyle S;\mathcal{T}}\) is positive or negative depending on the sign of the partial integral of the difference \(f_{R_1\vert h}(r) f_{R}(h){-} f_{R_{\scriptscriptstyle S}\vert h}(r) f_{H_{\scriptscriptstyle S}}(h)\) of joint probability densities, which depends on the parameters \(\lambda, \sigma\) and \(\alpha\).

Similarly, using (58 ) and (61 ), the difference \[\begin{align} p^c_{\scriptscriptstyle\mathcal{V};\mathcal{T}}- p^c_{\scriptscriptstyle t;\mathcal{T}} &= \int_0^\infty {\mathbb{1}}{\left\{h<\left({(1-\tau)}/{\tau\sigma^2}\right)^\frac{1}{\alpha}\right\}} f_{H_{\scriptscriptstyle\mathcal{V}}}(h){\rm d}h\nonumber\\ & \;- \int_{0}^{(\tau \sigma^2)^{-\frac{1}{\alpha}}} \!\!\!\! \! \int_{\left(\frac{h^{-\alpha}}{\tau}{-}\sigma^2\right)^{-\frac{1}{\alpha}}\vee h}^\infty f_{R_{\scriptscriptstyle S}\vert h}(r){\rm d}r f_{H_{\scriptscriptstyle S}}(h){\rm d}h,\nonumber \end{align}\] is positive or negative depending on the parameters. \(\square\)

12 Comparison of interference↩︎

 

12.1 Theorem 7 (with fading)↩︎

For \(I_{\scriptscriptstyle\mathcal{V}}\geq_{\textrm{L}} I_{t} \geq_{\textrm{L}} I_{\scriptscriptstyle S}\) we compare the Laplace transforms \[\begin{align} \mathbb{E}^0\left[e^{-\gamma I_{\scriptscriptstyle\mathcal{V}}}\right] &= \mathbb{E}^0_{H_{\scriptscriptstyle\mathcal{V}}}\!\!\left[ \frac{\mu}{\mu{+}\gamma H_{\scriptscriptstyle\mathcal{V}}^{-\alpha}}e^{-2\pi\lambda\int_{H_{\scriptscriptstyle\mathcal{V}}}^\infty\frac{\gamma r^{-\alpha}}{\mu+\gamma r^{-\alpha}}r{\rm d}r}\right], \label{eq:int-1} \end{align}\tag{76}\] \[\begin{align} \mathbb{E}^0\left[e^{-\gamma I_t}\right] &= \mathbb{E}^0_{R}\left[e^{-2\pi\lambda\int_{R}^\infty\frac{\gamma r^{-\alpha}}{\mu+\gamma r^{-\alpha}}r{\rm d}r}\right],\nonumber \end{align}\] \[\begin{align} \mathbb{E}^0\left[e^{-\gamma I_{\scriptscriptstyle S}}\right] &= \mathbb{E}^0_{H_{\scriptscriptstyle S}}\left[ e^{-2\pi\lambda\int_{H_{\scriptscriptstyle S}}^\infty\frac{\gamma r^{-\alpha}}{\mu+\gamma r^{-\alpha}}r{\rm d}r}\right],\nonumber \end{align}\] similarly to the proof of Theorem 6 in Subsection [subsection:CF], in the case of an environment with fading, by comparing the densities of \(H_{\scriptscriptstyle\mathcal{V}}, H_{\scriptscriptstyle S}\) and \(R\), using the function \(G\) analogous to (70 ) as \[\begin{align} G(h)&:= e^{-\lambda\pi h^2 \kappa(\gamma h^{-\alpha}/\mu, \alpha)}, \end{align}\] using the function \(\kappa\) from (85 ), along with the fact that \(\frac{\gamma}{\mu+\gamma h^{-\alpha}}\leq 1\), for all \(h\geq 0\).

To prove \(I_{\scriptscriptstyle\mathcal{I}}\geq_{\textrm{L}} I_{\scriptscriptstyle I}\), we compare \[\begin{align} \mathbb{E}^0\left[e^{-\gamma I_{\scriptscriptstyle I}}\right] &= \mathbb{E}^0_{H_{\scriptscriptstyle I}} \left[\frac{\mu}{\mu+\gamma H_{\scriptscriptstyle I}^{-\alpha}} e^{-2\pi\lambda\int_{H_{\scriptscriptstyle I}}^\infty\frac{\gamma u^{-\alpha}}{\mu+\gamma u^{-\alpha}}u{\rm d}u}\right],\nonumber \end{align}\] \[\begin{align} \mathbb{E}^0\left[e^{-\gamma I_{\scriptscriptstyle\mathcal{I}}}\right] &= \mathbb{E}^0_{H_{\scriptscriptstyle\mathcal{I}}} \left[\left(\frac{\mu}{\mu+\gamma H_{\scriptscriptstyle\mathcal{I}}^{-\alpha}}\right)^2 e^{-2\pi\lambda\int_{H_{\scriptscriptstyle\mathcal{I}}}^\infty\frac{\gamma u^{-\alpha}}{\mu+\gamma u^{-\alpha}}u{\rm d}u}\right], \nonumber \end{align}\] from (47 ) and (53 ), by comparing the densities of \(H_{\scriptscriptstyle I}\) and \(H_{\scriptscriptstyle\mathcal{I}}\), and using the \(G\) function as \[\begin{align} G(h) &= \frac{\mu}{\mu+\gamma h^{-\alpha}} e^{-2\pi\lambda\int_h^\infty\frac{\gamma u^{-\alpha}}{\mu+\gamma u^{-\alpha}}u{\rm d}u}, \label{eq:gamma-intG} \end{align}\tag{77}\] along with the fact that \(\frac{\gamma}{\mu+\gamma h^{-\alpha}}\leq 1\), for all \(h\geq 0\).

For proving \(\mathcal{T}_{\scriptscriptstyle{\mathcal{V}}}\geq_{\textrm{st}} \mathcal{T}_{t} \geq_{\textrm{st}} \mathcal{T}_{\scriptscriptstyle S}\), we compare \[\begin{align} \mathbb{P}^0(\mathcal{T}_{\scriptscriptstyle\mathcal{V}} \leq x)&=\mathbb{E}^0_{H_{\scriptscriptstyle\mathcal{V}}}\left[\left(1{-}e^{-\mu H_{\scriptscriptstyle\mathcal{V}}^{\alpha}x}\right) e^{-2\lambda\pi K_\alpha(\mu,x, H_{\scriptscriptstyle\mathcal{V}})}\right],\nonumber \end{align}\] \[\begin{align} \mathbb{P}^0(\mathcal{T}_t \leq x)&=\mathbb{E}^0_{R}\left[e^{-2\lambda\pi K_\alpha(\mu,x, R)}\right],\nonumber \end{align}\] \[\begin{align} \mathbb{P}(\mathcal{T}_{\scriptscriptstyle S} \leq x)&=\mathbb{E}_{H_{\scriptscriptstyle S}}\left[ e^{-2\lambda\pi K_\alpha(\mu,x, H_{\scriptscriptstyle S})}\right],\nonumber \end{align}\] by again comparing the densities of \(H_{\scriptscriptstyle\mathcal{V}}, H_{\scriptscriptstyle S}\) and \(R\). In the proof we use the function \(G\) similarly to (70 ) as \(G(h):=e^{-2\lambda\pi K_\alpha(\mu,x, h)}\) and the fact that \((1-e^{-\mu h^\alpha x})\leq 1\) for any \(h\geq 0\).

For \(\mathcal{T}_{\scriptscriptstyle\mathcal{I}} \geq_{\textrm{st}} \mathcal{T}_{\scriptscriptstyle I}\), we compare \[\begin{align} \mathbb{P}^0(\mathcal{T}_{\scriptscriptstyle I} \leq x)&=\mathbb{E}^0_{H_{\scriptscriptstyle I}}\left[\left(1{-}e^{-\mu H_{\scriptscriptstyle I}^{\alpha}x}\right)\!\;e^{-2\lambda\pi K_\alpha(\mu,x,H_{\scriptscriptstyle I})}\right],\nonumber \end{align}\] \[\begin{align} \mathbb{P}^0(\mathcal{T}_{\scriptscriptstyle\mathcal{I}} \leq x)&{=}\mathbb{E}^0_{H_{\scriptscriptstyle\mathcal{I}}}\left[\left(1{-}e^{-\mu H_{\scriptscriptstyle\mathcal{I}}^{\alpha}x}\right)^2 \!\!\! e^{-2\lambda\pi K_\alpha(\mu,x,H_{\scriptscriptstyle\mathcal{I}})}\right],\nonumber \end{align}\] by comparing the densities of \(H_{\scriptscriptstyle I}\) and \(H_{\scriptscriptstyle\mathcal{I}}\), and using the \(G\) function as \(G(h):= \left(1{-}e^{-\mu h^{\alpha}x}\right)e^{-2\lambda\pi K_\alpha(\mu,x, h)}\).\(\square\)

12.2 Theorem 7 (without fading)↩︎

The comparison of the interference at different typical epochs in the no-fading scenario can be proved using same type of comparisons as in the case with fading and using the proof of Theorem 6 in no-fading case. For the first part, \(I_{\scriptscriptstyle\mathcal{V}}\geq_{\textrm{L}} I_{t} \geq_{\textrm{L}} I_{\scriptscriptstyle S}\), we compare \[\begin{align} \mathbb{E}^0\left[e^{-\nu I_{\scriptscriptstyle\mathcal{V}}}\right] &= \mathbb{E}^0_{H_{\scriptscriptstyle\mathcal{V}}}\left[e^{-\nu H_{\scriptscriptstyle\mathcal{V}}^{-\alpha} -\pi\lambda L_\nu(H_{\scriptscriptstyle\mathcal{V}}, \alpha)}\right],\nonumber \end{align}\] \[\begin{align} \mathbb{E}^0\left[e^{-\nu I_{t}}\right] = \mathbb{E}^0_{R}\left[e^{ -\pi\lambda L_\nu(R, \alpha)}\right],\nonumber \end{align}\] \[\begin{align} \mathbb{E}^0\left[e^{-\nu I_{\scriptscriptstyle S}}\right] &= \mathbb{E}^0_{H_{\scriptscriptstyle S}}\left[e^{-\pi\lambda L_\nu(H_{\scriptscriptstyle S}, \alpha)}\right],\nonumber \end{align}\] by a similar comparison of the densities of \(H_{\scriptscriptstyle\mathcal{V}}, R\) and \(H_{\scriptscriptstyle S}\), using the function \(G(h)= e^{-\pi\lambda L_{\nu}(h,\alpha)}\), where \(L_{\nu}(r,\alpha)\) is as defined in (88 ). Also for proving \(I_{\scriptscriptstyle\mathcal{I}}\geq_{\textrm{L}} I_{\scriptscriptstyle I}\), we compare \[\begin{align} \mathbb{E}^0\left[e^{-\nu I_{\scriptscriptstyle I}}\right] &= \mathbb{E}^0_{H_{\scriptscriptstyle I}}\left[e^{-\nu H_{\scriptscriptstyle I}^{-\alpha} -\pi\lambda L_\nu(H_{\scriptscriptstyle I}, \alpha)}\right],\nonumber \end{align}\] \[\begin{align} \mathbb{E}^0\left[e^{-\nu I_{\scriptscriptstyle\mathcal{I}}}\right] &= \mathbb{E}^0_{H_{\scriptscriptstyle\mathcal{I}}}\left[e^{-2\nu H_{\scriptscriptstyle\mathcal{I}}^{-\alpha} -\pi\lambda L_\nu(H_{\scriptscriptstyle\mathcal{I}}, \alpha)}\right], \nonumber \end{align}\] using the densities of \(H_{\scriptscriptstyle I}\) and \(H_{\scriptscriptstyle\mathcal{I}}\) and the function \(G(h)= e^{-\nu h^{-\alpha}-\pi\lambda L_{\nu}(h,\alpha)}\). \(\square\)

13 Proofs: Scale invariance↩︎

 

13.1 Theorem 8 (with fading)↩︎

Recall the coverage probabilities for the typical handover case in (32 ) with \(\sigma=0\) as \[\begin{align} p_{\scriptscriptstyle\mathcal{V}}^c &= \frac{1}{1{+}\tau}\mathbb{E}^0_{H_{\scriptscriptstyle\mathcal{V}}}\left[ e^{-\pi\lambda H_{\scriptscriptstyle\mathcal{V}}^2 \kappa(\tau,\alpha)}\right] = \frac{1}{1{+}\tau} (1+\kappa(\tau,\alpha))^{-3/2},\nonumber \end{align}\] using the Laplace transform of \(H_{\scriptscriptstyle I}^2\) from Proposition 3. The coverage probability \(p_{\scriptscriptstyle\mathcal{V}}^c\) in this case depends only on \(\tau, \alpha\). By a similar computation we can show that \[p_{t}^c = (1+\kappa(\tau,\alpha))^{-1}and p_{\scriptscriptstyle S}^c = (1+\kappa(\tau,\alpha))^{-{\frac{1}{2}}},\] both of which depend only on \(\tau, \alpha\). This shows the scale invariance property of the coverage probabilities with respect to the parameter \(\lambda,\mu\).

For the case of typical max interference epoch, the coverage probability with \(\sigma=0\) is \[\begin{align} p^c_{\scriptscriptstyle I} &{=} \frac{2\tau^{-2/\alpha}}{\alpha}\int_0^\tau \!\frac{z^{2/\alpha-1}}{1{+}z}\mathbb{E}^0_{H_{\scriptscriptstyle I}}\!\! \left[e^{-\pi\lambda H_{\scriptscriptstyle I}^2 \kappa(z, \alpha)}\right]{\rm d}z. \label{eq:si2} \end{align}\tag{78}\] from (50 ). Using the density of \(H_{\scriptscriptstyle I}\), after exchanging the integrals in (78 ), we have \[\begin{align} p^c_{\scriptscriptstyle I} &{=} \frac{2\tau^{-2/\alpha}}{\alpha}\int_0^\tau \!\frac{z^{2/\alpha-1}}{1{+}z} (1+\kappa(z, \alpha))^{-3/2} {\rm d}z, \label{eq:si4} \end{align}\tag{79}\] using the Laplace transform of \(H_{\scriptscriptstyle I}^2\) from Proposition 3. The last expression for \(p^c_{\scriptscriptstyle I}\) in (79 ) is independent of \(\lambda, \mu\). Similarly we can show from (54 ) that \[\begin{align} p_{\scriptscriptstyle\mathcal{I}}^c =\frac{2\tau^{-2/\alpha}}{\alpha}\int_0^\tau \!\frac{z^{2/\alpha-1}}{(1{+}z)^2} \left(1{+}\kappa(z, \alpha)\right)^{-5/2}{\rm d}z, \label{eq:si3} \end{align}\tag{80}\] which is independent of \(\lambda,\mu\) and hence we have the result. The same scale invariance can be shown to hold for the tropical case as well. \(\square\)

13.2 Theorem 8 (without fading)↩︎

At the typical handover we have \[\begin{align} p^c_{\scriptscriptstyle\mathcal{V}} &=\mathbb{P}^0\left(H_{\scriptscriptstyle\mathcal{V}}^{-\alpha}/{I_{H_{\scriptscriptstyle\mathcal{V}}}}\geq \tau \right).\nonumber \end{align}\] Since the ratio \(\frac{H_{\scriptscriptstyle\mathcal{V}}^{-\alpha}}{I_{H_{\scriptscriptstyle\mathcal{V}}}}\) is independent of \(\lambda\), so is the coverage probability. The same reasoning applies for the scale invariance of the coverage probabilities in the four other cases:

• typical time: \(p^c_{t} =\mathbb{P}^0\left(R^{-\alpha}/{I_R}\geq \tau \right)\),

• typical MS: \(p^c_{\scriptscriptstyle S} =\mathbb{P}^0\left(H_{\scriptscriptstyle S}^{-\alpha}/{I_{H_{\scriptscriptstyle S}}}\geq \tau \right)\),

• typical MI: \(p^c_{\scriptscriptstyle I} =\mathbb{P}^0\left(H_{\scriptscriptstyle I}^{-\alpha}/{I_{H_{\scriptscriptstyle I}}}\geq \tau \right)\),

• typical mI: \(p^c_{\scriptscriptstyle\mathcal{I}} =\mathbb{P}^0\left(H_{\scriptscriptstyle\mathcal{I}}^{-\alpha}/{I_{H_{\scriptscriptstyle\mathcal{I}}}}\geq \tau \right)\),

from (57 ), (59 ), (62 ) and (65 ), respectively. For typical mS-MI, from (58 ), we have \[\begin{align} p^c_{\scriptscriptstyle\mathcal{V};\mathcal{T}} &{=} \int_0^\infty {\mathbb{1}}_{h^{-\alpha}>\tau h^{-\alpha}} f_{H_{\scriptscriptstyle\mathcal{V}}}(h){\rm d}h = \begin{cases} 1if\tau<1\\ 0if\tau= 1, \end{cases}\nonumber \end{align}\] since \(\tau\leq 1\). At a typical time from (60 ), for any \(\tau\geq 0\) we have \[\begin{align} p^c_{\scriptscriptstyle{t;\mathcal{T}}} &= \int_{0}^\infty \mathbb{P}\left(R_1>\tau^{1/\alpha} r\right) f_{R}(r){\rm d}r \nonumber\\ &=\int_{0}^\infty \int_{\tau^{1/\alpha}r\vee r}^\infty 2\lambda\pi h \;e^{-\lambda\pi (h^2-r^2)} {\rm d}h f_{R}(r){\rm d}r.\nonumber \end{align}\] For \(\tau\leq 1\), we have \(\tau^{1/\alpha}\leq 1\) and hence \(p^c_{t;\scriptscriptstyle{\mathcal{T}}} =1\) in this case. For \(\tau>1\), we have \(\tau^{1/\alpha} > 1\) and \[\begin{align} p^c_{\scriptscriptstyle{t;\mathcal{T}}} &=\int_{0}^\infty \int_{\tau^{1/\alpha}r}^\infty 2\lambda\pi h \;e^{-\lambda\pi (h^2-r^2)} {\rm d}h f_{R}(r){\rm d}r\nonumber\\ &=\int_{0}^\infty \int_{\lambda\pi\tau^{2/\alpha}r^2}^\infty e^{-z} e^{\lambda\pi r^2} {\rm d}z f_{R}(r){\rm d}r\nonumber\\ &=\int_{0}^\infty e^{-\lambda\pi r^2(\tau^{2/\alpha}-1)} f_R(r) {\rm d}r\nonumber\\ &=\int_{0}^\infty e^{-\lambda\pi r^2\tau^{2/\alpha}} 2\lambda\pi r {\rm d}r = \tau^{-2/\alpha}.\nonumber \end{align}\] We can similarly prove from (60 ) that \(p^c_{\scriptscriptstyle S;\mathcal{T}}\) is also independent of \(\lambda\), as follows: \[\begin{align} p^c_{\scriptscriptstyle S;\mathcal{T}} &= \int_{0}^\infty \mathbb{P}\left(R_{\scriptscriptstyle S}>\tau^{1/\alpha} h\right) f_{H_{\scriptscriptstyle S}}(h){\rm d}h \nonumber\\ &=\int_{0}^\infty \int_{\tau^{1/\alpha} h \vee h}^\infty f_{R_{\scriptscriptstyle S}\vert H_{\scriptscriptstyle S}=h}(r){\rm d}r f_{H_{\scriptscriptstyle S}}(h){\rm d}h.\nonumber \end{align}\] For \(\tau\leq 1\), we have \(p^c_{\scriptscriptstyle S;\mathcal{T}}=1\). For \(\tau>1\) we have \[\begin{align} p^c_{\scriptscriptstyle S;\mathcal{T}} &= \int_{0}^\infty \int_{\tau^{1/\alpha} h}^\infty f_{R_{\scriptscriptstyle S}\vert H_{\scriptscriptstyle S}=h}(r){\rm d}r f_{H_{\scriptscriptstyle S}}(h){\rm d}h\nonumber\\ &= \int_{0}^\infty \int_{\tau^{1/\alpha} h}^\infty \frac{\lambda^{{\frac{1}{2}}} r}{(r^2-h^2)^{{\frac{1}{2}}}} e^{-\lambda\pi(r^2-h^2)}{\rm d}r f_{H_{\scriptscriptstyle S}}(h){\rm d}h\nonumber\\ &= \int_{0}^\infty \int_{\lambda\pi(\tau^{2/\alpha}-1) h^2}^\infty \frac{1}{2\sqrt{\pi}} z^{-1} e^{-z} {\rm d}z f_{H_{\scriptscriptstyle S}}(h){\rm d}h\nonumber\\ &= \frac{1}{2\sqrt{\pi}}\mathbb{E}^0_{H_{\scriptscriptstyle S}}\left[\Gamma\left(0, \lambda\pi (\tau^{2/\alpha}-1)H^2_{\scriptscriptstyle S}\right)\right].\nonumber \end{align}\] In case of \(\tau\leq 1\), at the typical max-interference, we have from (67 ) that \[\begin{align} p^c_{{\scriptscriptstyle I ;\mathcal{T}}} &=\mathbb{P}^0\left(H_{\scriptscriptstyle I}\geq R_{\scriptscriptstyle I}\tau^{1/\alpha}\right)\nonumber\\ &=\int_0^\infty \mathbb{P}^0(h^{-2}R^2_{\scriptscriptstyle I}\leq \tau^{-2/\alpha}) f_{H_{\scriptscriptstyle I}}(h){\rm d}h\nonumber\\ &=\int_0^\infty \tau^{-2/\alpha} f_{H_{\scriptscriptstyle I}}(h){\rm d}h = \tau^{-2/\alpha},\nonumber \end{align}\] using Corollary 3. The final expression is independent of \(\lambda\). For the typical min interference, from (68 ) we have \[\begin{align} p^c_{\scriptscriptstyle\mathcal{I};\mathcal{T}} &= \mathbb{E}^0_{H_{\scriptscriptstyle\mathcal{I}},R_{\scriptscriptstyle\mathcal{I}}}\left[F_{I_{H_{\scriptscriptstyle\mathcal{I}}}}\left( R_{\scriptscriptstyle\mathcal{I}}^{-\alpha}/\tau\right) \right]\nonumber\\ &=\mathbb{P}^0\left(H_{\scriptscriptstyle\mathcal{I}}^{-\alpha}\leq R_{\scriptscriptstyle\mathcal{I}}^{-\alpha}/\tau\right)\nonumber\\ &=\int_0^\infty \mathbb{P}^0\left(h^{-2}R^2_{\scriptscriptstyle\mathcal{I}}\leq \tau^{-2/\alpha}\right) f_{H_{\scriptscriptstyle\mathcal{I}}}(h){\rm d}h\nonumber\\ &=\int_0^\infty \tau^{-2/\alpha} f_{H_{\scriptscriptstyle\mathcal{I}}}(h){\rm d}h = \tau^{-2/\alpha},\nonumber \end{align}\] using Corollary 5, where the last term is independent of \(\lambda\). \(\square\)

14 Other attenuation functions↩︎

 

14.1 Bounded path-loss function↩︎

In the case of bounded attenuation function \(\ell(r)=(1+r)^{-\alpha}\) as in [baf], given \(H=h\) we have the coverage probability as \[\begin{align} \mathbb{P}(\mathcal{S}>\tau\vert H=h) &= \mathbb{E}_{I_{h}}\left[\mathbb{P}(\mathcal{S}>\tau\vert I_h)\right]\nonumber\\ &=\mathbb{E}_{I_{h}}\left[\mathbb{P}\left(\rho >\tau (1+h)^{\alpha}\left(\sigma^2+I_{h}\right)\right)\right]\nonumber\\ &=\mathbb{E}_{I_{h}}\left[e^{-\mu\tau (1+h)^{\alpha}\left(\sigma^2+I_{ h}\right)}\right]\nonumber\\ &= e^{-\mu\tau (1+h)^{\alpha}\sigma^2}\mathcal{L}_{I_{h}}(\mu\tau (1+h)^{\alpha}).\nonumber \end{align}\] For example, in the case of signal handover \(I_h:=\rho (1+h)^{-\alpha}+\sum_{i} \rho_i (1+D_i)^{-\alpha}\). Computing the PGFl with respect to the point process \(\eta_h\) we have the Laplace transform of \(I_h\) for any \(\gamma\geq 0\) as \[\begin{align} \mathbb{E}_{I_{h}}\left[e^{-\gamma I_{h}}\right] &= \frac{\mu}{\mu+\gamma(1+h)^{-\alpha}} e^{-2\pi\lambda\int_{h}^\infty\frac{\gamma(1+r)^{-\alpha}}{\mu+\gamma(1+r)^{-\alpha}}r{\rm d}r}. \label{eq:BA1} \end{align}\tag{81}\] Hence for \(\gamma=\mu\tau (1+h)^{\alpha}\), we re-write the integral in the exponent in (81 ) as \[\int_{h}^\infty\frac{\tau}{\tau+ (1+r)^\alpha(1+h)^{-\alpha}} r{\rm d}r := M_\alpha(h,\tau)\] and we have the required Laplace transform as \[\begin{align} \mathbb{E}_{I_{h}}\left[e^{-\mu\tau (1+h)^{\alpha} I_{h}}\right] &= \frac{1}{1+\tau} e^{-2\lambda\pi M_\alpha(h,\tau)}.\nonumber \end{align}\] The coverage probability is \[p^c_{\scriptscriptstyle\mathcal{V}}= \frac{1}{1+\tau} \mathbb{E}^0_{H_{\scriptscriptstyle\mathcal{V}}}\left[e^{-\mu\tau (1+H_{\scriptscriptstyle\mathcal{V}})^{\alpha}\sigma^2} e^{-2\lambda\pi M_\alpha(H,\tau)}\right].\] Similarly for the typical time and typical MS, we have \[p^c_{\scriptscriptstyle t}= \mathbb{E}^0_{R}\left[e^{-\mu\tau (1+R)^{\alpha}\sigma^2} e^{-2\lambda\pi M_\alpha(R,\tau)}\right],\] \[p^c_{\scriptscriptstyle S}= \mathbb{E}^0_{H_{\scriptscriptstyle S}}\left[e^{-\mu\tau (1+H_{\scriptscriptstyle S})^{\alpha}\sigma^2} e^{-2\lambda\pi M_\alpha(H_{\scriptscriptstyle S},\tau)}\right].\] A comparison result similar to Theorem 6 can be shown to hold true, which is \(p^c_{\scriptscriptstyle\mathcal{V}}\leq p^c_{\scriptscriptstyle t}\leq p^c_{\scriptscriptstyle S}\). Similarly, for the other two typical epochs \[\begin{align} p_{\scriptscriptstyle I}^c &{=}E^0_{H_{\scriptscriptstyle I}, R_{\scriptscriptstyle I}}\!\! \left[Q_\alpha(H_{\scriptscriptstyle I}, R_{\scriptscriptstyle I},\tau) e^{-\mu\tau (1+R_{\scriptscriptstyle I})^{\alpha}\sigma^2}\right.\nonumber\\ &\times \left. e^{-2\pi\lambda\int_{H_{\scriptscriptstyle I}}^\infty Q_\alpha(R_{\scriptscriptstyle I},u,\tau)u{\rm d}u}\right],\nonumber \end{align}\] \[\begin{align} p_{\scriptscriptstyle\mathcal{I}}^c &{=}E^0_{H_{\scriptscriptstyle\mathcal{I}}, R_{\scriptscriptstyle\mathcal{I}}}\!\! \left[\left(Q_\alpha(H_{\scriptscriptstyle\mathcal{I}}, R_{\scriptscriptstyle\mathcal{I}},\tau)\right)^2 e^{-\mu\tau (1+R_{\scriptscriptstyle\mathcal{I}})^{\alpha}\sigma^2}\right.\nonumber\\ &\times \left. e^{-2\pi\lambda\int_{H_{\scriptscriptstyle\mathcal{I}}}^\infty Q_\alpha(R_{\scriptscriptstyle\mathcal{I}},u,\tau)u{\rm d}u}\right],\nonumber \end{align}\] where \(Q_\alpha(x,y,\tau):=\frac{(1+x)^\alpha}{(1+x)^\alpha+\tau (1+y)^\alpha}\). It can be proved similarly that \(p_{\scriptscriptstyle\mathcal{I}}^c \leq p_{\scriptscriptstyle I}^c\).

14.2 Step attenuation function↩︎

Consider the step attenuation function \(\ell(r)=p {\mathbb{1}}_{r\leq d}\) as in [step-fun]. The SINR is defined as \[\mathcal{S}\equiv SINR:= \frac{\rho \, p {\mathbb{1}}_{H\leq d}}{\sigma^2+\sum_{i} \rho_i\, p {\mathbb{1}}_{H\leq D_i\leq d}}.\] Then the coverage probability with respect to the Palm probability distribution of epoch of interest is \[\begin{align} p^c(\tau,\mu,\lambda,\alpha)&= \mathbb{E}^0_{H}[\mathbb{P}(\mathcal{S}>\tau\vert H)], \label{eq:st1} \end{align}\tag{82}\] where \(f_H\) is the density of the distance \(H\) to the serving station. Conditioned on \(H=h\) the inner term (82 ) is \[\begin{align} \mathbb{P}(\mathcal{S}>\tau\vert h) &= \mathbb{E}_{I_{h}}\left[\mathbb{P}(\mathcal{S}>\tau\vert h, I_h)\right] \left(1-e^{-\lambda\pi d^2}\right)\nonumber\\ &=\mathbb{E}_{I_{h}}\left[\mathbb{P}\left(\rho >\frac{\tau}{p} \left(\sigma^2+I_{h}\right)\right)\right]\left(1-e^{-\lambda\pi d^2}\right) \nonumber\\ &=\mathbb{E}_{I_{h}}\left[e^{-\frac{\mu\tau}{p}\left(\sigma^2+I_{ h}\right)}\right] \left(1-e^{-\lambda\pi d^2}\right)\nonumber\\ &= e^{-\frac{\mu\tau\sigma^2}{p}}\!\! \left(1{-}e^{-\lambda\pi d^2}\right)\! \mathbb{E}_{I_{h}}\!\left[e^{-\frac{\mu\tau}{p} I_h}\right]. \nonumber \end{align}\] Using the PGFl of the max-shot noise of a Poisson point process, we get \[\begin{align} \mathbb{E}_{I_{h}}\left[e^{-\frac{\mu\tau}{p} I_h}\right]&= \begin{cases} \frac{1}{1+\tau} e^{-2\lambda\pi(d-h)\tau/(1+\tau)}, \text{ for typical ms-MI,}\\ e^{-2\lambda\pi(d-h)\tau/(1+\tau)}, \text{ for typical MS,}\\ e^{-2\lambda\pi(d-h)\tau/(1+\tau)}, \text{ for typical time}. \end{cases}\nonumber \end{align}\] As a result the coverage probabilities are \[\begin{align} p^c(\tau,\mu,\lambda,\alpha) &= \begin{cases} \frac{C}{1+\tau} \mathbb{E}^0_{H_{\scriptscriptstyle\mathcal{V}}}\left[e^{\gamma H_{\scriptscriptstyle\mathcal{V}}}\right], \text{ for typical ms-MI,}\\ C \mathbb{E}^0_{H_{\scriptscriptstyle S}}\left[e^{\gamma H_{\scriptscriptstyle S}}\right], \text{ for typical MS,}\\ C \mathbb{E}^0_{R}\left[e^{\gamma R}\right], \text{ for typical time}, \end{cases}\nonumber \end{align}\] where \(\gamma= 2\lambda\pi\tau/(1+\tau)\) and \[C\equiv C(\tau,d,\mu,\lambda,\sigma):= e^{-\frac{\mu\tau\sigma^2}{p}}\left(1-e^{-\lambda\pi d^2}\right) e^{-2\lambda\pi \tau/(1+\tau)}.\] Even though we have \(H_{\scriptscriptstyle\mathcal{V}}\geq_{\textrm{mgf}} R\geq_{\textrm{mgf}} H_{\scriptscriptstyle S}\) from Lemma 3, this shows that no distinct ordering holds true among \(p_{\scriptscriptstyle\mathcal{V}}^c, p_{t}^c\) and \(p_{\scriptscriptstyle S}^c\), for all values of \(\tau\). Similarly using \[SINR:= \frac{\rho \, p {\mathbb{1}}_{R\leq d}}{\sigma^2+\sum_{i} \rho_i\, p {\mathbb{1}}_{H\leq D_i\leq d}},\] with \(R\in \{R_{\scriptscriptstyle I}, R_{\scriptscriptstyle\mathcal{I}}\}\) and \(H\in \{H_{\scriptscriptstyle I}, H_{\scriptscriptstyle\mathcal{I}}\}\) at typical mI and MI, respectively, one can show different ordering between \(p^c_{\scriptscriptstyle\mathcal{I}}\) and \(p^c_{\scriptscriptstyle I}\) depending for different values of \(\tau\), even though we have \(H_{\scriptscriptstyle\mathcal{I}}\geq_{\textrm{mgf}} H_{\scriptscriptstyle I}\) and \(R_{\scriptscriptstyle\mathcal{I}}\geq_{\textrm{mgf}} R_{\scriptscriptstyle I}\), from Lemma 3.

It is also evident that in the interference limited regime, the coverage probabilities are not scale invariant with respect to \(\lambda\), as the constant \[C(\tau,d,\mu,\lambda,\sigma)\vert_{\sigma=0}= \left(1-e^{-\lambda\pi d^2}\right) e^{-2\lambda\pi \tau/(1+\tau)}\] is not. On the other hand, the \(\mu\)-scale invariance holds true.

15 Special integrals↩︎

15.1 The integral in (36 )↩︎

We determine the integral in (36 ) of the form \(\int_{h}^\infty\frac{\tau h^{\alpha}}{r^{\alpha}+\tau h^{\alpha}}r{\rm d}r\) for some \(h\), \(\tau\) and \(\alpha\) positive. Taking a change of variable \(\tau^{-2/\alpha} h^{-2}r^2=z\) we obtain that \[\begin{align} \int_{h}^\infty\frac{\tau h^{\alpha}}{r^{\alpha}+\tau h^{\alpha}}r{\rm d}r &= {\frac{1}{2}}\tau^{2/\alpha} h^2 \int_{\tau^{-2/\alpha}}^\infty\frac{1}{1+z^{\alpha/2}} {\rm d}z\nonumber\\ &= {\frac{1}{2}}h^2 \kappa(\tau,\alpha),\nonumber \end{align}\] where \[\kappa(\tau,\alpha):=\tau^{2/\alpha} \int_{\tau^{-2/\alpha}}^\infty\frac{1}{1+z^{\alpha/2}} {\rm d}z. \label{eq:kappa}\tag{83}\]

15.2 The integral (76 )↩︎

We compute the integral \(\int_h^\infty\frac{\gamma r^{-\alpha}}{\mu+\gamma r^{-\alpha}}r{\rm d}r\) in the exponent in (76 ) by a change of variable \((\gamma/\mu)^{-2/\alpha} r^2=z\) \[\begin{align} \nonumber\\ &= \frac{1}{\mu} \int_h^\infty\frac{\gamma/\mu}{\gamma/\mu+ r^{\alpha}}r{\rm d}r \nonumber\\ &= \frac{1}{2\mu} (\gamma/\mu)^{2/\alpha}\int_{(\gamma/\mu)^{-2/\alpha}h^2}^\infty (1+ z^{\alpha/2})^{-1}{\rm d}z\tag{84}\\ &= \frac{h^2}{2\mu} (\gamma h^{-\alpha}/\mu)^{2/\alpha}\int_{(\gamma h^{-\alpha}/\mu)^{-2/\alpha}}^\infty (1+ z^{\alpha/2})^{-1}{\rm d}z\nonumber\\ &= \frac{1}{2\mu} h^2 \kappa(\gamma h^{-\alpha}/\mu, \alpha), \tag{85} \end{align}\] from (83 ). Note from (84 ) that \(h^2 \kappa(\gamma h^{-\alpha}/\mu, \alpha)\) is a non-increasing function of \(h\) for fixed \(\alpha, \gamma\) and \(\mu\).

15.3 The integral in (49 )↩︎

For the integral \(\int_h^\infty\frac{\tau r^{\alpha}}{u^{\alpha}+\tau r^{\alpha}}u{\rm d}u\) in (49 ), by the change of variable \(\tau^{-2/\alpha} r^{-2}u^2=z\) we have \[\begin{align} \int_h^\infty\!\!\! \frac{\tau r^{\alpha}}{u^{\alpha}+\tau r^{\alpha}}u{\rm d}u &{=} \frac{\tau^{2/\alpha} r^2}{2}\!\! \int_{\tau^{-2/\alpha} (r/h)^{-2}}^\infty\!\frac{1}{z^{\alpha/2}+1}{\rm d}z\nonumber\\ &{=} \frac{h^2}{2} (\tau (r/h)^\alpha)^{2/\alpha} \!\! \int_{\tau (r/h)^\alpha)^{-2/\alpha}}^\infty\!\frac{1}{z^{\alpha/2}+1}{\rm d}z\nonumber\\ &\stackrel{(\ref{eq:kappa})}{=} \frac{h^2}{2} \kappa(\tau (r/h)^\alpha, \alpha). \label{eq:kappar-h} \end{align}\tag{86}\]

15.4 The integral in (37 )↩︎

For the integral \(\int_{h}^{\infty}e^{-\mu r^\alpha x}r{\rm d}r\) in (37 ) we have \[\begin{align} \int_{h}^{\infty}e^{-\mu r^\alpha x}r{\rm d}r &= \frac{1}{\alpha} \int_{\mu h^\alpha x}^{\infty}e^{-z} \left(\frac{z}{\mu x}\right)^{2/\alpha} \frac{1}{z}{\rm d}z\nonumber\\ &= \frac{1}{\alpha} (\mu x)^{-2/\alpha} \int_{\mu h^\alpha x}^{\infty} z^{2/\alpha-1}e^{-z}{\rm d}z\nonumber\\ &= \frac{1}{\alpha} (\mu x)^{-2/\alpha} \Gamma\left(2/\alpha,\mu h^\alpha x\right)\nonumber\\ &:=K_\alpha(\mu,x, h), \label{eq:kalpha} \end{align}\tag{87}\] where \(\Gamma(\cdot,\cdot)\) is the upper incomplete gamma function defined as \(\Gamma(a,b):=\int_b^\infty z^{a-1} e^{-z}{\rm d}z\)

15.5 The integral in (?? )↩︎

The integral \(\int_{h}^\infty\left(1-e^{-\nu r^{-\alpha}}\right) r{\rm d}r\) in (?? ) is computed as \[\begin{align} \nonumber\\ &= \frac{1}{\alpha} \nu^{2/\alpha}\int_0^{\nu h^{-\alpha}}\left(1-e^{-z}\right) z^{-\frac{2}{\alpha}-1} {\rm d}z\nonumber\\ &= -\frac{1}{2} h^2 (1-e^{-\nu h^{-\alpha}}) + \frac{1}{2} \nu^{2/\alpha}\gamma\left(1-\frac{2}{\alpha}, \nu h^{-\alpha}\right) \nonumber\\ &:= {\frac{1}{2}}L_\nu(h, \alpha), \label{eq:NFI} \end{align}\tag{88}\] where \(\gamma(\cdot,\cdot)\) is the lower incomplete gamma function defined as \(\gamma(a,b):=\int_0^b z^{a-1} e^{-z}{\rm d}z\).

Acknowledgement↩︎

The authors are thankful to the ERC-NEMO grant, under the European Union’s Horizon 2020 research and innovation program, grant agreement number 788851 to INRIA Paris. This research was also funded in part by the France 2030 BPI “5G NTN mmWave” project to Télécom Paris.

References↩︎

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1. ¹Télécom Paris, ²INRIA Paris and ³Ecole Normale Superieure Paris.
   E-mail: sanjoy.jhawar@telecom-paris.fr.↩︎

2. E-mail: francois.baccelli@inria.fr.↩︎