April 02, 2024

The Internet of Things (IoT) is a communication scheme which allows various objects to exchange several types of information, enabling functions such as home automation, production management, healthcare, etc. Moreover, energy-harvesting (EH) technology is considered for IoT environment in order to reduce the need for management and enhance maintainability. However, since environments considering outdoor elements such as pedestrians, vehicles and drones have been on the rise recently, it is important to consider mobility when designing an IoT network management scheme. In order to handle this challenge, prior research has made an attempt to solve this problem via variational autoencoder (VAE) and backward-pass rate evaluation method. In this article, we propose a guided-mutation genetic algorithm (GMGA) to derive a sub-optimal relaying topology for IoT systems considering energy-harvesting. Furthermore, we propose a mobility-aware iterative relaying topology algorithm, which calculates the sub-optimal relaying topology of current time frame using the topology result of the previous one. Simulation results verify that our proposed scheme effectively solves formulated IoT network problems compared to other conventional schemes, and also effectively handles IoT environments in terms of mobility.

G.P. Kam : Guided-Mutation Genetic Algorithm for Mobile IoT Network Relay G.P. Kam : Guided-Mutation Genetic Algorithm for Mobile IoT Network Relay

Genetic algorithm, Mutation rate modulation, Internet of Things, Mobility, TDMA system, Energy harvesting, Relay.

The Internet of Things (IoT) has opened up a world of possibilities in several areas such as smart grid, home automation, smart transportation, etc [1]. With an ability for distant things to exchange duplex information in a wide area, the limitations of distance and time for various fields have been reduced. One of the applications of IoT is the smart cities, which can use public resources more efficiently and improve the quality of services for citizens while reducing operational costs through device connectivity [2]. In order to maintain such system, an energy harvesting (EH) technology provides an effective power management methodology in terms of usability and resilience, using power beacons (PBs) to supply energy to network nodes by transmitting solar or RF energy [3]. However, since the energy distribution and the position of nodes may be uneven under the variety of distances between the nodes and the PBs, finding the optimal relaying topology under these circumstances is complicated.

Moreover, since mobile nodes such as vehicles continuously alter positions over time, recalculations of relaying topology according to these changes are necessary. Otherwise, the relaying topology does not properly reflect the channel characteristics of the network, eventually resulting in a deterioration in the overall performance of the network [4]. Given these challenges, finding the optimal topology for this kind of problem is NP-hard and requires extensive computation time [5]. Besides, several methods such as deep learning and genetic algorithms have been applied to address such optimization problems effectively. In [6], deep learning is used to predict ergodic capacity of IoT network. Also in [7], genetic algorithm (GA) is used for the optimization of node location in high-density wireless sensor networks.

In this paper, we propose a guided-mutation genetic algorithm (GMGA) and mobility-aware iterative relaying topology algorithm to derive a sub-optimal relaying topology by solving the formulated problem under IoT network and EH environment. Numerical analysis confirms that not only our proposed methodology outperforms than conventional applicable methods for our formulated problem, but also our proposed methodology is applicable for a real-world implementation considering mobility.

Recent studies have focused on developing relaying schemes to calculate the optimal relaying topology. In [8], an energy-efficient relaying scheme for wireless sensor networks (WSNs) was explored, emphasizing relaying methods in scenarios where event detection occurs among numerous nodes. Additionally, [9] devised a ‘Teaching Learning’ based optimization algorithm to navigate the optimal path through a selection of nodes for relaying.

Furthermore, there is also some studies applying machine learning on relaying schemes. [10] attempted to find a relaying topology using a supervised learning, which is actually inefficient in complex scenarios. Since calculating relaying topology is an NP-hard problem, application of supervised learning is often limited due to the requirement for extensive training data. To overcome this issue, [11] calculated a sub-optimal relaying topology using variational autoencoder (VAE) with novel evaluation method which does not require a training dataset, and proposed iterative balancing time slot allocation algorithm; yet, the process reached its limit due to time-consuming nature of VAE. There are also other studies to solve the problem through reinforcement learning [12]. However, the lack of proven convergence and the extensive time required to discover the sub-optimal path remain significant challenges [13].

In addition to research on stationary nodes, research on mobile nodes has also been conducted in WSNs. In [14], the connectivity within heterogeneous 5G mobile systems is investigated, with an analysis of how connectivity changes based on the characteristics of mobile nodes and network parameters. In order to investigate connectivity, they proposed three algorithms for the cluster analysis of WSNs: k-means, algorithm of the foreign element, fuzzy C-means algorithm. [15] studied an intrusion prevention framework to improve secure routing in mobile IoT environments using WSN, focusing on network lifetime, data reliability, and security. [16] proposed a random waypoint model incorporating random orientation of user equipment, applied to analyze handover rates in indoor LiFi networks. These studies on mobile nodes have not proceeded with the relaying topology due to the considerable amount of time required. In summary, despite the development of various schemes, persistent limitations have necessitated ongoing research in this area.

In connectivity-related issues, genetic algorithm (GA) is frequently employed across various applications within the telecommunications division. [17] researched network security for intrusion detection using GA and support vector machines (SVMs), demonstrating a higher detection rate compared to other SVM-based intrusion detection algorithms. [18] proposed a novel method combining an improved genetic algorithm with a deep belief network (DBN) to enhance intrusion detection in IoT environments. GA has also been applied for ad-hoc optimization, as seen in [19], where GA was used for UAV path planning to identify the optimal UAV relay, and as seen in [20], which applied GA to solve the area coverage problem in WSN. [21] represents a case where GA was utilized to efficiently compute dynamic QoS parameters in IoT systems concerning computation time and optimality. Furthermore, [7] developed a MOONGA algorithm to propose a method for identifying the optimal node positions in fixed relaying topology. However, node locations are often constrained by practical needs or geographical limitations in reality, necessitating algorithms that calculate the relaying topology while holding the node positions fixed.

Since the method of applying GA to a problem varies depending on the problem, special modulations are applied to GA. [22] applied ranking group selection, direction-based crossover, and normal mutation to enhance the search capabilities of algorithm and population diversity, thereby improving solutions to complex optimization problems. [23] increased search efficiency for continuous constrained optimization problems through two-direction crossover and grouped mutation. [24] experimentally demonstrated that scheduling problems could be solved using dispatching rules for population initialization. However, these modulations are limited to the problems proposed in each paper, and to solve our connectivity problem, a modulated GA with reduced computation time and for discrete values is necessary. Thus, we aim to solve the nonlinear and NP-hard problem of finding a sub-optimal relaying topology by applying GA with modulation.

The main contributions of this paper are threefold:

First, we prove that for the Iterative Balancing time slot allocation algorithm which allocates time slots in order to maximize minimum bits/Hz, equivalence of maximum and minimum bits/Hz equals to the maximization of minimum bits/Hz using dual problem and KKT conditions.

Second, we propose a guided-mutation genetic algorithm (GMGA), which effectively handles both scalability and non-linearity of the formulated IoT environment based on EH. By mutating the node connections with a probability based on the cost of each links, our GMGA shows superior performance results than conventional algorithm in less computation time.

Third, we propose a mobility-aware iterative relaying topology algorithm to handle environments considering mobility, which reduces repetitive calculations and derive sub-optimal topology that alters with movement of the nodes, thus enhancing adaptability into the real-world environments such as outdoor transportation.

The remainder of this paper consists of the following. In Section II, we establish the system model and formulate an IoT optimization problem, which is a main problem in this paper. In Section III, we prove the validity of IB time slot allocation algorithm and propose a GMGA in order to calculate relaying topology. In addition, we propose a mobility-aware iterative relaying topology algorithm based on the formulated system model. In Section IV, we provide a detailed explanation of simulation parameters and conduct simulations to confirm the sub-optimality of our proposed scheme by comparing performance with other conventional schemes. Finally, in Section V, we provide the key findings of our research and conclude the paper with a short summary.

This subsection describes a system model. The basic configuration of the system model is inspired by approach of [11]. In our system model, we consider \(N_d\) IoT devices (nodes) and \(N_b\) power beacons (PBs) which are uniformly distributed within a circle with a radius R, centered around a single data packet destination(packet sink). By labeling the nodes as 1,2,...,\(N_d\) and the sink as \(N_{d}+1\), the set associated with the nodes is \(\mathbb{N}_d = \{1,2,...,N_d\}\) and the set associated with the overall communicating nodes is \(\mathbb{N} = \{1,2,...,N_{d}+1\} = \mathbb{N}_{d} \cup \{N_{d}+1\}\), respectively.

Furthermore, we consider time division multiple access (TDMA) system in our system model, since TDMA system has abilities of high energy efficiency and an on-demand feature. Those abilities are necessary for IoT devices operating with energy harvesting (EH) and offer flexibility in resource allocation for various devices [25]. In our TDMA-based system model, each node transmits data to the higher node defined by relaying topology in assigned time slot within time frame \(T\) on the basis of TDMA system.

The time frame \(T\) in the TDMA system is divided into \(N_d\) time slots, and the time slot assigned to a node \(i\) is denoted as \(t_i\)(\(i \in \mathbb{N}_d\)). The time slot \(t_i\), allocated for node \(i\) to send the data to the upper node, is defined by the following conditions: \[\begin{align} \sum_{i\in\mathbb{N}_d}{t_i}=T,\,\, t_i\in (0,T)\,\, \forall i\in\mathbb{N}_d. \end{align}\]

Additionally, EH based on radio frequency (RF) is considered to supply energy to the nodes in our system model. For an RF power source, the harvested energy delivered to a *\(k\)*th node is \(E_{eh,k}=T\sum_{n\in\mathbb{N}_b}{P_b |h_{n,k}|^2 d^{-\alpha}_{n,k}}\), where \(P_b\) is power transmitted by PB, \(h_{n,k}\) is a channel between *\(n\)*th PB and *\(k\)*th node following \(\mathcal{CN}(0,1)\), \(d_{n,k}\) is distance between *\(n\)*th PB and *\(k\)*th node, and \(\alpha\) is a path loss exponent, respectively. Assuming that all the energy delivered to the *\(k\)*th node is consumed for transmission of data within its time slot, the amount of bits per frequency (bits/Hz), \(r_{k}(\mathbf{c},\mathbf{t})\), of *\(k\)*th node transmitting to a parent node under TDMA system is given by the following according to Shannon’s capacity:

\[\begin{align} r_{k}(\mathbf{c},\mathbf{t})&=\sum_{n\in\mathbb{N}}{c_{k,n}t_{k}} \log_{2}{(1+\frac{E_{k}|h_{k,n}|^2 d_{k,n}^{-\alpha}}{t_k N})}, \\ c_{k,n}&= \begin{aligned} \begin{cases} 1, \quad if\,\, n\text{th
node is parent of } k\text{th node}, \\ 0, \quad \text{otherwise}. \end{cases} \end{aligned}
\end{align}\] where \(\mathbf{t}\) is an allocated time slot vector, \(E_k\) is energy of *\(k\)*th node by EH which is proportional to \(E_{eh,k}\), and \(N\) is noise power, respectively. However, since it is necessary to transmit not only own data but also data from child nodes, the amount of bits/Hz budget for transmitting
data of *\(k\)*th node denoted as \(R_k(\mathbf{c},\mathbf{t})\) is as follows: \[\begin{align}
R_k(\mathbf{c},\mathbf{t})=\sum_{n\in\mathbb{N}}{c_{k,n}t_{k}} \log_{2}{(1+\frac{A_{k,n}}{t_k})} \nonumber\\ -\sum_{n'\in\mathbb{N}_d}{c_{n',k}t_{n'}} \log_{2}{(1+\frac{A_{n',k}}{t_{n'}})}, \label{f95calculate}
\end{align}\tag{1}\] where \(A_{k,n}=E_{k}|h_{k,n}|^2 d_{k,n}^{-\alpha}/{N}\) and the signal to noise ratio (SNR) is \(\Gamma_{k,n}=A_{k,n}/t_k\) when transmitting node is
*\(k\)*th node and receiving node is *\(n\)*th node.

For a simulation of mobile nodes, we assume that nodes are moving within the circle according to the Random Waypoint Model (RWM) which was adopted in mobile Ad-hoc network (MANET) simulation of [26]. Note that in our simulation, PBs are stationary; however, applying mobility to them is not difficult and does not critically impact the simulation results. Since mobility models do not consider whether the waypoint of the next step is inside the boundary of the circle, movements derived from some mobility models could violate the boundary condition. However, RWM is free from this consideration due to the convexity of the circle, resulting the derived waypoint of the inbound point from RWM is also inbound.

Moreover, we define a unit mobility simulation time \(T_u\) which corresponds to the update period of node positions. In our system model, \(T_u\) is defined as an integer multiple of \(T\)(\(T_u=kT\), \(k \in \mathbb{Z}^{+}\)). Furthermore, all nodes move at a constant speed \(v_c\) within the given circle with a radius \(R\). In the real-world, nodes might have varying speeds over time, but such consideration on our system model causes only minor impact in this paper.

In this subsection, we formulate our target problem based on the system model defined in the subsection above. Our objective is to maximize \(R_k(\mathbf{c},\mathbf{t})\) while each node transmits data to the sink within a limited time frame, \(T\) in the system model. In order to increase the bits/Hz in a systematic aspect rather than an individual aspect, it is important to enhance the capacities for all nodes since the minimum \(R_k(\mathbf{c},\mathbf{t})\) limits overall performance of the system. Therefore, the main objective to enhance the performance of the system model is to maximize the minimum value among all \(R_k(\mathbf{c},\mathbf{t})\) of nodes.

However, since \(R_k(\mathbf{c},\mathbf{t})\) takes into account both time slot \(\mathbf{t}\) and connectivity \(\mathbf{c}\), we include constraints on \(\mathbf{t}\) and \(\mathbf{c}\) into the main problem. In our problem, \(\mathbf{t}\) corresponds to the allocated time slots within the time frame T, where the sum of allocated time slots for all nodes equals to T. Additionally, in our system model, every node has a single uplink route connected to the sink. Considering all these constraints, our final formulated problem is as follows:

\[\begin{align} \text{(P1)} \quad \max_{\mathbf{c},\mathbf{t}} \, &\min_{k} \quad R_{k}(\mathbf{c},\mathbf{t})\tag{2}\\ \textrm{s.t.} \quad & \sum_{n \in \mathbb{N}_d} t_n =T, \tag{3}\\ & \sum_{n'\in \mathbb{N}} C_{n,n'} =1\,\, \forall n \in \mathbb{N}_d,\tag{4}\\ & (\mathbf{C}^{N_d})_{n,N_d+1} = 1\,\, \forall n \in \mathbb{N}_d \tag{5} . \end{align}\]

where \(\mathbf{C}\) is extended version of \(\mathbf{c}\) satisfying \[\begin{align} &\mathbf{C}_{n,n'}=\mathbf{c}_{n,n'}\quad \forall n \in \mathbb{N}_d,\, n' \in \mathbb{N},\tag{6} \\ &\mathbf{C}_{N_d+1,n'} = 0 \quad \forall n' \in \mathbb{N}_d, \tag{7}\\ &\mathbf{C}_{N_d+1,N_d+1} = 1. \tag{8} \end{align}\]

Then, (6 )-(8 ) provides a mathematical approach to connection, while (4 ) ensures that each node has a single uplink to a parent node, and (5 ) guarantees that there are no cycles in the system, and every node ultimately transmits data to the sink.

However, finding (\(\mathbf{c}, \mathbf{t}\)) which maximizes minimum of \(R_{k}(\mathbf{c},\mathbf{t})\) is challenging due to following reasons: First, the presence of logarithmic operations in Shannon’s capacity. Second, the NP-hardness of finding the optimal \(\mathbf{c}\) since the number of possible candidate of \(\mathbf{c}\) is \((N_d+1)^{N_d-1}\) in our formulated problem.

In this paper, we address each challenge from three aspects. First, an Iterative Balancing (IB) time slot allocation algorithm is used in order to calculate optimal \(\mathbf{t}\) in given \(\mathbf{c}\) [11]. We also prove a validity of IB time slot allocation algorithm to increase the reliability of our formulated problem. Second, we propose a GMGA method to calculate sub-optimal solution of \(\mathbf{c}\) in reasonable computation time. Finally, we propose a mobility-aware iterative relaying topology algorithm by applying mobility in our system model in order to consider the real-world application.

In this section, we propose a GMGA in order to calculate sub-optimal solution of \(\mathbf{c}\), which modulates the mutation rate of GA based on a cost of each links. We also propose a mobility-aware iterative relaying topology algorithm in order to handle wireless channel status changing over time without repetitive calculations. It uses the final result from a prior simulation frame in the current simulation frame. Moreover, in IB time slot allocation algorithm, we prove that equivalence of maximum \(R_k(\mathbf{t})\) and minimum \(R_k(\mathbf{t})\) achieves the maximization of minimum \(R_k(\mathbf{t})\) with given \(\mathbf{c}\) in order to strengthen the theoretical foundation.

In this subsection, we describe IB time slot allocation algorithm assigning time slot to each node from the restricted time source \(T\), regarding TDMA system. When \(\mathbf{c}\) is given, the IB time slot allocation algorithm works by allocating time \(\Delta\) which is subtracted from a node with maximum \(R_k(\mathbf{t})\), to time slot of node whose \(R_k(\mathbf{t})\) is minimum, while maintaining the sum of time slots same. After finishing one iteration of time slot allocation, \(\Delta/2\) is substituted for \(\Delta\). The convergence of the algorithm which is the difference between maximum \(R_k(\mathbf{t})\) and minimum \(R_k(\mathbf{t})\) converges to value smaller than \(\epsilon_1\) is proven in [11].

However, although decrease of difference between maximum and minimum ensures the equilibrium of \(R_k(\mathbf{t})\), it does not guarantee maximization of minimum \(R_k(\mathbf{t})\). In this paper, we prove that maximization of minimum \(R_k(\mathbf{t})\) with constraints (3 ), (4 ), (5 ) is achieved when \(R_1(\mathbf{t})=R_2(\mathbf{t})=...=R_{N_d}(\mathbf{t})\). The proof of this equivalence is given as follows.

**Proposition 1**. *Let us define the amount of bits/Hz *the *\(k\)*th node itself can transmit as \(R_k(\mathbf{t})\) with given \(\mathbf{c}\) and \(k^*=\mathop{\mathrm{argmin}}\limits_{k}{R_k(\mathbf{t})}\). Then, a condition to maximize \(R_{k^*}(\mathbf{t})\) where \(t_1+t_2+...+t_{N_d}=T\) is \(R_1(\mathbf{t})=R_2(\mathbf{t})=...=R_{N_d}(\mathbf{t})\).**

*Proof.* First, we consider the characteristic of \(R_k(\mathbf{t})\) function. Partial derivative of \(R_k(\mathbf{t})\) with respect to \(t_k\),
denoted by \(\nabla_{t_k} R_k(\mathbf{t})\), is calculated as follows:

\[\begin{align} \label{Rprime} \nabla_{t_k} R_k(\mathbf{t})&=\frac{\partial{R_k(\mathbf{t})}}{\partial{t_k}} \nonumber\\
&=\sum_{n\in\mathbb{N}}{\frac{c_{k,n}}{\ln{2}}}{(-\frac{A_{k,n}}{t_k+A_{k,n}}+\ln{(1+\frac{A_{k,n}}{t_k})})}.
\end{align}\tag{9}\] Since \(\ln{(1+x)}>x/(1+x)\) when \(x>0\), \(\nabla_{t_k} R_k(\mathbf{t})>0\) holds, which means that \(R_k(\mathbf{t})\) is monotonic increasing function for \(t_k\). This result implies that as more time is allocated to *\(k\)*th node, the amount of data
that the *\(k\)*th node transmit to the upper node increases.

In order to make our problem into a minimizing problem, let us define \(S_k(\mathbf{t}) = -R_k(\mathbf{t})\). Since (9 ) guarantees that \(R_k(\mathbf{t})\) is monotonic increasing function for \(t_k\), \(S_k(\mathbf{t})\) is monotonic decreasing function for \(t_k\): \[\begin{align} \label{Sprime} \nabla_{t_k}S_k(\mathbf{t})<0, \end{align}\tag{10}\] and our formulated problem is as follows: \[\begin{align} \text{(P2)}\quad \mathop{\mathrm{argmin}}_{\mathbf{t}}\, \max_{k}\quad &S_k(\mathbf{t}) \\ \textrm{s.t.} \quad &\sum_{j \in \mathbb{N}_d} {t_j}=T. \end{align}\] By adding inequality terms about the maximization of \(S_k(\mathbf{t})\), we can remove an inner maxima: \[\begin{align} \text{(P3)}\quad \mathop{\mathrm{argmin}}_{\mathbf{t}}\quad &S_{k^*}(\mathbf{t}) \\ \textrm{s.t.} \quad &S_{k^*}(\mathbf{t})\geq S_k(\mathbf{t})\,\, \forall\, k\in\mathbb{N}_d\backslash\{k^*\}, \\ &\sum_{j \in \mathbb{N}_d} {t_j}=T. \end{align}\] Note that we exclude a condition when \(k=k^*\) at the inequality constraint in order to simplify the calculation and because it does not affect the overall problem. Since \(S_k(\mathbf{t})\) is not convex for \(\mathbf{t}\), convex optimization method is not applicable. Thus, we convert the problem into a dual problem using Lagrange function and find the optimal solution of primal problem using Karush–Kuhn–Tucker (KKT) conditions.

Lagrange function of the problem (P3) is as follows: \[\begin{align} L(\mathbf{t},\boldsymbol{\lambda},\nu)&=S_{k^*}(\mathbf{t})+\sum_{i \in \mathbb{N}_d \backslash \{k^*\}}{\lambda_i(S_i(\mathbf{t})-S_{k^*}(\mathbf{t}))}\nonumber\\ &+\nu (\sum_{j \in \mathbb{N}_d} {t_j}-T), \end{align}\]

and dual problem is as follows: \[\label{dualproblem} \begin{align} \max_{\boldsymbol{\lambda},\nu}\,\,\min_{\mathbf{t}}\quad&L(\mathbf{t},\boldsymbol{\lambda},\nu)\\ \textrm{s.t.} \quad &\boldsymbol{\lambda} \succcurlyeq \mathbf{0}. \end{align}\tag{11}\]

To find the optimal solution of primal problem (P3), KKT conditions are considered: \[\begin{align} &\text{(P4-1)}\quad \nabla_\mathbf{t}L(\mathbf{t}^*,\boldsymbol{\lambda}^*,\nu^*)=\mathbf{0}, \\ &\text{(P4-2)}\quad \lambda^*_i (S_i(\mathbf{t^*})-S_{k^*}(\mathbf{t^*}))=0 \,\,\,\forall\, i \in \mathbb{N}_d\backslash\{k^*\},\\ &\text{(P4-3)}\quad S_i(\mathbf{t^*})-S_{k^*}(\mathbf{t^*}) \leq 0 \,\,\,\forall\, i \in \mathbb{N}_d\backslash\{k^*\},\\ &\text{(P4-4)}\quad \sum_{j \in \mathbb{N}_d} {t^*_j}-T=0,\\ &\text{(P4-5)}\quad \lambda^*_i \geq 0 \,\,\,\forall\, i \in \mathbb{N}_d\backslash\{k^*\}, \end{align}\] where \((\mathbf{t}^*,\boldsymbol{\lambda}^*, \nu^*)\) satisfies KKT conditions. If KKT conditions are satisfied, strong duality holds ensuring \(\mathbf{t}^*\) is the optimal solution of the primal problem.

Partial derivative of \(L(\mathbf{t},\boldsymbol{\lambda},\nu)\) about \(t_x\) is represented by: \[\label{gradLaboutt} \begin{align} \frac{\partial{L}}{\partial{t_x}}&= \begin{cases} \lambda_x \nabla_{t_x}S_x(\mathbf{t})+\nu, \,\,& if\,\, x\neq k^*, \\ \\ (1-\sum\limits_{i \in \mathbb{N}_d \backslash \{k^*\}}{\lambda_i})\nabla_{t_{k^*}}S_{k^*}(\mathbf{t})+\nu, \,\,& if\,\, x = k^*. \end{cases} \end{align}\tag{12}\] In order to solve (P4-1), we put \({\partial{L}}/{\partial{t_x}}\) as \(0\) and solve a simultaneous equation about \(\lambda^*_x\) and \(\nu^*\): \[\begin{align} \lambda^*_x &= \frac{\frac{1}{\nabla_{t_x} S_x(\mathbf{t}^*)}}{\sum\limits_{i \in \mathbb{N}_d}\frac{1}{\nabla_{t_i} S_i(\mathbf{t}^*)}}\tag{13}, \\ \nu^* &= -\frac{1}{\sum\limits_{i\in\mathbb{N}_d}\frac{1}{\nabla_{t_i} S_i(\mathbf{t}^*)}}, \tag{14} \end{align}\] where \(x \in \mathbb{N}_d\backslash\{k^*\}\), without consideration of division by zero due to (10 ). (P4-5) is also guaranteed by (10 ) and (13 ). Since (13 ) stands for \(\lambda^*_i \neq 0\), (P4-2) is simplified as: \[\begin{align} \label{KKTresult} S_i(\mathbf{t}^*)-S_{k^*}(\mathbf{t}^*)=0 \,\,\,\forall\, i \in \mathbb{N}_d\backslash\{k^*\}, \end{align}\tag{15}\] which is also satisfying (P4-3). In other words, (15 ) is a sufficient condition for both (P4-2) and (P4-3). The KKT conditions ensure that obtained solution minimizes the primal problem, not the the solution exists. Referring to [11], convergence of time slot allocation algorithm regardless of \(\epsilon_1\) value is proven. By setting \(\epsilon_1 \rightarrow 0, \epsilon_2 \rightarrow 0\), difference between maximum and minimum bits/Hz also converges to zero by the squeeze theorem with constant time slot summation. Therefore, we ensure that \(\mathbf{t}^*\) satisfying (15 ) and (P4-4) exists.

Finally, \((\mathbf{t^*}, \boldsymbol{\lambda}^*,\nu^*)\) is the solution of KKT conditions (P4), which is also the optimal solution of the primal problem (P3) and the dual problem (11 ). KKT conditions about \(\mathbf{t}^*\) are (P4-4) and (15 ), which is the requirements to satisfy (P3). In addition, \(R_k(\mathbf{t})\) is \(-S_k(\mathbf{t})\). Thus, we conclude that when \(R_1(\mathbf{t}^*)=R_2(\mathbf{t}^*)=...=R_{N_d}(\mathbf{t}^*)\), \(\mathbf{t}^*\) maximizes the minimum of \(R_k(\mathbf{t})\) where \(t_1+t_2+...+t_{N_d}=T\). ◻

In this subsection, we propose a GMGA in order to find a sub-optimal solution of \(\mathbf{c}\) in our formulated problem, which modulates mutation rate regarding the cost of the links between nodes.

Finding a relaying topology based on exhaustive search method requires searching from \((N_{d}+1)^{(N_d-1)}\) possible candidates, making it an NP-hard problem and implausible for real-world application. Despite previous attempts to address this issue, practical application remains challenging due to extended computation times and comparably low minimum bits/Hz than optimal solution. For instance, the approach using a variational autoencoder (VAE) in [11] aimed to train layers by devising loss function evaluation method called PT-EVM and minimizing it to discover sub-optimal solution of \(\mathbf{c}\). However, this method faced challenges in practical implementation due to time-consuming nature of training neural networks.

In parallel with these challenges, in prior approach, GA is also commonly used as a solver for NP-hard problems in a wide area. Such application is also applied to our formulated problem, since our system model takes the form of a tree structure, which can be divided into branches that is easy to evaluate whether it helps overall performance or not. By defining genes of chromosome as the index of the parent node, GA can handle our formulated problem by selecting the beneficial branches and inheriting them to the next generation.

With these advantages, GA operates by selecting the best-performing candidates from a generation and applying crossover and mutation to pass on their traits to the next generations. Throughout this process, GA compares and refines high-score expected candidates rather than evaluating every possible candidate. Since our formulated problem has requirements including all nodes being connected to a sink and having only one parent node through an uplink, we add some details in our proposed GA scheme. Detailed explanations of four components(evaluation, selection, crossover, and mutation) of our scheme are as follows:

First, in the evaluation part, the candidates are evaluated based on their measured performances. The evaluation part called at line 4 of Alg. 2 which is defined from line 15 to 20 signifies the evaluation for the first generation, while line 10 represents the evaluation for the other generations. Since our objective is to maximize the \(R_{\text{min}}\), we designate the \(R_{\text{min}}\) value obtained through time slot allocation algorithm as a score of a candidate.

Second, in the selection part called at line 5 and 11 of Alg. 2 which is defined from line 21 to 28, all candidates are sorted in descending order, and a specific number of high-scored candidates are then selected as parents for the next generation.

Third, in the crossover part called at line 9 of Alg. 2 which is defined from line 29 to 38, the chromosomes of two parent candidates are combined, creating a new offspring chromosome with new attributes. Crossover occurs at \(n_c\) points, and in the Alg. 2, we provide an example for the case where \(n_c=2\).

Finally, in the mutation part called at line 9 of Alg. 2 which is defined from line 39 to 48, the genes of the individual are mutated into random values under a mutation probability \(p_m\), in order to explore other possibilities which cannot be induced from crossover operations. After completing the crossover or mutation part, a tree validation test is conducted at the end of each process to satisfy the constraints (4 ) and (5 ) in (P1). In addition, we include elitism in our algorithm to ensure the enhancement of performance as described in line 8 of Alg. 2.

Above mentioned descriptions are the main structure of GA for application to our formulated problem. The basic form of GA uses the uniform mutation rate, which means that when a node A is selected to be mutated, the probability of another node being selected as an uplink for node A is uniform. However, the uniform mutation rate leads to an excess of low-performance candidates in our formulated problem, such as nodes connecting directly to the sink without sufficient energy or to the distant nodes. Furthermore, uniform mutation rate results in some performance related problems, including an increment of candidates which will not be selected in the evaluation part, reducing the variation between parents and offsprings. The issue of topology remaining unchanged throughout generations mentioned above can lead to very early stopping under the mistaken decision that the algorithm has reached an optimal topology, resulting in premature outcomes.

In our proposed GMGA method, the mutation rate is adjusted to induce mutations toward more reasonable node connections. Specifically, it is adjusted to be inversely proportional to the link cost so that the probability of a specific node being chosen as an uplink is varied. Through this adjustment, reasonable mutations such as mutating towards adjacent nodes are expected. The time slot used for calculating link cost is derived from the time slots of parents, since it is closer to the optimal time slot than time slot obtained by uniformly dividing the time frame. Detailed explanation of mutation rate \(\mu_{i,j}\), which is the conditional probability of uplink of node i mutating to node j given that node i mutates, is as follows: \[\begin{align} \mu_{i,j}&=P(\text{node i uplinks to node j | node i mutates})\\ &=\frac{t_i\log_{2}{(1+\frac{A_{i,j}}{t_i})}}{\sum_{j \in \mathbb{N}}{t_i\log_{2}{(1+\frac{A_{i,j}}{t_i})}}} \end{align}\] where \(\mathbf{t}\) is the time slot allocation of parents.

The guided-mutation results in less computation time than basic GA with uniform mutation rate. Besides, in order to boost our scheme more for the real-world application, we devise several tunings as follows:

First, we include the case of connecting all nodes directly to the sink in the first generation, as described in line 2 of Alg. 2. One of the advantages of GA is the ability to include ‘promising’ candidates in the learning process without modulation of the algorithm scheme, allowing them to compete with other candidates. Typical example of ‘promising’ candidates is a case of direct connection of all nodes to the sink. If all nodes have sufficient energy, it is better to connect directly to the sink since relaying is detrimental. Therefore, in the first generation, we include a case of direct connection to the sink, considering scenarios where there are enough PBs or nodes densely clustered around PBs. While this tuning could be perceived as disrupting exploration, the possibility of a random tree showing higher performance than direct connect is rare. Moreover, since only one candidate among several candidates in the first generation is modified, it has minimal impact on the exploration.

Second, we start time slot allocation from the time slot of parents except first generation in order to calculate time slot of offsprings faster, since the topologies of offsprings are similar to the topologies of parents. This approach is experimentally proven to be more time-efficient than starting calculations from a uniform time slot. Additionally, scores of all topologies generated during the learning process are stored, so that if a score calculated in a past generation is redundantly required in the current generation, the stored value is retrieved without repetitive calculations. Since time slot allocation algorithm and calculation of rate are the most time consuming part of our algorithm, the storage of calculation is effective in reducing the computation time.

Finally, in the evaluation part, using a time slot allocation algorithm with small \(\epsilon_1\) requires more iterations to minimize the difference between the maximum and minimum \(R_k(\mathbf{c},\mathbf{t})\). Meanwhile, in the evaluation process, precise \(R_k(\mathbf{c},\mathbf{t})\) values are not required. Instead, our objective is to compare which topology is better. Therefore, we define \(\epsilon'_1\) which is bigger than \(\epsilon_1\) for the time slot allocation in the evaluation part, allowing faster comparison and tolerant convergence. Note that the \(\epsilon'_1\) value is sufficiently small, so that we determine that time slot allocation is done for comparison purpose when the difference between the maximum and minimum values of \(R_k(\mathbf{c},\mathbf{t})\) fell below \(\epsilon'_1\).

In this subsection, we describe a relaying topology algorithm regarding mobile nodes.

Previously investigated relaying topology algorithms are hard to apply in the real-world which involves mobile nodes. Some methodologies with a rapid computation result in insufficient capacity for transmit their data due to an inefficient topology. Conversely, some methodologies showing good performance fail to ascertain a topology before the subsequent simulation frame. This trade-off between computational speed and performance impedes the adoption of relaying topologies in the real-world with mobile nodes since the nodes unable to operate in the calculated topology which is sub-optimal for the current time frame. However, GA has a simple structure facilitating calculations and ensures performance improvements over generations. Unlike other algorithms, GA can start calculations from the previously calculated result while avoiding the redundant calculation. Thus, by using these profits, we utilize GA which has relatively fast calculation and good performance.

As described above, the merits of GA are incorporating promising candidates into the learning process and refining the candidates to find a sub-optimal solution of \(\mathbf{c}\), iteratively. Thus, since initiating the calculation with a relaying topology which is close to the optimum enhances the performance, we assign the final relaying topology results from the previous frame to the first generation of current frame for the mobility simulation. The reason of using the relaying topology result from the previous frame is as follows.

While nodes move during a unit time interval, relocated nodes are akin to the pre-movement state. Additionally, the topology of the system model only depends on a distribution of the nodes if other conditions are unchanged during the unit time interval. Therefore, the changes in relaying topology across consecutive simulation frames are minimal. By incorporation of the final relaying topology results from the previous frame into the first generation of the current frame, it provides a more reasonable starting point than a random tree. Note that the incorporation of final results from the previous frame has a minimal impact on exploration, similar to the above mentioned incorporation of the direct connection case.

Including all of our considerations, flow chart of our mobility-aware iterative relaying topology algorithm is depicted in Fig. 3.

In this subsection, we define the system model parameters for evaluating the performance of our proposed scheme.

In our system model, nodes are distributed within a circle with a radius \(R\) of 0.5 km. Each node transmits data under a TDMA system with a time frame \(T\) of 100 milliseconds. For the energy harvesting model, the transmit power of the Power Beacon (PB) is set to 1W. The energy harvesting model is assumed as a linear model, where the energy of node is 0.7 times the harvested energy sent from the PBs. The wireless fading channel, represented by \(|h_{k,n}|^2\), follows an exponential random variable with a unit mean. The path loss exponent \(\alpha\) is 3, the bandwidth is 125 kHz, and the noise figure \(NF\) is 6dB in our system model, respectively. The Noise power \(N\) is calculated under \(N=-174+NF+10\log_{10}{BW}\).

Next, we set the hyper-parameters for our GA as follows. First, the size of the first generation \(n_{\text{origin}}\) and the size of the parent generation \(n_{\text{parent}}\) are both 5. The size of the offspring generation \(n_{\text{offspring}}\) is set to 50. Next, the number of crossover points is assumed to be \(n_c= 2\), and the mutation probability to be \(p_m = 0.05\).

In the evaluation part, the tolerant value of \(\epsilon_1\) which is \(\epsilon'_1\) is set to \(10^{-3}\) bits/Hz. For the time slot allocation algorithm which is used to calculate the exact performance of an algorithm, \(\epsilon_1\) is set to \(10^{-6}\) bits/Hz, and \(\epsilon_2\) is set to \(10^{-7}\) sec. Finally, we set the stop condition to halt the GA if \(R_{\text{min}}\) remains unchanged for \(200/N_{d}-4\) generations.

For the mobility-aware system model, parameters are set as follows. First, the speed of each node is set to \(6.42\) m/s, which is the average speed of vehicles in Seoul, Korea [27]. Second, the staying time of RWM is set to zero in order to assume worst case regarding mobility. If a node arrives at its destination, the node waits until the termination of current frame and sets the next random destination. Finally, the total simulation time is 3 minutes, and the unit simulation time is 20 seconds. Thus, the number of simulation frame for the given number of nodes, PBs and scheme is 10.

To compare the performance of our GMGA with the performance of other schemes, we choose 4 different schemes and optimal solution (if applicable) case along with our proposed scheme:

**Optimal solution (Opt)**which exhaustively compares the performance of all possible trees in order to find the optimal solution, and allocates time slot using IB time slot allocation algorithm.**Direct connect scheme (Dir)**which connects every node to the sink and allocates time slot using time slot allocation algorithm.**MST scheme (MST)**which makes a Minimum Spanning Tree (MST) based on the link cost denoted as \(1/\left( t_i \log _{2} (1+\Gamma_{i,j})\right)\) starting from the sink with uniform time slot allocation, and allocates time slot using time slot allocation algorithm.**Greedy scheme (Greedy)**which establishes initial connections by linking all nodes directly to the sink and iteratively selects a node \(i\) randomly and connects it to the node with the highest achievable rate, determined by \(\min(t_i \log_{2} (1+\Gamma_{i,j}), \mathbf{B}_{j})\) \(\forall j \in \mathbb{N}_d\), and allocates time slot using time slot allocation algorithm.**VAE scheme (VAE)**which decides the topology with VAE and PT-EVM scheme [11], and allocates time slot using time slot allocation algorithm.**Proposed scheme (Prop)**which decides the topology with GMGA, and allocates time slot using time slot allocation algorithm.

All considered schemes are implemented on Matlab R2023a, using a computer with an Intel Core i7-12700 and 32GB of memory, without GPU acceleration.

Fig. 4 shows the change in relaying topology obtained by proposed scheme as all nodes are moving under RWM. Each sub-figure 4 (a), 4 (b), 4 (c) shows the sample distribution of 10 nodes when the simulation time is 0, 20, 40 seconds, respectively. The relaying topology of each distribution is changing over time for high \(R_\text{min}\) of changing distribution since relaying topology is dependent on the distribution of nodes.

Fig. 5 shows the computation time of considered schemes with respect to the number of nodes from \(N_d=5\) to \(25\) with an interval of 2. The computation time of exhaustive search method is excessive for real-world application, suggesting the need for an alternative scheme. Our proposed GMGA takes less computation time than the VAE scheme while takes more computation time than the other conventional schemes. Unlike VAE scheme using back propagation for learning, the GMGA choose the best relaying topology among candidates, requiring less computation time than VAE scheme. In addition, GMGA avoids redundant scoring of relaying topology, decreasing the usage of laborious time slot allocation algorithm.

Fig. 6 (a) to Fig. 6 (c) show the \(R_{\text{min}}\) of considered schemes for a stationary simulation with respect to \(N_b \in \{1,2,3\}\) when \(N_d \in \{5,6,7\}\). Each data point of Fig. 6 (a) to Fig. 6 (c) is an averaged performance results of 30 random distributions of nodes and PBs. Note that value of \(R_k(\mathbf{c},\mathbf{t})\) can be transformed into a rate by multiplying \(\text{BW}/T\) which is constant in our test environment. In addition, \(R_k(\mathbf{c},\mathbf{t})\) is calculated by the IB time slot allocation algorithm whose \(\epsilon_1=10^{-6}\) so that the difference between the maximum and minimum \(R_k(\mathbf{c},\mathbf{t})\) is negligible. Our proposed scheme shows the superior performance among considered schemes for every condition. Moreover, the results of our proposed scheme overlaps the results of exhaustive search scheme, which means that our proposed scheme finds the optimal relaying topology when \(N_b \in \{1,2,3\}\), \(N_d \in \{5,6,7\}\).

Fig. 7 (a) to Fig. 7 (c) show the \(R_{\text{min}}\) of considered schemes with respect to \(N_b \in \{1,2,3,4,5\}\) when \(N_d \in \{10,20,30\}\). Each data point of Fig. 7 (a) to Fig. 7 (c) is also an averaged performance results of 30 random distributions of nodes and PBs. As in the previous case, our proposed scheme shows superior performance among the considered schemes for every condition. Note that simulation using exhaustive search method is not conducted due to excessive computation time.

As shown in Fig. 6 and Fig. 7, our proposed scheme shows the superior performance compared to the above mentioned conventional schemes, able to find the optimal solution. We suggest the reason of the superior performance as follows: The guided mutation guide the connection of node into more rational way. The optimal relaying topology is likely to have connections from one node to another node where the cost is lower rather than higher. While connecting to a node with higher cost can be beneficial in a global aspect, it is rare for the final optimal result. Therefore, in the GMGA, the conditional probability \(\mu_{i,j}\) given that a mutation occurs is likely to be higher than \(1/N_d\), which is the conditional probability of randomly selecting a node. Let us assume that \(i\)th node connects to \(j\)th node which is an optimal connection. Since there is no way to know whether this connection is optimal during a learning process, mutations occasionally occur at \(i\)th node. For the uniform mutation rate, the conditional probability of the \(i\)th node connecting to a different node instead of the \(j\)th node due to a misjudgment is \(1-1/N_d\). However, for the GMGA, the conditional probability of misjudgment is \(1-\mu_{i,j}\), which is lower than \(1-1/N_d\).

Fig. 8 (a) to Fig. 8 (c) show the \(R_\text{min}\) of considered schemes for a mobility simulation with respect to \(N_b \in \{1,2,3\}\) when \(N_d \in \{5,10,20\}\). Each data point of Fig. 8 (a) to Fig. 8 (c) is an averaged performance results of 30 random distributions of nodes and PBs during mobility simulation. In other words, each data point is an average of \(30\) random distributions consisting of \(10\) simulation frames, totaling \(300\) simulation results. Fig. 8 (a) to Fig. 8 (c) show that our proposed scheme has a superior performance among considered schemes for every condition. Note that the number of nodes is selected as 5, 10, and 20 in order to achieve similar coverage to stationary simulations and to prevent excessive computation time while conducting simulations, which requires about ten times more calculations than stationary simulations.

Fig. 9 (a) to Fig. 9 (c) show the computation time of considered schemes for a mobility simulation with respect to \(N_d \in \{5,10,20\}\) when \(N_b \in \{1,2,3\}\). Each data point of Fig. 9 (a) to Fig. 9 (c) is also an averaged performance results of 30 random distributions of nodes and PBs during mobility simulation. Fig. 9 (a) to Fig. 9 (c) show that as the number of nodes increases, computation time increases due to the increased complexity of the problem. In addition, it shows that the computation time of our proposed algorithm is shorter than the VAE scheme and longer than the other schemes. The computation time of the direct connect scheme is shortest since it only calculates a single time slot allocation of a defined topology. On the other hand, the computation time of the VAE scheme is the longest since it calculates numerous weights by back propagation. The blue dashed line of Fig. 9 (a) to Fig. 9 (c) indicates the unit simulation time, specified as 20 seconds in this test environment. The result shows that the computation time of our proposed scheme is shorter than the unit simulation time, which means that calculation of the relaying topology of the current frame without accumulating calculation demands is possible. Therefore, a mobility-aware iterative relaying topology algorithm is appropriate to find a sub-optimal relaying topology in the real-world.

As shown in Fig. 8 and Fig. 9, our proposed scheme shows the superior performance compared to the above mentioned conventional schemes. The reason we suggest is an inheritance of a final result, while other conventional schemes are not considered to start calculations from a given topology. However, our proposed scheme can start calculation from the given topology which is the final topology of previous simulation frame in our consideration, enabling the calculation from an advantageous status. If starting computation of the relaying topology from a random distribution is considered, potential area of each node corresponds to a region with a radius of \(R\), with an area of \(R^{2}\pi\). On the other hand, if starting computation from the final result of the previous simulation frame is considered, maximum potential area of each node is \(v_c T_u\) away from the previous location of node, with an area of \((v_c T_u)^2\pi\). Note that \((v_c T_u)^2\pi\) is a maximum value considering a node which locates nearby the edge of simulation circle with a radius R. Therefore, compared to the random distribution, calculation starting from the previous simulation frame has a positional uncertainty of \(({v_c T_u}/{R})^2\) which is about 6.59% in our parameter configuration. Furthermore, since this value pertains to a single node, starting calculations from the final result of previous simulation frame significantly reduces uncertainty in the mobile IoT system with a large number of nodes. Thus, fewer changes in the topology are expected than starting calculations from a random node distribution, leading to a superior performance and reduced computation time.

In this study, we formulated a system model which includes randomly distributed IoT devices and power beacons and assumed a TDMA system and energy harvesting. We proved the validity of iterative balancing time slot allocation algorithm using the KKT condition. We proposed a GMGA which modulates the mutation rate inversely proportional to the link cost for reasonable connection to overcome the limitations of a normal genetic algorithm; an excess of low-performance candidates, such as direct sink connections of low-powered nodes or connections to the distant nodes. We also proposed a mobility-aware iterative relaying topology algorithm which starts the calculation from the final result of the previous frame. To show the validity of our proposed algorithms, we conducted a stationary simulation environment and a mobility simulation environment. We also provided simulation results of our proposed scheme compared with conventional schemes and an exhaustive search scheme, which searches the optimal solution based on brute-force method. Finally, we observed and confirmed that our proposed scheme outperforms other conventional schemes in terms of both performance and computation time, giving us a glimpse of the feasibility of real-world adoption.

Gyupil Kam received a B.S degree in Electrical Engineering from Korea Advanced Institute of Science and Technology(KAIST). Since June 2023, he has been a research officer at the Agency for Defense Development (ADD), South Korea. His research interests include telecommunication, genetic algorithm, internet of things (IoT), wireless networks, and machine learning.

Kiseop Chung received a B.S degree in Electrical and Computer Engineering from Seoul National University, Seoul, South Korea, in 2022. Since June 2022, he has been a research officer at the Agency for Defense Development (ADD), South Korea. His research interests include internet of things (IoT), wireless networks, unsupervised machine learning, embedded systems, and hardware architecture.

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