Sensor clustering using a K-means algorithm in combination with optimized unmanned aerial vehicle trajectory in wireless sensor networks

Citation: Tran, T.-N.; Nguyen, T.-L.; Hoang, V.T.; Voznak, M. Sensor Clustering Using a K-Means Algorithm in Combination with Optimized Unmanned Aerial Vehicle Trajectory in Wireless Sensor Networks. Sensors 2023,23, 2345. https://doi.org/10.3390/s23042345 Academic Editor: Alberto Gotta Received: 3 January 2023 Revised: 23 January 2023 Accepted: 16 February 2023 Published: 20 February 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). sensors Article Sensor Clustering Using a K-Means Algorithm in Combination with Optimized Unmanned Aerial Vehicle Trajectory in Wireless Sensor Networks Thanh-Nam Tran 1,† , Thanh-Long Nguyen 2,*,† , Vinh Truong Hoang 3and Miroslav Voznak 4 1Data Science Laboratory, Faculty of Information Technology, Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam 2Faculty of Information Technology, Ho Chi Minh City University of Food Industry, Ho Chi Minh City 700000, Vietnam 3Faculty of Computer Science, Ho Chi Minh City Open University, Ho Chi Minh City 700000, Vietnam 4Faculty of Electrical Engineering and Computer Science, VSB-Technical University of Ostrava, 17. listopadu 2172/15, 708 00 Ostrava, Czech Republic *Correspondence: [email protected] † These authors contributed equally to this work. Abstract: We examine a general wireless sensor network (WSN) model which incorporates a large number of sensors distributed over a large and complex geographical area. The study proposes solutions for a flexible deployment, low cost and high reliability in a wireless sensor network. To achieve these aims, we propose the application of an unmanned aerial vehicle (UAV) as a flying relay to receive and forward signals that employ nonorthogonal multiple access (NOMA) for a high spectral sharing efficiency. To obtain an optimal number of subclusters and optimal UAV positioning, we apply a sensor clustering method based on K -means unsupervised machine learning in combination with the gap statistic method. The study proposes an algorithm to optimize the trajectory of the UAV, i.e., the centroid-to-next-nearest-centroid (CNNC) path. Because a subcluster containing multiple sensors produces cochannel interference which affects the signal decoding performance at the UAV, we propose a diagonal matrix as a phase-shift framework at the UAV to separate and decode the messages received from the sensors. The study examines the outage probability performance of an individual WSN and provides results based on Monte Carlo simulations and analyses. The investigated results verified the benefits of the K-means algorithm in deploying the WSN. Keywords: wireless sensor network (WSN); unnamed aerial vehicle (UAV); optimal UAV positioning; K-means clustering; gap statistic method; centroid-to-next-nearest-centroid (CNNC) trajectory 1. Introduction Deployments of wireless sensor networks (WSNs) are increasing because of their beneficial applications. For example, WSNs can be deployed to monitor or collect environmental data (meteorological information such as precipitation, wind speed and direction, air pressure, humidity, temperature, etc.) in remote or difficult terrain [ 1 – 6 ]. A major challenge in deploying a WSN is distributing a large number of wireless sensors over a large and complex geographical area. Wireless sensors are generally low cost, have low power consumption and are highly flexible in their application. However, transmitting a signal from a wireless sensor directly to a control centre presents an important challenge [ 7 ], especially if a large number of wireless sensors are deployed to directly collect data. Using terrestrial infrastructure for the purpose of collecting data from wireless sensors is impractical because of high deployment costs and a low flexibility. A potential solution to this problem is using an environmental monitoring system which dispatches an unmanned aerial vehicle (UAV) to a geographical area to retrieve the collected sensor data. Sensors 2023,23, 2345. https://doi.org/10.3390/s23042345 https://www.mdpi.com/journal/sensors Sensors 2023,23, 2345 2 of 25 To deploy a WSN, Heinzelman et al. [ 8 ] proposed a low-energy adaptive clustering hierarchy (LEACH), a clustering method now considered the most well-known clustering protocol for WSNs. In a hierarchical topology, clusters contain two types of node: cluster members and cluster heads. Member nodes are grouped into different clusters, and in each cluster, a single node is designated a cluster head. The cluster head has the most important role in the cluster, tasked with receiving signals from cluster members and forwarding those signals to other cluster heads [9] or the base station [10]. UAVs have gained increasing consideration as aerial relays which deliver mobility and on-demand wireless connections in areas with complex topography and no network coverage. A UAV’s online time, however, is limited by its own on-board energy limitations. The evolution of UAV-assisted WSNs is compelling the scientific community to search for new ways of performing energy harvesting (EH) from external power sources to prolong the online time of UAVs. A variety of effective solutions have been proposed, grouped according to two main types of technique, namely, simultaneous wireless information and power transfer (SWIPT) and techniques for determining the optimal positions for the UAV. Radio frequency EH shows promise as a potential solution for UAV-assisted WSNs. Initial studies on radio frequency EH were used in a technology termed wireless power transfer (WPT) to recharge the wireless sensors in the WSN. A new radiofrequency EH technique, termed SWIPT, introduced significant benefits to WPT [ 11 ]. Many authors have studied SWIPT over the last two decades [ 12 – 15 ], investigating the performance difference between time switching and power splitting in SWIPT protocols [ 16 ]. The current study applied a time-switching protocol because it supports a phase for EH. Applied to all beyond 5G/6G wireless communications, nonorthogonal multiple access (NOMA) provides massive connections, low latency and high reliability [ 14 , 17 – 21 ]. In the current study, we took advantage of NOMA’s benefits, applying NOMA at the UAV to superimpose coding of the data received from wireless sensors and to forward this superimposed signal to a mobile data centre. 1.1. Motivation The use of machine learning in practical applications is escalating. The authors in [6] applied an artificial neural network (ANN) for sensor clustering. By contrast, some wireless sensors in the study in [ 6 ] were clustered as separate single-member clusters. In [ 22 ], the authors proposed a distanceand energy-constrained K -means clustering scheme (DEKCS) for cluster head selection to prolong the lifetime of underwater WSNs. With this new clustering algorithm, a prospective cluster head was selected according to its position in the cluster and its residual battery level. The authors dynamically updated residual energy thresholds set for prospective cluster heads to ensure that the network fully depleted its energy before disconnection. In this manner, cluster heads could be drained of energy and become inactive/dead sensors. The current study applied the K -means algorithm and the gap statistic method first introduced in [ 23 ] to obtain an optimal number of subclusters. To our best knowledge, the gap statistic method has not been applied for WSN clustering in any previous study. In [ 24 ], the authors examined a UAV-assisted data collection WSN. The UAV’s trajectory was optimized by applying the travelling salesman problem. Note that in [ 1 ], the UAV in the proposed network visited every wireless sensor, while in [ 24 ], the optimal serving order for sensors was determined according to a standard travelling salesman problem algorithm, which can be optimally solved with the efficient cutting-plane method (i.e., the shortest path from the start point to the end point). The authors also proposed an algorithm which used the pattern search method to solve the problem of optimizing the UAV position and sensor uploading power. In [ 1 ], the UAV could be exhausted as a consequence of long flight distances. The authors in [ 24 ] used a UAV that navigated the shortest path from the start point to end point, but it consequently ignored/missed some wireless sensors. In another study, the authors addressed the UAV’s trajectory problem by jointly optimizing the UAV’s velocity, hovering positions and visiting sequence [ 25 ]. The Sensors 2023,23, 2345 3 of 25 scientific community is very interested in studying UAVs’ trajectories for the significant potential gains in aerial network performance. Researchers have applied several types of trajectory, for example, straight trajectory [ 26 , 27 ], circular trajectory [ 10 , 28 , 29 ] and spiral trajectory [ 30 – 32 ]. The authors in [ 25 ] introduced an interesting UAV trajectory scheme (Figure 4), where the UAV visited all N monitoring areas and then found suitable positions to transmit the collected data. In [ 25 ], the UAV collected data in four stages: (i) UAV data collection flight, (ii) UAV data collection processing, (iii) UAV data transmission flight and (iv) UAV data transmission processing. That UAV’s operating schedule is illustrated in Figure 5 [25]. The current study proposes the use of a UAV for its high mobility, quick implementation and low cost. The main drawback to a small-sized, lightweight aircraft such as a UAV, however, is its limited on-board energy. UAVs are therefore not suitable for flying close to each sensor to collect data, as proposed in [ 25 ]. The current study therefore investigated the application of a K -means algorithm to cluster wireless sensors into multiple, optimized subclusters. 1.2. Contribution Inspired by the studies mentioned in the previous section, we employed a UAV as an aerial relay to provide a sustainable, functional solution for a WSN. The main contributions of the current study are: • The use of three-dimensional Cartesian coordinates for a WSN which contains a random number of randomly distributed wireless sensors. • The decomposition of the UAV trajectory optimization into two subproblems: (i) the global WSN cluster is divided into multiple subclusters whose number is optimized with unsupervised machine learning which applies K -means clustering in combination with the gap statistic method; (ii) a centroid-to-next-nearest-centroid algorithm is then applied to find the shortest path for travel through every subcluster. • An analysis of the system performance of the WSN over Rayleigh distributions and a presentation of the derived closed-form expressions for the outage probability at the UAV and mobile base station. • Outage probability results for the UAV and mobile base station derived from Monte Carlo simulations and verified with an analysis. The remainder of the paper is organized as follows: Section 2introduces the WSN model, wireless sensor clustering algorithm, joint UAV trajectory, free-space channel modelling, and joint UAV operating schedule; Section 3provides an analysis of the WSN’s performance based on outage probability and presents the closed form expressions for outage probability at the UAV and mobile base station; Section 4examines and plots the investigated results; Section 5discusses conclusions. For clarity, Table 1presents the notation used in the paper. Table 1. List of important notations. Notations Describe Conditions NRandom number of sensors 20 ≤N≤50 K Optimal number of clusters given by the K -means algorithm, where the number of subclusters K is optimized kmin ≤K≤N kmin Snn th sensor node, where a lower value for n has higher priority n={1, . . . , N} CGlobal wireless sensor cluster C⊃Sns.t. ∀n∈N,|C|=N Sensors 2023,23, 2345 4 of 25 Table 1. Cont. Notations Describe Conditions Ckkth subcluster k={1, . . . , K},Nk=|Ck| S(i,k) nn th sensor is i th member of the k th subcluster, where a lower value for ihas higher priority i={1, . . . , NK},NK=|Ck| TGlobal transmission time period T∈Z+ tUAV time period t=mod(T,K)∨K ASn,AU,AB Number of antennae at the sensors, UAV and mobile base station ASn≥1, AU≥1, and AB≥1 εPath-loss exponent factor ε≥2 ˜ CVisited cluster set Updated after the UAV visits the centroid of a subcluster Ckas given by ˜ C←˜ C∪Ck HSn,U,HU,B Precoding fading channel matrices from sensors to the UAV and from the UAV to the mobile base station HSn,U∈CASn×AU and HU,B∈CAU×AB have sizes of ASn×AUand AU×AB, respectively σSn,U,σU,BChannel gains σSn,U=Eh(.,.) Sn,U 2,σU,B=Eh(.,.) U,B 2 αS(i,k) n Power allocation factor for sensor Sn , indexed i th in subcluster CkαS(1,k) n+. . . +αS(Nk,k) n =1 and αS(1,k) n >. . . >αS(Nk,k) n PSn,PU,PB Respective power domains at the sensors, UAV and mobile base station BLet PS1=. . . =PSNdB RPredefined bit-rate threshold for sensors bps/Hz γU−xS(i,k) n ,γB−xS(i,k) n SINR reached at UAV U and B when message xS(i,k) n of sensor Snis decoded SIC decodes the message with the biggest power allocation factor by treating other messages and AWGN as interference RU−xS(i,k) n ,RB−xS(i,k) n Instantaneous bit rate reached at UAV U and mobile base station B when message xS(i,k) n of sensor Sn is decoded bps/Hz OPU,OPB Outage probabilities at UAV U and mobile base station B 0 ≤OPU≤ 1, 0 ≤OPB≤ 1, a lower outage probability result is better performance 2. WSN Model The current study examines a general WSN with a randomly distributed number of wireless sensors. Figure 1depicts a WSN with a random number of sensors N=42 positioned at the Cartesian coordinate (x,y,z) in three dimensions. Let us assume that a mobile base station B is positioned at B(0, 0, 0) and each wireless sensor Sn for n={1, . . . , N} is positioned randomly at coordinate Sn(x,y, 0), where x={0.1, . . . , 1}and y={0.1, . . . , 1} as shown in Table 2. For simplicity, we assume that the wireless sensors and mobile base station are positioned relative to a flat earth. Definition 1. We denote the global set C as containing all wireless sensors. |C| returns N , the total number of wireless sensor nodes (i.e., |C|=N ). Let us assume that the data observations (i.e., wireless sensor positioning) are clustered into K subclusters, i.e., C⊇C1∪. . . ∪CK and N=|C|=|C1|+. . . +|CK|=∑K k=1Nk, where Nk=|Ck|. Figure 1illustrates a random distribution of wireless sensors. Each wireless sensor is allocated a given index by the subscript n , where n={1, . . . , N} and a lower index n has a higher priority. For clarity, sensor Sn has a higher priority than sensor Sn+1 (e.g., sensor S1 has a higher priority than sensor S2). Sensors 2023,23, 2345 5 of 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x-axis 0 0.2 0.4 0.6 0.8 1 y-axis S1 S2S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 S16 S17 S18 S19 S20 S21 S22 S23 S24 S25 S26 S27 S28 S29 S30 S31 S32 S33 S34 S35 S36 S37 S38 S39 S40 S41 S42 Mobile base station Wireless sensors Figure 1. Random positioning of sensor nodes, where 20 ≤N≤50. Table 2. Wireless sensor positions are distributed randomly. Sensors x-Coordinate y-Coordinate Sensors x-Coordinate y-Coordinate S11 0.3 S20.2 0.8 S30.8 0.8 S40.8 0.6 S50.3 0.2 S61 0.5 S70.3 0.7 S80.8 0.3 S90.4 0.6 S10 0.4 0.8 S11 0.4 0.1 S12 1 0.2 S13 0.5 0.2 S14 0.7 0.3 S15 0.9 0.5 S16 0.3 0.4 S17 0.8 0.1 S18 0.9 0.3 S19 0.5 1 S20 0.9 0.8 S21 0.2 0.3 S22 0.6 0.2 S23 0.1 0.1 S24 0.6 0.7 S25 0.7 0.2 S26 0.9 0.2 S27 0.9 0.6 S28 0.7 0.6 S29 0.2 0.7 S30 1 0.9 S31 1 1 S32 0.6 1 S33 0.9 0.4 S34 1 0.4 S35 1 0.7 S36 0.8 0.2 S37 0.5 0.1 S38 0.2 0.9 S39 0.4 0.4 S40 0.5 0.9 S41 0.1 0.7 S42 0.5 0.4 Note: Two or more wireless sensors will never occupy the same position; each position is allocated only one wireless sensor, as illustrated in Figure 1. 2.1. WSN Clustering Remark 1. Because the optimization problem is complex, we propose breaking it down into several subproblems and observing the random distribution of wireless sensors over a large geographical area, as illustrated in Figure 1. The global wireless sensor cluster can be divided into multiple subclusters. The number of subclusters can be optimized by applying the gap statistic method, and the wireless sensors can be assigned to a subcluster using the K -means algorithm. To solve these problems, we propose a solution in Proposition 1. Sensors 2023,23, 2345 6 of 25 Proposition 1. The optimal number of subclusters is yielded as follows: • Observing the latitudes ( x -axis) and longitudes ( y -axis) of the wireless sensors, we determine the optimal number of subclusters K←koptimal . The gap statistic method is applied to the number of subclusters k to compute the corresponding total within the intracluster variation Wk, i.e., the sum of squares function, given by Wk= k ∑ κ=1 1 2|Cκ|∑ Si,Sj∈Cκ dSi,Si, (1) where κ={1, . . . , k} , |Cκ| returns the number of wireless sensor nodes in cluster Cκ and dSi,Sj is the squared Euclidean distance of all pairwise sensor nodes in the cluster Cκ for Si,Sj∈Cκ and i6=j . It is important to note that we assume kmin = 4and kmax =|C| kmin =N kmin , where |C| is the number of observations (or the number of sensors within the global cluster C ). Let us briefly consider factors kmin and kmax . We define kmin = 4to prevent a uniform distribution of wireless sensors positions throughout the area; the gap statistic method thus returns the optimal number of clusters koptimal = 1. For example, Figure 2a,b in [ 23 ] plots the distribution of sensors spread throughout a region and the corresponding optimal number of clusters at K= 1, respectively. However, we also define kmax =N kmin to prevent each wireless sensor owning a private cluster, a problem that would lead to a UAV visiting every wireless sensor to collect data. • Reference data sets Ω with a random uniform distribution are generated. Each reference data set ω of these reference data sets Ω is clustered with a variable number of clusters k={kmin, . . . , kmax} . The corresponding total is computed within the intracluster variation Wκω given in the dispersion metrics for κ={1, . . . , k}and ω={1, . . . , Ω}. • The estimated gap statistic is computed as the deviation of the observed Wk value from its expected value Wκω under the null hypothesis Gap(k)=1 Ω Ω ∑ ω=1 log(Wκω)−log(Wk) . Let l=1 Ω Ω ∑ ω=1 log(Wκω) . The standard deviation (sd) of the statistics is then computed, given by sdk=s1 Ω Ω ∑ ω=1 (log(Wκω)−l)2. • Using the gap statistic method, the smallest value of κ is selected as the optimal number of clusters, the gap statistic being within one standard deviation of the gap statistic at κ+ 1, given koptimal =min{k} and Gap(k)≥Gap(k+1)−θk+1 , where θk+1=sdk+1q1+1 Ω . For example, Figure 2indicates the optimal number of clusters at koptimal = 4, determined by the gap statistic algorithm according to the randomly positioned sensor nodes shown in Figure 1. The K -means algorithm was used to calculate the position for each centroid, with an optimal number of clusters K←koptimal . The computed centroids of four subclusters ( K= 4) are listed in Table 3. Figure 3illustrates all wireless sensor nodes after clustering to K subclusters. After clustering, each sensor node is grouped into a subcluster; for example, S3 1indicates that sensor S1is a member of subcluster C3(Figure 3). Table 3. Centroids after clustering. Centroids x-Axis y-Axis Centroids x-Axis y-Axis C10.38 0.24 C20.3636 0.8 C30.8769 0.3 C40.8875 0.75 Sensors 2023,23, 2345 7 of 25 4 5 6 7 8 9 10 11 Number of Clusters -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Gap Values Optimal number of sub-clusters K=4 Figure 2. Optimized number of subclusters using the gap statistic method, the optimal number of clusters at K=4 satisfying the first maximum standard error. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x-axis 0 0.2 0.4 0.6 0.8 1 y-axis S5 1 S11 1 S13 1 S16 1 S21 1 S22 1 S23 1S37 1 S39 1S42 1 S2 2 S7 2 S9 2 S10 2 S19 2 S24 2 S29 2 S32 2 S38 2S40 2 S41 2 S1 3 S6 3 S8 3 S12 3 S14 3 S15 3 S17 3 S18 3 S25 3S26 3 S33 3S34 3 S36 3 S3 4 S4 4 S20 4 S27 4 S28 4 S30 4 S31 4 S35 4 Cluster C1Cluster C2Cluster C3Cluster C4 Figure 3. Sensor clustering using the K-means algorithm, with optimal number of clusters K=4. Sensors 2023,23, 2345 8 of 25 2.2. Joint UAV Trajectory Definition 2. We introduce a novel joint UAV trajectory algorithm to compute the centroid to next nearest centroid. Problem 1. Operating as a flying relay, the UAV has the advantage of a high mobility and is able to fly close to wireless sensors to receive and forward a superimposed signal. This, however, leads to a long flight path, and the other wireless sensors must wait to be served. We minimized the flight time/path of the UAV according to the cluster centroid positions shown in Table 3. How to obtain the shortest flight path is outlined in Proposition 2. Proposition 2. The centroid-to-next-nearest-centroid trajectory was computed as follows: • Step 1: To determine the nearest centroid from the mobile base station B , we calculate the smallest pairwise Cartesian distance from the mobile base station to each subcluster centroid. • Step 2: The UAV selects the next nearest cluster centroid. In this case, the UAV considers candidate centroids without regard to any of the previously selected cluster centroids in ˜ C . It is important that the centroids contained in the visited set ˜ C be removed from the candidate list to prevent the UAV returning to the previous subcluster ˜ C . The UAV repeats Step 2 (i.e., C\˜ C6=∅) until the list of candidate subclusters is empty (i.e., C\˜ C=∅). Without loss of generality, we examined a single round trip of the UAV. Table 4lists the next nearest subcluster centroids determined from the above selection strategy. The results in Figure 3indicate that subcluster C1 was the nearest to the mobile base station B compared to the other subclusters. Subcluster C1 was therefore selected at block period time T= 1. The UAV visited subcluster C1 first to collect data from all sensor members in subcluster C1 . The visited set ˜ C←˜ C∪C1 was then updated. After all data from the sensor members in subcluster C1 were collected, the UAV selected subcluster C3 because it contained the next nearest subcluster centroid. The UAV then visited subcluster C3 at global time period T= 2 to collect data from all sensor members in the subcluster. The visited set ˜ C←˜ C∪C3 was again updated. The UAV continued to follow this procedure, selecting the next nearest centroid and updating the visited set, until all data have been collected from each subcluster. In this manner, the UAV followed the shortest possible flight path, as shown in Figure 4. After travelling through all K subclusters ( ˜ C≡C ) and collecting all data from wireless sensor members in each subcluster, the UAV’s task was complete and it returned to the mobile base station. For the real-time application of a UAV-assisted WSN, the UAV would repeat the round trips summarized in Table 5. Table 4. Pairwise centroid-to-centroid distance based on Cartesian distances. C1C2C3C4 C10 0.5602 0.5005 0.7195 C30.5005 0.7166 0 0.4501 C40.7195 0.5262 0.4501 0 C20.5602 0 0.7166 0.5262 Observing Table 4, notice that numbers with inclined lines (e.g., 0 ) and numbers with bold (e.g., 0.5005 ) mean visited clusters and next nearest clusters. For clarity, when UAV visited cluster C1 (row with C1 ), the UAV selects the next-nearest cluster (i.e., C3 ) and ignores cluster C1 . Next, the UAV visited cluster C3 (row with C3 ), the UAV selects the next-nearest cluster (i.e., C4 ) and ignores clusters C1 and C3 . The remaining rows in Table 4 have the same meaning. Sensors 2023,23, 2345 9 of 25 Figure 4. Joint UAV trajectory and the shortest path based on the centroid-to-next-nearest-centroid distance given by Algorithm 1(i.e., C1→C3→C4→C2). Algorithm 1 K -means clustering for the optimal number of subclusters and shortest path determined from a centroid to the next nearest centroid Input: Generate a wireless sensor network with a number N of randomly positioned wireless sensors; Output: An optimal number of subclusters Kand subcluster centroids; 1: Initialize variables kmin =4, kmax =N kmin ; 2: Attempt ∀k={kmin, . . . , kmax} to find the optimal number K of subclusters, computed according to Proposition 1; 3: Find the centroid positions for Ksubclusters by applying K-means clustering; 4: Compute the pairwise distances between the mobile base station B and subcluster centroids; 5: Select the nearest centroid and update ˜ C; 6: while C∩˜ C6=|C|do 7: Compute the pairwise distances between the current centroid and other centroids; 8: Select the nearest centroid and then update ˜ C. 9: end while 10: return the number of subclusters K , centroid positions Ck(x,y) for k∈K and the shortest path. Sensors 2023,23, 2345 16 of 25 Algorithm 3 Calculate the outage probability at the mobile base station from (18) for transmission block tover Rayleigh distributions. Input: Initialize the parameters as in Table 1and randomly generate 10 6 samples of each fading channel over a Rayleigh distribution; Output: Simulate (Sim) the results for outage probability at the mobile base station B; 1: for k=1 to the optimal number Kof the subcluster do 2: for i=1 to the number of sensors Nkdo 3: Calculate the SINR at the UAV Ufrom (8); 4: Calculate the achievable maximum instantaneous bit-rate at the UAV U from (9); 5: Calculate the minimum-maximum instantaneous bit-rate threshold at the UAV U from (9); 6: Calculate the SINR at the mobile base station from (13) or (14); 7: Calculate the achievable maximum bit-rate at the mobile base station from (15); 8: Calculate the achievable minimum-maximum bit-rate at the mobile base station from (15); 9: Initialize variable count ←0; 10: for l=1 to 106samples do 11: if  min   min xS(i,k) n ∈Xk max [AS×AU]RU−xS(i,k) n (T,N,K), min xS(i,k) n ∈Xk max [AU×AB]RB−xS(i,k) n (T,N,K)   ≥ R then 12: count ←count +1; 13: end if 14: end for 15: OPB−x(i,k) Sn (T,N,K)=1−count 106; 16: end for 17: OPB(T,N,K)=1 Nk Nk ∑ k=1 OPB−x(i,k) Sn (T,N,K); 18: end for 19: return Dependent outage probability at the mobile base station OPB(T,N,K); Remark 3. The outage probability at the mobile base station in transmission block t is given by (18) from Theorem 1and expressed in novel closed-form as follows: OP(i) B(T,N,K)=max(ASAU ∑ ψ=0 (−1)ψ(ASAU)! ψ!(ASAU−ψ)!exp−ψγ min{ρSnσSn,U}, AUAB ∑ ψ=0 (−1)ψ(AUAB)! ψ!(AUAB−ψ)!exp−ψγ βρUσU,B), (19) s.t. βi=αS(i,k) n−γ Nk ∑ j=i+1 αS(j,k) n, (20) β=min i={1,...,Nk}{βi}, (21) where SINR threshold γ=22R−1. See Appendix Dfor the proof. 4. Numerical Results and Discussion In this section, we examine the individual WSN and discuss the results of the study. For the purposes of the analysis and the Monte Carlo simulations, a random number of wireless sensors N was generated and randomly distributed according to the positions il- Sensors 2023,23, 2345 17 of 25 lustrated in Figure 1. Unless specified otherwise, we assumed that the mobile base station’s position was at coordinate B(0, 0, 0) and that the UAV’s position U(x,y, 1) determined by K -means clustering had a fixed altitude at z= 1. The number of antennae equipped at the wireless sensors, UAV and mobile base station was ASn=AU=AB= 2. The K -means algorithm determined the optimal number of subclusters as K= 4. The path-loss exponent factor was ε= 4. The list of pairwise distances from each subcluster centroid to the mobile base station was dA2G C1,B= 0.494, dA2G C2,B= 0.8788, dA2G C3,B= 0.9268 and dA2G C4,B= 1.1620. The nearest subcluster to the mobile base station was therefore C1 . The UAV selected subcluster C1 for the global period T={1, 5, 9, 13, . . .} . At global period T={2, 6, 10, 14, . . .} , the UAV then selected the next nearest subcluster to its current subcluster C1 , i.e., subcluster C3 , because distances dA2A C1,C3=0.5005 <dA2A C1,C2=0.5602 <dA2A C1,C4=0.7195 . At global period T={3, 7, 11, 15, . . .} , the UAV again selected the next nearest subcluster to subcluster C3 , i.e., subcluster C4 , since distances dA2A C3,C4= 0.4501 <dA2A C3,C4= 0.7166. At global period T={4, 8, 12, 17, . . .} , the UAV selected the next nearest subcluster to subcluster C4 , i.e., subcluster C2 , because the final distances dA2A C4,C2= 0.5262. The UAV thus selected the shortest trajectory C1→C3→C4→C2 . Without loss of generality and for simplicity, we assumed that λ1=λ2=λ3=t 3 for a single round trip of the UAV and global period T={1, 2, 3, 4}. 4.1. Numerical Results Figure 7a–d plot the outage probabilities at the UAV at the point when it decoded the received signals from wireless sensors in subclusters C1 , C3 , C4 and C2 , respectively. The bit-rate threshold for all wireless sensors was R= 1.5 bps/Hz. The outage probabilities at the majority of wireless sensors were very similar as SNR ρSn→∞ ; however, the graphs in Figure 7a indicate that the outage probability of sensor S23 was worse than the outage probability at the other sensors of the same subcluster C1 . Figure 3indicates that wireless sensor S23 was the farthest from the subcluster centroid C1 ( dG2A S(7,1) 23 = 1.0479). The results verified the efficiency of the K -means algorithm. It is important that the bit-rate threshold R for the wireless sensors was set to R= 1.5 bps/Hz. However, the UAV successfully decoded most of the messages from the wireless sensors, achieving a high outage probability performance (Figure 7). We conclude that the outage probability performance of the majority of wireless sensors in each subcluster was equal since they were evenly distributed around the subcluster’s centroid. The Monte Carlo simulations given by (16) were also verified by the analysis results given by (17). Next, we examined the results for the mobile base station and obtained its outage probability performance at the points when the UAV visited subclusters C1 , C3 , C4 and C2 and forwarded the superimposed signals, given by (11) , to the base station (Figure 8a–d) . The outage probability performance of the mobile base station was poorer than the outage probability performance at the UAV (Figure 7a–d), even though the bit-rate threshold was set to R= 0.1 bps/Hz. This may have been because the UAV was deployed with NOMA and therefore, the sensors were forced to share the power domain to transmit the messages in the superimposed signal. This means that the last member in the subcluster was allocated a very small power allocation factor, given by (12) . These power allocation factors are presented in Table 6. Subcluster C3 contained N3= 13 wireless sensors, and the last wireless sensor in C3 ( S(13,3) 36 ) was allocated the lowest power allocation factor ( αS(13,3) 36 ). Therefore, despite a global optimization of the subclusters and the positions of the cluster centroids by the K -means algorithm, the large number of wireless sensors in the subcluster unfortunately led to unsatisfactory results. Sensors 2023,23, 2345 18 of 25 0 1 2 3 4 5 6 7 8 9 10 SNR (dB) 10-1 100 Outage Probability (Sim) Sensor S5(Sim) Sensor S11 (Sim) Sensor S13 (Sim) Sensor S16 (Sim) Sensor S21 (Sim) Sensor S22 (Sim) Sensor S23 (Sim) Sensor S37 (Sim) Sensor S39 (Sim) Sensor S42 (Ana) Sensor S5(Ana) Sensor S11 (Ana) Sensor S13 (Ana) Sensor S16 (Ana) Sensor S21 (Ana) Sensor S22 (Ana) Sensor S23 (Ana) Sensor S37 (Ana) Sensor S39 (Ana) Sensor S42 (a) 0 1 2 3 4 5 6 7 8 9 10 SNR (dB) 10-1 100 Outage Probability (Sim) Sensor S1(Sim) Sensor S6 (Sim) Sensor S8(Sim) Sensor S12 (Sim) Sensor S14 (Sim) Sensor S15 (Sim) Sensor S17 (Sim) Sensor S18 (Sim) Sensor S25 (Sim) Sensor S26 (Sim) Sensor S33 (Sim) Sensor S34 (Sim) Sensor S36 (Ana) Sensor S1 (Ana) Sensor S6(Ana) Sensor S8 (Ana) Sensor S12 (Ana) Sensor S14 (Ana) Sensor S15 (Ana) Sensor S17 (Ana) Sensor S18 (Ana) Sensor S25 (Ana) Sensor S26 (Ana) Sensor S33 (Ana) Sensor S34 (Ana) Sensor S36 (b) 0 1 2 3 4 5 6 7 8 9 10 SNR (dB) 10-1 100 Outage Probability (Sim) Sensor S3(Sim) Sensor S4 (Sim) Sensor S20 (Sim) Sensor S27 (Sim) Sensor S28 (Sim) Sensor S30 (Sim) Sensor S31 (Sim) Sensor S35 (Ana) Sensor S3(Ana) Sensor S4 (Ana) Sensor S20 (Ana) Sensor S27 (Ana) Sensor S28 (Ana) Sensor S30 (Ana) Sensor S31 (Ana) Sensor S35 (c) 0 1 2 3 4 5 6 7 8 9 10 SNR (dB) 10-1 100 Outage Probability (Sim) Sensor S2(Sim) Sensor S7 (Sim) Sensor S9(Sim) Sensor S10 (Sim) Sensor S19 (Sim) Sensor S24 (Sim) Sensor S29 (Sim) Sensor S32 (Sim) Sensor S38 (Sim) Sensor S40 (Sim) Sensor S41 (Ana) Sensor S2 (Ana) Sensor S7(Ana) Sensor S9 (Ana) Sensor S10 (Ana) Sensor S19 (Ana) Sensor S24 (Ana) Sensor S29 (Ana) Sensor S32 (Ana) Sensor S38 (Ana) Sensor S40 (Ana) Sensor S41 (d) Figure 7. Outage probability at the UAV for the UAV’s subcluster trajectory sequence ( a ) C1 , ( b ) C3 , (c)C4and (d)C2. 0 1 2 3 4 5 6 7 8 9 10 SNR (dB) 10-2 10-1 100 Outage Probability (Sim) Sensor S5(Sim) Sensor S11 (Sim) Sensor S13 (Sim) Sensor S16 (Sim) Sensor S21 (Sim) Sensor S22 (Sim) Sensor S23 (Sim) Sensor S37 (Sim) Sensor S39 (Sim) Sensor S42 (Ana) Sensor S5(Ana) Sensor S11 (Ana) Sensor S13 (Ana) Sensor S16 (Ana) Sensor S21 (Ana) Sensor S22 (Ana) Sensor S23 (Ana) Sensor S37 (Ana) Sensor S39 (Ana) Sensor S42 (a) 0 1 2 3 4 5 6 7 8 9 10 SNR (dB) 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Outage Probability (Sim) Sensor S1(Sim) Sensor S6 (Sim) Sensor S8(Sim) Sensor S12 (Sim) Sensor S14 (Sim) Sensor S15 (Sim) Sensor S17 (Sim) Sensor S18 (Sim) Sensor S25 (Sim) Sensor S26 (Sim) Sensor S33 (Sim) Sensor S34 (Sim) Sensor S36 (Ana) Sensor S1 (Ana) Sensor S6(Ana) Sensor S8 (Ana) Sensor S12 (Ana) Sensor S14 (Ana) Sensor S15 (Ana) Sensor S17 (Ana) Sensor S18 (Ana) Sensor S25 (Ana) Sensor S26 (Ana) Sensor S33 (Ana) Sensor S34 (Ana) Sensor S36 (b) Figure 8. Cont. Sensors 2023,23, 2345 19 of 25 0 1 2 3 4 5 6 7 8 9 10 SNR (dB) 10-1 100 Outage Probability (Sim) Sensor S3(Sim) Sensor S4 (Sim) Sensor S20 (Sim) Sensor S27 (Sim) Sensor S28 (Sim) Sensor S30 (Sim) Sensor S31 (Sim) Sensor S35 (Ana) Sensor S3(Ana) Sensor S4 (Ana) Sensor S20 (Ana) Sensor S27 (Ana) Sensor S28 (Ana) Sensor S30 (Ana) Sensor S31 (Ana) Sensor S35 (c) 0 1 2 3 4 5 6 7 8 9 10 SNR (dB) 10-1 100 Outage Probability (Sim) Sensor S2(Sim) Sensor S7 (Sim) Sensor S9(Sim) Sensor S10 (Sim) Sensor S19 (Sim) Sensor S24 (Sim) Sensor S29 (Sim) Sensor S32 (Sim) Sensor S38 (Sim) Sensor S40 (Sim) Sensor S41 (Ana) Sensor S2 (Ana) Sensor S7(Ana) Sensor S9 (Ana) Sensor S10 (Ana) Sensor S19 (Ana) Sensor S24 (Ana) Sensor S29 (Ana) Sensor S32 (Ana) Sensor S38 (Ana) Sensor S40 (Ana) Sensor S41 (d) Figure 8. Outage probability at the mobile base station for the UAV’s subcluster trajectory sequence (a)C1, (b)C3, (c)C4and (d)C2. 4.2. Discussion The outage probability performance at the mobile base station was strongly affected by several factors, such as the UAV’s transmit power PU , the distance of the UAV from the mobile base station dU,B and the number Nk of messages transmitted in the superimposed signals. The UAV was not able to increase the transmit power PU , however, because of its power limitations. A large number of antennae at both the wireless sensors and the UAV could not be equipped since the constraints for a small size, light weight and low cost did not permit it. It was also not possible to reduce the distance from the UAV to the mobile base station because of obstructions in the terrain. To address these conditions, we equipped a larger number of antennae at the mobile base station, as the mobile base station incorporated a generator and energy was not a significant problem. Therefore, equipping AB= 32 antennae at the mobile base station instead of the same number at the UAV ASn=AU=AB= 2 (Figure 8a–d) yielded the results in Figure 9a–d. It is clear that the outage probability improved significantly at the mobile base station for AB= 32 while ASn=AU= 2. It is also clear that the outage probability performance at the mobile base station when the UAV visited subcluster C1 improved greatly since this subcluster C1 was closer than the other subclusters. The outage probabilities at the other subclusters also improved as the SNR ρU→∞. 0 1 2 3 4 5 6 7 8 9 10 SNR (dB) 10-3 10-2 10-1 100 Outage Probability (Sim) Sensor S5(Sim) Sensor S11 (Sim) Sensor S13 (Sim) Sensor S16 (Sim) Sensor S21 (Sim) Sensor S22 (Sim) Sensor S23 (Sim) Sensor S37 (Sim) Sensor S39 (Sim) Sensor S42 (Ana) Sensor S5(Ana) Sensor S11 (Ana) Sensor S13 (Ana) Sensor S16 (Ana) Sensor S21 (Ana) Sensor S22 (Ana) Sensor S23 (Ana) Sensor S37 (Ana) Sensor S39 (Ana) Sensor S42 (a) 0 1 2 3 4 5 6 7 8 9 10 SNR (dB) 10-3 10-2 10-1 100 Outage Probability (Sim) Sensor S1(Sim) Sensor S6 (Sim) Sensor S8(Sim) Sensor S12 (Sim) Sensor S14 (Sim) Sensor S15 (Sim) Sensor S17 (Sim) Sensor S18 (Sim) Sensor S25 (Sim) Sensor S26 (Sim) Sensor S33 (Sim) Sensor S34 (Sim) Sensor S36 (Ana) Sensor S1 (Ana) Sensor S6(Ana) Sensor S8 (Ana) Sensor S12 (Ana) Sensor S14 (Ana) Sensor S15 (Ana) Sensor S17 (Ana) Sensor S18 (Ana) Sensor S25 (Ana) Sensor S26 (Ana) Sensor S33 (Ana) Sensor S34 (Ana) Sensor S36 (b) Figure 9. Cont. Sensors 2023,23, 2345 20 of 25 0 1 2 3 4 5 6 7 8 9 10 SNR (dB) 10-3 10-2 10-1 100 Outage Probability (Sim) Sensor S3(Sim) Sensor S4 (Sim) Sensor S20 (Sim) Sensor S27 (Sim) Sensor S28 (Sim) Sensor S30 (Sim) Sensor S31 (Sim) Sensor S35 (Ana) Sensor S3(Ana) Sensor S4 (Ana) Sensor S20 (Ana) Sensor S27 (Ana) Sensor S28 (Ana) Sensor S30 (Ana) Sensor S31 (Ana) Sensor S35 (c) 0 1 2 3 4 5 6 7 8 9 10 SNR (dB) 10-3 10-2 10-1 100 Outage Probability (Sim) Sensor S2(Sim) Sensor S7 (Sim) Sensor S9(Sim) Sensor S10 (Sim) Sensor S19 (Sim) Sensor S24 (Sim) Sensor S29 (Sim) Sensor S32 (Sim) Sensor S38 (Sim) Sensor S40 (Sim) Sensor S41 (Ana) Sensor S2 (Ana) Sensor S7(Ana) Sensor S9 (Ana) Sensor S10 (Ana) Sensor S19 (Ana) Sensor S24 (Ana) Sensor S29 (Ana) Sensor S32 (Ana) Sensor S38 (Ana) Sensor S40 (Ana) Sensor S41 (d) Figure 9. Improved outage probability at the mobile base station equipped with AB= 32 antennae for the UAV trajectory subcluster sequence (a)C1, (b)C3, (c)C4and (d)C2. 5. Conclusions This study presented a general WSN containing a randomly distributed number of wireless sensors with three-dimensional Cartesian coordinates. To improve the WSN’s performance, we applied a K -means algorithm and gap statistic method to optimize sensor clustering into a number of subclusters K . The UAV’s trajectory was calculated with an algorithm which determined the shortest path between the subcluster centroids. The aims of the study were achieved (i.e., flexible deployment, low cost and high reliability) through the effective proposed solutions, and the results were verified with both Monte Carlo simulations and theoretical analysis. Although the study provided some benefits from the application of the K -means algorithm for wireless sensor clustering, some problems still persisted that can be studied in future work. Future studies can investigate the problems with (1) fragmented power resources created by an imbalance in the number of subcluster sensors and (2) some clusters covering a larger geographic area than others as a result of sparsely distributed sensors. As a potential solution, we propose dividing the network into larger clusters when the number of sensors reaches a certain threshold. Author Contributions: T.-N.T. and T.-L.N. proposed ideas and formulation of overarching research aims, designed of WSN model, applied mathematical to analyze the proposed model, wrote the initial draft. V.T.H. contributed to preparation and presentation of the published work by those from the original research group, specifically critical review, commentary and revision—including preand post-publication stages. M.V. contributed to evolution of overarching research goals, management activities to annotate, oversight and leadership responsibility for the research activity planning and execution, including mentorship external to the core team, acquisition of the financial support for the project leading to this publication. All authors have read and agreed to the published version of the manuscript. Funding: This research received funding from the Ministry of Education, Youth and Sports under grant Reg. No. SP2021/25 and partially under the Large Infrastructures for Research, Experimental Development and Innovations project Reg. No. LM2018140. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data used in this study were randomly generated. Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design of the study, the collection, analysis and interpretation of data, writing of the manuscript, or the decision to publish the results. Sensors 2023,23, 2345 21 of 25 Abbreviations A2A air-to-air A2G air-to-ground AWGN additive white Gaussian noise CDF cumulative distribution function CSI channel state information EH energy harvesting FR flying relay G2A ground-to-air MIMO multi-input multioutput NOMA nonorthogonal multiple access PDF probability density function SIC successive interference cancellation SINR signal-to-interference-plus-noise ratio SNR signal-to-noise ratio SWIPT simultaneous wireless information and power transfer UAV unmanned aerial vehicle WPT wireless power transfer WSN wireless sensor network Appendix A The probability density function (PDF) and cumulative distribution function (CDF) of the Rayleigh distribution are expressed, respectively, as: f|hsrc,des|2(x)=1 σsrc,des exp−x σsrc,des , (A1) and F|hsrc,des|2(x)=1−exp−x σsrc,des , (A2) where hsrc,des 2 are random independent variables, i.e., x in (A1) and (A2) . In addition, σsrc,des is the expected channel gain, where σsrc,des =Ehhsrc,des 2i between the source (src) and destination (des). Appendix B We studied an individual WSN (Figure 1) which contained an optimal number of subclusters K=4 (Figure 2) and wireless sensors allocated according to those subclusters (Figure 3). A UAV travelled the shortest path from each subcluster centroid through all subclusters (Figure 4). The results for the outage probability at the UAV indicated a high performance (Figure 7a–d); however, the outage probability at the mobile base station was comparatively poorer (Figure 8a–d). We hypothesized that the MIMO technique could improve the OP performance and therefore equipped a larger number of antennas at the mobile base station ( AB= 32); the results showed a significant improvement in outage probability at the mobile base station (Figure 8a,c,d), except when the mobile base station decoded the message of the last sensor S(13,3) 36 of subcluster C3 , which demonstrated a worse performance (Figure 8b), probably because the subcluster contained |C3|= 13 sensors. Another individually generated WSN (Figure A1a) contained an optimal number of K= 10 subclusters (Figure A1b) and wireless sensors allocated according to those subclusters (Figure A1c). This configuration also contained an optimal UAV trajectory (Figure A1d). It is important to note that the subclusters depicted in Figure A1c contained fewer sensors than the subclusters in Figure 3. The wireless sensors depicted in Figure A1c were consequently better served than the wireless sensors in Figure 3. However, fewer sensors in each subcluster, and thus a greater number of subclusters, led to a longer UAV flight path. Sensors 2023,23, 2345 22 of 25 For example, the results in Figure 4indicate that the UAV’s period to visit all subclusters was t= 4; the results in Figure A1d, however, indicate a UAV period of t= 10. We conclude that fewer sensors in each subcluster deliver better results, but at the expense of the UAV consuming more energy to travel longer distances. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x-axis 0 0.2 0.4 0.6 0.8 1 y-axis S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 S16 S17 S18 S19 S20 S21 S22 S23 S24 S25 S26 S27 S28 S29 S30 S31 S32 S33 S34 S35 S36 S37 S38 S39 S40 Mobile base station Wireless sensors (a) 4 5 6 7 8 9 10 Number of Clusters -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 Gap Values Optimal number of sub-clusters K=10 (b) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x-axis 0 0.2 0.4 0.6 0.8 1 y-axis S3 1S12 1 S17 1 S5 2 S27 2S35 2 S10 3 S25 3 S8 4 S22 4 S23 4 S36 4 S15 5 S24 5S29 5 S40 5 S2 6 S14 6 S18 6 S19 6 S21 6 S7 7 S13 7 S28 7 S1 8 S33 8S38 8 S20 9 S30 9 S31 9 S32 9S37 9 S4 10 S6 10 S9 10 S11 10 S16 10 S26 10 S34 10 S39 10 Cluster C1 Cluster C2 Cluster C3 Cluster C4 Cluster C5 Cluster C6 Cluster C7 Cluster C8 Cluster C9 Cluster C10 (c) (d) Figure A1. The randomly distributed WSN ( a ), determined optimal number of subclusters K ( b ), division into subclusters (c) and centroid-to-next-nearest-centroid trajectory (d). Appendix C By substituting the SINR given by (8) into (9) and then substituting (9) into Theorem 1, as given (16), we thus obtain the independent outage probability at the UAV: OPU−x S(i,k) n =1−Prmaxn|HSn,U|2o≥γ ρSn. (A3) From the precoding matrix |HSn,U|2given (2) and the PDF given by (A1), we obtain OPU−x S(i,k) n =1−  1− ASnAU ∑ ψ=0 (−1)ψASnAU ψ+∞ Z γ/ρSn 1 σSn,U exp−ψx σSn,Udx   = ASnAU ∑ ψ=0 (−1)ψ(ASnAU)! ψ!(ASnAU−ψ)!exp−ψγ ρSnσSn,U. (A4) Note that Equation (A4) evaluates the independent outage probability at the UAV at the point when the UAV decodes the received signal from wireless sensor S(i,k) n in Sensors 2023,23, 2345 23 of 25 subcluster Ck unsuccessfully. The outage probability at the UAV is generally expressed as OPU(T,N,K)=1 Nk Nk ∑ k=1 OPU−x(i,k) Sn (T,N,K). Appendix D Expression (18) is rewritten as follows: OPB(T,K,N)=1−Pr   min xS(i,k) n ∈Xk max [ASn×AU]RU−xS(i,k) n (T,N,K)≥ R, min xS(i,k) n ∈Xk max [AU×AB]RB−xS(i,k) n (T,N,K)≥ R   =max   1−Pr   min xS(i,k) n ∈Xk max [ASn×AU]RU−xS(i,k) n (T,N,K)≥ R   , 1−Pr   min xS(i,k) n ∈Xk max [AU×AB]RB−xS(i,k) n (T,N,K)≥ R     . (A5) Let ϑ= 1 −Pr   min xS(i,k) n ∈Xk max [ASn×AU]RU−xS(i,k) n (T,N,K)≥ R   . By substituting the SINR given by (8) into (9) and then substituting (9) into ϑ, we obtain ϑ=1−     1−min i={1,...,Nk}          ASnAU ∑ ψ=0 (−1)ψASnAU ψ+∞ Z γ/ρsn 1 σS(i,k) n,U exp −ψx σS(i,k) n,U!dx              =min i={1,...,Nk}          ASnAU ∑ ψ=0 (−1)ψ(ASnAU)! ψ!(ASnAU−ψ)! +∞ Z γ/ρsn 1 σS(i,k) n,U exp −ψx σS(i,k) n,U!dx         = ASnAU ∑ ψ=0 (−1)ψ(ASnAU)! ψ!(ASnAU−ψ)!exp    −ψγ ρsnmin i={1,...,Nk}nσS(i,k) n,Uo    . (A6) Similarly, let ξ= 1 −Pr   min xS(i,k) n ∈Xk max [AU×AB]RB−xS(i,k) n (T,N,K)≥ R   . By substituting the SINR given by (13) or (14) into (15) and then substituting (15) into ξ , we obtain Sensors 2023,23, 2345 24 of 25 ξ=1−Pr   min xS(i,k) n ∈Xk max [AU×AB]RB−xS(i,k) n (T,N,K)≥ R   =1−           1−min i={1,...,Nk}                    AUAB ∑ ψ=0 (−1)ψAUAB ψ+∞ Z γ,ρU αS(i,k) n−γ Nk ∑ j=i+1 αS(j,k) n! 1 σU,B exp−ψx σU,Bdx                              = AUAB ∑ ψ=0 (−1)ψ(AUAB)! ψ!(AUAB−ψ)!exp      −ψγ ρUmin i={1,...,Nk}(αS(i,k) n−γ Nk ∑ j=i+1 αS(j,k) n)σU,B       . (A7) From expression (A5) , we conclude that the outage probability at the mobile base station is independent since it belongs to the outage probability at UAV. The study verified that ϑ<ξ and therefore, the outage probability at the mobile base station given by (A5) refers to OPB(T,N,K)=max{ϑ,ξ}=ξ . Observing the individual WSN model depicted in Figure 4, a possible explanation is that the UAV was close to the wireless sensors, which thus strongly owned the channel state information (CSI) hSn,U , even though they simply transmitted their own messages xSn under their own power domain PSn . However, the UAV travelled a long distance since the subcluster was far from the mobile base station; therefore, it forwarded messages in the superimposed signal by sharing the power domain PU , and thus OPB(T,N,K)=max{ϑ,ξ}=ξ . To improve OPB(T,N,K) , both ϑ and ξ must be improved. To improve ϑ , the number of antennae at the wireless sensors or their transmit power must be increased. However, wireless sensors have certain constraints, such as having a low cost and low power. These solutions are therefore not practical or even obtainable. To address these conditions, we attempted to improve ξ by increasing the number of antennae at the mobile base station ( AB= 32). 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