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Lifetime analysis and improvement of an amplify-and-forward wireless relay network Li, Shuotong 2014

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Lifetime Analysis and Improvement of anAmplify-and-Forward Wireless Relay NetworkbyShuotong LiA THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Electrical and Computer Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)September 2014c© Shuotong Li, 2014iiAbstractIn a wireless relay network, the source and the destination are not able to communicatedirectly with each other because they are out of radio communication range. Instead,they communicate with the aid of intermediate node(s) which relay the signals.In the first part of the thesis, we analyze the lifetime distribution in a variable gainamplify-and-forward (VG-AF) opportunistic wireless relay network (OWRN) in the pres-ence of Rayleigh fading. Two different methods are proposed to derive the probabilitydensity function (pdf) of the network lifetime. The first method is the Pearson method,which is useful in obtaining approximate expressions for probability distributions. Usinga relatively short simulation for the network lifetime of interest, we obtain estimates ofthe first four moments of the network lifetime. Based on these moments, a fairly accurateapproximation of the lifetime distribution is derived. The second method is based on thecentral limit theorem (CLT). Based on the fact that there are sufficient transmissions inone lifetime of a network, the lifetime distribution is close to normal distribution. Toverify our methods, we use large Monte Carlo simulations to obtain good approximationsfor the network lifetime distribution.In the second part of the thesis, we analyze the lifetime distribution in the presenceiiiof Weibull fading as well as Nakagami fading. First, exact outage probability expressionsfor both cases with the opportunistic amplify-and-forward relaying strategy are derived.Simulations results verify our theoretical analysis. Using the techniques in the first part,we then obtain the lifetime distributions for both cases with the methods used in thefirst part of the thesis.In the third part of the thesis, an algorithm based on an energy saving adaptivetransmit power threshold is proposed to improve the relay network lifetime. Calculatedbased on the residual energy information, QoS requirements and the number of relays,this transmit power threshold is adaptive to the residual energy value of each relay.Simulation results verify that this algorithm prolongs the lifetime of the relay networkcompared with a number of other existing relay selection strategies while satisfying theQoS requirements.ivPrefaceThis thesis is original, unpublished, independent work by the author, Shuotong Li.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Wireless Relay Network . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Wireless Sensor Network . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Network Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Relay Selection Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 WRN Lifetime Distribution under Rayleigh Fading . . . . . . . . . . . 7vi2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Approximating Network Lifetime PDF Using the Pearson Method . . . . 142.4.1 Pearson Systems and Monte Carlo Methods . . . . . . . . . . . . 152.4.2 Lifetime Analysis with Pearson Method . . . . . . . . . . . . . . . 162.4.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . 162.5 Approximating Network Lifetime PDF Using the Central Limit Theorem 202.5.1 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . 222.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 WRN Lifetime Distribution under Weibull and Nakagami Fading . . 323.1 Motivations and Contributions . . . . . . . . . . . . . . . . . . . . . . . . 323.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.3 System Outage Probability Analysis . . . . . . . . . . . . . . . . . . . . . 343.3.1 Weibull Fading Channel Analysis . . . . . . . . . . . . . . . . . . 343.3.2 Nakagami Fading Channel Analysis . . . . . . . . . . . . . . . . . 373.4 Lifetime Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 An Adaptive Relay Transmit Power Scheme for Improving Lifetime 504.1 Motivations and Contributions . . . . . . . . . . . . . . . . . . . . . . . . 50vii4.2 Adaptive Relay Transmit Power Limited Algorithm . . . . . . . . . . . . 524.3 Optimal Selection Algorithm Analysis . . . . . . . . . . . . . . . . . . . . 544.4 Performance Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . 675.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70viiiList of Figures2.1 System model of an AF WRN with K relays. . . . . . . . . . . . . . . . 112.2 Amplify-and-forward opportunistic scheme for our system model. . . . . 132.3 Network lifetime pdf with Rayleigh fading, K = 3, e = [10, 10, 10] J. (a)Case 1: Nsamples = 104. (b) Case 2: Nsamples = 107. . . . . . . . . . . . . 172.4 Network lifetime pdf with Rayleigh fading, K = 5, e = [10, 10, 10, 10, 10]J. (a) Case 1: Nsamples = 104. (b) Case 2: Nsamples = 107. . . . . . . . . . 182.5 Network lifetime pdf with Rayleigh fading, K = 3, e = [30, 30, 30] J. (a)Case 1: Nsamples = 104. (b) Case 2: Nsamples = 107. . . . . . . . . . . . . 192.6 Network lifetime pdf with Rayleigh fading, K = 3, e = [5, 10, 15] J. (a)Case 1: Nsamples = 104. (b) Case 2: Nsamples = 107. . . . . . . . . . . . . 202.7 Ratio of total residual energy to total initial energy at all relays as afunction of total initial energy at all relays with K = 3, K = 4, K = 5, K= 10. All the relays have equal initial energy. . . . . . . . . . . . . . . . 24ix2.8 Ratio of total residual energy to total initial energy at all relays as afunction of total initial energy at all relays with K = 3. (a) Case 1:e = [e, e, e], (b) Case 2: e = [0.9e, e, 1.1e]. (c) Case 3: e = [0.8e, e, 1.2e].(d) Case 4: e = [0.7e, e, 1.3e], where e = E0/3. . . . . . . . . . . . . . . . 252.9 Lifetime distribution for WRN under Rayleigh fading channels with K =3, initial relay energy e = [50, 50, 50] J. . . . . . . . . . . . . . . . . . . . 272.10 Lifetime distribution for WRN under Rayleigh fading channels with K =5, initial relay energy e = [50, 50, 50, 50, 50] J. . . . . . . . . . . . . . . . 282.11 Lifetime distribution for WRN under Rayleigh fading channels with K =3, initial relay energy e = [25, 50, 75] J. . . . . . . . . . . . . . . . . . . . 292.12 Lifetime distribution for WRN under Rayleigh fading channels with K =3, initial relay energy e = [9, 9, 9] kJ. . . . . . . . . . . . . . . . . . . . . 302.13 Computer time required to obtain the pdf using the Pearson method, theCLT method and pure simulation. K = 3 and relays have equal initialenergy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.1 Simulation and theoretical results (3.7) for the single path outage proba-bility against relay transmit power under Weibull fading (Case 1: ω1k =ω2k = 1, n1k = n2k = 2, Case 2: ω1k = ω2k = 1, n1k = n2k = 3). . . . . . . 383.2 Simulation and theoretical results for the single path outage probabilityagainst relay transmit energy under Nakagami fading (Case 1: λ1k = λ2k =1, m1k = m2k = 1, Case 2: λ1k = λ2k = 1, m1k = m2k = 2). . . . . . . . . 40x3.3 Lifetime distribution for Weibull fading (Ω1k = Ω2k = 1, m1k = m2k = 3)with K = 3 and e0 = 50 J using the Pearson and CLT methods . . . . . 423.4 Lifetime distribution for Weibull fading (Ω1k = Ω2k = 1, m1k = m2k = 3)with K = 5 and e0 = 50 J using the Pearson and CLT methods . . . . . 433.5 Lifetime distribution for Weibull fading (Ω1k = Ω2k = 1, m1k = m2k = 3)with K = 7 and e0 = 50 J using the Pearson and CLT methods . . . . . 443.6 Lifetime distribution for Weibull fading (Ω1k = Ω2k = 1, m1k = m2k = 3)with K = 3 and e0 = 10 J using the Pearson and CLT methods . . . . . 453.7 Lifetime distribution for Nakagami fading (λ1k = λ2k = 1, m1k = m2k = 2)with K = 3 and e0 = 50 J using the Pearson and CLT methods . . . . . 463.8 Lifetime distribution for Nakagami fading (λ1k = λ2k = 1, m1k = m2k = 2)with K = 5 and e0 = 50 J using the Pearson and CLT methods . . . . . 473.9 Lifetime distribution for Nakagami fading (λ1k = λ2k = 1, m1k = m2k = 2)with K = 7 and e0 = 50 J using the Pearson and CLT methods . . . . . 483.10 Lifetime distribution for Nakagami fading (λ1k = λ2k = 1, m1k = m2k = 2)with K = 3 and e0 = 10 J using the Pearson and CLT methods . . . . . 494.1 Stochastic shortest path of relaying scheme for WRN with two relays. . . 564.2 Network lifetime with Rayleigh fading channels, two relays, each with equalinitial energy, end-to-end SNR threshold of 0.25 and L = 8 (10, 20,..., 80mW). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57xi4.3 Average network lifetime with Rayleigh fading, two relays, end-to-end SNRthreshold of 0.25 and L = 8 (10, 20,..., 80 mW). Case 1: e = [e/6, 5e/6],Case 2: e = [e/4, 3e/4], Case 3: e = [e/3, 2e/3], Case 1: e = [e/2, e/2],where e = E0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.4 Running time for optimal selection algorithm as a function of total initialenergy E0 with three relays, end-to-end SNR threshold of 0.25 and L = 8transmit power level L = 8 (10, 20,..., 80 mW) (Case 1: e = [e, e, e], Case2: e = [0.5e, e, 1.5e], where e = E0/3). . . . . . . . . . . . . . . . . . . . . 594.5 Running time for optimal selection algorithm as a function of transmitpower levels with three relays, end-to-end SNR threshold of 0.25 and initialenergy of 5000 mJ at each relay. For transmit power level L = 1 (80 mW),L = 2 (40, 80 mW), L = 4 (20, 40, 60, 80 mW) and L = 8 (10, 20,..., 80mW). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.6 Running time for optimal selection algorithm as a function of relay num-bers with initial energy of 3000 mJ at each relay, end-to-end SNR thresholdof 0.25 and transmit power level L = 8 (10, 20,..., 80 mW). . . . . . . . . 614.7 Average lifetime for five relaying schemes (MTP, MOP, Fixed Psoft, Adap-tive Psoft and optimal selection) with K = 3 and η = 0.15. . . . . . . . . 634.8 Average lifetime for five relaying schemes (MTP, MOP, Fixed Psoft, Adap-tive Psoft and optimal selection) with K = 5 and η = 0.15. . . . . . . . . 64xii4.9 Average lifetime for five relaying schemes (MTP, MOP, Fixed Psoft, Adap-tive Psoft and optimal selection) with K = 3 and η = 0.2. . . . . . . . . . 654.10 Average Poutage of four relaying schemes (MTP, MOP, Fixed Psoft andAdaptive Psoft) with K = 3 and η = 0.15. . . . . . . . . . . . . . . . . . 66xiiiList of AcronymsWRN Wireless Relay NetworkCSI Channel State InformationREI Residual Energy InformationWSN Wireless Sensor NetworkAF Amplify-and-ForwardFG-AF Fixed-Gain Amplify-and-ForwardVG-AF Variable-Gain Amplify-and-ForwardSNR Signal-to-Noise-RatioSOP System Outage ProbabilityQoS Quality of ServiceMTP Minimum Transmission PowerMRE Maximum Residual EnergyxivMEI Maximal Energy-efficiency IndexMOP Minimum Outage ProbabilityPMF Probability Mass FunctionPDF Probability Density FunctionCLT Central Limit TheoremAWGN Additive White Gaussian NoisexvAcknowledgementsI owe my deepest gratitude to my research supervisor, Dr. Cyril Leung, whose encour-agement, guidance and support made my graduate study productive and stimulating. Iappreciate the time we spent together at UBC. Without his help, this thesis would nothave been possible. Special thanks to my senior, Seyed Ali Mousavifar, for sharing hisexpertise in relay sensor networks and for his precious assistance in my research. Mysincere thanks also go to the other colleagues in the data communications group for theircompany and knowledge. I offer my regards and blessings to all of those who supportedme in any respect during the completion of my master program. Finally, I would like totake this opportunity to thank my parents for their love and encouragement. To them, Idedicate this thesis.1Chapter 1IntroductionThis chapter first presents some background information on wireless relay networks. Theconcepts of wireless sensor network, network lifetime and relay selection strategy are alsointroduced. The structure of the thesis is given at the end of this chapter.1.1 Wireless Relay NetworkA wireless relay network (WRN) is a type of wireless network which has been commonlystudied. It consists of some relays which are used for assist in communications betweena source and a destination. The main idea is that communication is achieved by relayingpackets from the source to the destination with the help of one or more intermediatenodes based on some relaying scheme. Whether or not to transmit a packet and whichrelay will transmit it are decided based on the channel state information (CSI) of thelinks, residual energy information (REI) of the relays and the relay selection strategythat is used. A WRN can be viewed as a combination of a broadcast channel (from thesource to the relays and the destination) and a multiple access channel (from both thesource and relays to the destination). It can provide broader coverage [1], wireless channelChapter 1. Introduction 2impairment mitigation [2], energy savings [3][4] and data rate improvement [5][6]. It iswidely used in many wireless communication systems, such as wireless sensor networks(WSNs), cellular networks and Ad Hoc networks.One of the most commonly used relaying methods is called Amplify-and-Forward(AF). In this relaying method, the relay simply amplifies the signal received in the lasttransmit time slot. It then forwards the signal to the next relay or the destinationwithout any further processing. Amplify-and-Forward has been studied in the context ofWRNs, analog network coding and estimation of the capacity of relay networks, offeringthe benefits of wireless relaying at a low implementation cost [7]. Two variations of AFrelaying are fixed-gain AF (FG-AF) and variable-gain AF (VG-AF).In FG-AF relaying (also known as CSI independent relaying), the relay gain is fixed.Because the relay doesn’t need to calculate a variable gain value, the energy of therelays is saved, which contributes to a longer lifetime. However, the end-to-end signal-to-noise-ratio (SNR) requirement may not be satisfied under poor channel conditions. Theanalysis for FG-AF is quite straightforward. The transmit power at a relay is knowngiven the power of the signal the relay receives.In VG-AF relaying (also known as CSI assisted relaying), to satisfy the SNR require-ment, the relay gain is chosen according to the channel conditions at the cost of someadditional processing. Compared to FG-AF, the analysis of VG-AF is more complicated.In [8], an expression for the probability mass function (pmf) of the VG-AF relay trans-mit power level for a two-hop opportunistic WRN in the presence of Rayleigh fading isChapter 1. Introduction 3derived.1.2 Wireless Sensor NetworkA wireless sensor network consists of a number of sensor nodes, which are spatiallydistributed and have the capability of sensing, processing, and communicating data. Itcan be used for applications ranging from remote monitoring to tracking target objects.A wireless sensor node typically consists of a sensing module, a processing module forprocessing the output of the sensing module and a wireless communication module fordata transmission. The sensor node usually has limited energy, often supplied by a smallbattery.The main constrains in a WSN are limited amount of energy, short communicationdistance and limited processing capabilities and storage of data. Unless its battery canbe continuously recharged, a sensor node can only operate for a limited time durationbefore it runs out of energy. Due to this limitation, a WSN should be designed to operatelong enough so that it can complete its tasks. The network lifetime is thus an importantsystem specification.1.3 Network LifetimeThe network lifetime of a WSN is defined as the number of packets that are successfullytransmitted before the network becomes inoperable. The point at which a WSN is con-Chapter 1. Introduction 4sidered to be inoperable varies somewhat. For example, it can be the instant when anyone sensor node fails to transmit, or alternatively, a certain fraction or number of nodesfail to transmit. For sensors in a well structured WSN, energy balance can be achievedin a certain area due to their similar behaviours [9]. When one sensor node fails, theremaining sensors in its neighbourhood are likely to run out of energy faster becausethey may assume some of the responsibilities of the failed node. Thus, the above twodefinitions of WSN being inoperable may not yield very different results [10].In this thesis, we define the lifetime as the number of packets successfully transmitteduntil the network becomes inoperable based on the system outage probability (SOP) [8].For a signal passing through a series of N relays, the end-to-end characteristics fromthe source to the destination depend on the N + 1 channel gains along the path. Theequivalent end-to-end SNR must satisfy a minimum threshold value, selected accordingto the target error rate. An outage occurs when the equivalent end-to-end SNR fallsbelow this threshold SNR. The system outage probability expression is given in [8]. Weconsider network inoperable if the network SOP exceeds the required threshold imposedby the quality of service (QoS) requirements.1.4 Relay Selection StrategyFour relay selection strategies have been studied in [11] in order to prolong the WSN life-time. they are the minimum transmission power (MTP) strategy, the maximum residualenergy (MRE) strategy, the maximal energy-efficiency index (MEI) strategy, and the min-Chapter 1. Introduction 5imum outage probability (MOP) strategy. In MTP, the relay with the lowest requiredtransmit power is selected regardless of the REI; in MRE, the relay with the largestdifference between its current residual energy and its transmit power is selected; in MEI,the relay with the largest ratio of its current residual energy and its transmit power isselected; in MOP, the relay which yields the smallest probability that an outage occursis selected.Numerical results of lifetime study of these four strategies for different initial relayenergies and different numbers of relays show that MOP and MEI perform better thanMTP and MRE for large initial relay energies, with MOP outperforming MEI slightly.Moreover, MRE performs slightly better than MTP for a small number of relays and lowinitial relay energies, but worse as the number of relays and the initial energies increase.In [12], a dynamic transmit power threshold is proposed for improving the WRNlifetime. The predicted outage probability is calculated based on an energy conservingalgorithm and is applied at each relay transmission. Numerical results show that in thisscheme, the network refrains from using up a large amount of energy to transmit underpoor channel conditions. Rather, it postpones its transmission to subsequent time slotswhen the channel condition is better. A substantial performance improvement is achievedwhen the number of relays is large.In [13], the optimal WRN lifetime is computed using a dynamic programming tech-nique. By constructing a path for each state that goes through the maximum averagenetwork lifetime from the initial state to the terminating state using CSI and REI, theChapter 1. Introduction 6optimal average network lifetime is obtained, where a state is the current residual ener-gies of the relays. Although difficult to implement in practice since it’s computationallycomplex, this scheme provides an upper bound on the WRN lifetime.1.5 Structure of the ThesisThe remainder of the thesis is organized as follows.In Chapter 2, we present two different methods to obtain the pdf of the lifetimeof a two-hop AF opportunistic wireless relay network with Rayleigh fading. Previouswork is first summarized. The system model and proposed schemes are then described.Numerical results of the accuracy of the algorithms are finally presented.In Chapter 3, we extend the results of chapter 2 to Weibull and Nakagami fadings.In so doing, we describe methods to determine the system outage probabilities which areneeded.In Chapter 4, we present an adaptive relay transmit power energy saving scheme.Numerical results are provided to illustrate the network lifetime improvements over someexisting schemes.Conclusions and some topics for future work are outlined in Chapter 5.7Chapter 2WRN Lifetime Distribution underRayleigh FadingIn this chapter, we study the lifetime distribution for a two-hop amplify-and-forwardopportunistic WRN under Rayleigh fading. We state the motivation and contributions,followed by the system model description. Then two different analysis methods to obtainthe lifetime pdf are described, and performance evaluation results are provided. Finally,a summary is given at the end of this chapter.2.1 MotivationWireless relaying is commonly used to prolong the network lifetime of WSNs [14][15][16][17][18].For example, in [19], a greedy approach is proposed. By exploiting both CSI and REIof individual nodes, this protocol maximizes the minimum residual energy across thenetwork at each packet transmission. This protocol, referred to as the max-min protocol,selects the sensor with the maximum energy-efficiency index, which is the sensor withthe largest difference between its current battery energy and the required transmissionChapter 2. WRN Lifetime Distribution under Rayleigh Fading 8energy. Numerical results show that this approach can improve the network lifetime sig-nificantly. In [20], a distributed max-min protocol, allowing nodes to determine whetherto transmit using its own channel state and residual energy, is proposed.In [11], four relay selection strategies are studied, namely, the minimum transmis-sion power (MTP) strategy, the maximum residual energy (MRE) strategy, the maximalenergy-efficiency index (MEI) strategy, and the minimum outage probability (MOP)strategy. Some strategies are proposed to minimize the average transmit power whilesome to balance energy usage. The results show that MOP and MEI performs betterthan MTP and MRE. Furthermore, MOP outperforms all other schemes, even thoughthe difference with MEI is small. MTP outperforms MRE and its performance is veryclose to those of MEI and MOP when the initial energy is large. All of the four strate-gies involve trade-offs between minimizing the transmit power and balancing the residualenergy at each relay node. The performance results in [11] are based on simulations.It is shown in [19], that for a WSN with total non-rechargeable initial energy E0, theaverage network lifetime E[L1], defined as the average number of time slots before thenetwork becomes inoperable, is given by:E[L1] =E0 − E[Er]λE[Et] + Pc, (2.1)where E[Er] is the expected value of the residual energy in the network when itbecomes inoperable; E[Et] is the expected value of the transmission energy; Pc is theconstant continuous power consumption over the whole network; λ is the average relayChapter 2. WRN Lifetime Distribution under Rayleigh Fading 9transmit rate defined as the number of data collections per unit time; and E[Et] is the ex-pected relay transmit energy consumed by all relays in a data transmission. For differentrelay selection strategies, channel fading models or lifetime definitions, E[Er] and E[Et]may be different.An expression for the probability mass function (pmf) of the relay transmission poweris derived in [8] for a two-hop AF opportunistic wireless relay network with Rayleighfading. The average transmit power per time slot is then computed and the expectedlifetime is estimated, with obtaining Pc and λ based on the specifications of the radiofrequency (RF) module and the data transmission protocol, and neglecting E[Er] sincethe residual energy is typically negligible compared with E0 in practice.2.2 ContributionsIn this chapter, we study the lifetime distribution of a two-hop AF opportunistic WRNin Rayleigh fading. Two different approaches are used to obtain the expressions for thepdf, based on the Pearson method and the Central Limit Theorem (CLT), respectively.The main contributions of this chapter are summarized as follows:• We apply the Pearson method to obtain an approximate expression for the pdf oflifetime distribution based on the statistical moments from the simulation data. Theapproach greatly reduces computer simulation time compared with conventionalsimulation methods.Chapter 2. WRN Lifetime Distribution under Rayleigh Fading 10• We study the residual energy at the instant when the network becomes inopera-ble using Monte Carlo method for different numbers of relays, initial energies anddifferent relaying strategies. Using the result and the CLT, we derive an approxi-mate expression for the pdf of the network.• Using simulations, we show both the Pearson and CLT methods can provide accu-rate approximation for the pdf of the network lifetime.The remainder of this chapter is organized as follows. Section 2.3 describes the systemmodel and the opportunistic AF scheme. The Pearson method is described and itsapplication to the lifetime analysis is presented in Section 2.4. Section 2.5 presents thetheoretical analysis of the CLT method as well as the simulation results. Finally, asummary of this chapter is given in Section 2.6.2.3 System ModelA two-hop AF WRN consists of K parallel paths shown in Fig 2.1; each path consists ofa single relay employing AF scheme. Assuming that both CSI and REI in the K pathsare available at the beginning of a packet transmission, the minimum energy required tomeet the end-to-end SNR requirement can be computed. The source first broadcasts thepacket to all the relays; based on the selection strategy, a certain relay is then selectedto amplify and forward the packet to the destination. In this chapter, the channel fadingin all links is assumed to be Rayleigh. Therefore, the channel link gains are independentChapter 2. WRN Lifetime Distribution under Rayleigh Fading 11	  Figure 2.1: System model of an AF WRN with K relays.complex Gaussian random variables (rvs). Let the channel gain, hSk, from the source tothe kth relay be distributed as CN (0, δ2Sk) and the channel gain, hkD, from the kth relayto the destination be distributed as CN (0, δ2kD). Additive white Gaussian noise (AWGN)with zeros mean and unit variance, i.e. CN (0, 1) , denoted as wSk and wkD respectively,is present in each channel.Chapter 2. WRN Lifetime Distribution under Rayleigh Fading 12The equivalent end-to-end SNR on the kth path can be expressed as [8],γeq(k) =PS|hSk|2Pk|h2kD|PS|hSk|2 + |Pk|h2kD|+ 1, (2.2)where PS is the transmit power at the source and Pk is the transmit power at thekth relay. The source transmit power is assumed to be constant while the relay transmitpower can be varied to satisfy the end-to-end SNR requirement. The relay transmitpower is limited to a maximum value, Pmax; the relay transmit power is discretized andcan take on one of L non-zero power levels {αl}Ll=1, 0 = α0 < α1 < α2 < ... < αL = Pmax.The relay transmit power, Pk, must satisfy the constraint Pk ≤ min{Pmax, ek}, where ekis the residual energy of the kth relay.Each relay computes the minimum transmission power level and the correspondingenergy required to meet the minimum end-to-end SNR value for a successful transmissionbased on the CSI. Assuming that each packet is transmitted in a time slot of unit lengthand the power and energy are equal in value.The kth relay path is in outage if γeq(k) ≥ γth, where γeq(k) is defined in (2.2) andγth is the minimum destination threshold SNR for a successful transmission. We say thata system outage occurs when all K relays fail to meet the SNR requirement. Assumingthat the SNRs on the K paths are independent, the system outage probability, Pout, canbe written asPout =K∏k=1Prk, (2.3)where Prk is the probability that the kth path is in outage and is given by [8].Chapter 2. WRN Lifetime Distribution under Rayleigh Fading 13When a system outage occurs, the system uses (2.3) to determine if the networkis inoperable. The network is considered to be inoperable if Pout > η, where η is thesystem outage probability threshold. The value chosen for η is based on the quality ofservice (QoS) requirement. A flowchart of the amplify-and-forward opportunistic schemeis illustrated in Fig 2.2.Figure 2.2: Amplify-and-forward opportunistic scheme for our system model.The relay selection strategy in this chapter is opportunistic scheduling (previouslyreferred to as Minimum Transmit Power (MTP) strategy). In this scheme, relay k∗ isselected if and only if Pk∗ is the smallest for all relays that satisfy γeq(k) ≥ γth andek ≥ Pk.The definition of network lifetime that we adopt is the number of packets successfullyChapter 2. WRN Lifetime Distribution under Rayleigh Fading 14transmitted before the network becomes inoperable [8]:E[L] = E[L1](1− Pout) =E0 − E[Er]E[Et] + E[Erx] + E[Esleep] + E[ECSI ]× (1− Pout), (2.4)where E[L1] is given in (2.1). The values for E[Erx], E[Esleep] and E[ECSI ] can be obtainedbased on the RF module used at the relay and data transmission protocol. Their typicalvalues are discussed in [8] and are usually negligible compared to the relay transmitenergy. In our study, E[Erx], E[Esleep] and E[ECSI ] are set to 0. The values for E[Er]and E[Et], depend on the relay selection strategy, channel fading models and lifetimedefinitions. For the AF opportunistic WSN, the pmf of Et is derived in [8] and E[Et] cantherefore be obtained. In addition, E[Er], varying with different numbers of relays andamounts of initial energy, is studied later in this chapter.2.4 Approximating Network Lifetime PDF Usingthe Pearson MethodThis section examines the use of the Pearson method to obtain an approximate expres-sions for the pdf of the network lifetime distribution. The Pearson method is summarized,followed by its application to our problem. Results to illustrate the accuracy of the ap-proach are presented in Section 2.4.3.Chapter 2. WRN Lifetime Distribution under Rayleigh Fading 152.4.1 Pearson Systems and Monte Carlo MethodsPearson systems are commonly used as solutions to a differential equation with sevendifferent families of curves, referred to as systems of distribution [18][21]. The use of thePearson method for approximating probability distribution in wireless communicationis introduced in [22]. A selection parameter κ, based on the first four moments of theempirical data, determines the type of the exact form of the curve. The most commontypes of curves are the type one, type four, and type six curves corresponding to thecases κ < 0, 0 < κ < 1 and κ > 1 respectively. The expressions for each type is given in[22].The Monte Carlo (MC) method is a computational algorithm that relies on repeatedrandom sampling to obtain numerical results. Typically, simulations are run many timesto obtain the unknown quantities. A particular pattern is usually followed by this method[23][24]:• Define the domain of the input data.• Generate the input data over the domain based on the probability distribution thatis related to the problem.• Perform the computation over the input data.• Obtain and analyze the results.MC methods are especially useful for simulating phenomena with significant uncertaintyin inputs. They are used to study systems with a large number of coupled degrees of free-Chapter 2. WRN Lifetime Distribution under Rayleigh Fading 16dom, in fields such as physical sciences, engineering, computer graphics, computationalbiology and statistics [25]. In the section, we apply Monte Carlo method to generatestatistics from small sample sizes under our system model conditions as well as simulatedlifetime distribution to compare against those obtained using our proposed methods.2.4.2 Lifetime Analysis with Pearson MethodThe average network lifetime for an opportunistic WRN is studied in [8]. However, thenetwork lifetime pdf was not considered. The procedure for using the Pearson methodto approximate the network lifetime rv, L, is as follows: first moment, (E[L]) can becomputed using (2.4); the other three moments are obtained from Monte Carlo simula-tions with small sample sizes. This is because it is found that the three moments can beobtained to an accuracy of better than 99% by using a sample size of 1000. Once thefirst four moments are obtained, an approximation of the network lifetime pdf can beobtained.2.4.3 Numerical ResultsIn this section, results are provided to illustrate the accuracy of the approximate networklifetime pdf obtained with the Pearson method. Fig 2.3 to 2.6 show the pdf of thelifetime distribution for an AF opportunistic WSN under Rayleigh fading with differentrelay numbers, K, and different initial energies at all relays, e, over Nsamples = 104and Nsamples = 107 iterations respectively to show the Pearson method and MC methodChapter 2. WRN Lifetime Distribution under Rayleigh Fading 17performances over different sizes of samples. Nsamples = 104 means the simulation isaveraged over 104 iterations, where an iteration is one network lifetime. It can be seenthat the pdf curves obtained using (1) the Pearson method and (2) pure MC simulationsagree closely.1500 1550 1600 1650 1700 1750 1800 185000.0020.0040.0060.0080.010.0120.014  Case 1, SimCase 2, SimCase 1, PearsonCase 2, PearsonFigure 2.3: Network lifetime pdf with Rayleigh fading, K = 3, e = [10, 10, 10] J. (a)Case 1: Nsamples = 104. (b) Case 2: Nsamples = 107.To illustrate the reduced running time for the Pearson method, we compare the dif-ferences of both Monte Carlo and Pearson methods when the sample number is reduced.In Fig 2.3, the Pearson approximation is not sensitive to Nsamples while the simulationis quite sensitive to the Nsamples. As is shown, the Pearson approximations using twodifferent sample sizes share the same curve as the simulation using the large sample sizewhile the simulation using the small sample size exhibits a noticeable degree of statis-Chapter 2. WRN Lifetime Distribution under Rayleigh Fading 183200 3250 3300 3350 3400 3450 3500 355000.0020.0040.0060.0080.010.012  Case 1, SimCase 2, SimCase 1, PearsonCase 2, PearsonFigure 2.4: Network lifetime pdf with Rayleigh fading, K = 5, e = [10, 10, 10, 10, 10] J.(a) Case 1: Nsamples = 104. (b) Case 2: Nsamples = 107.tical fluctuation. This shows that the Pearson method is able to obtain quite accurateapproximation with a relatively small sample size. Thus, the Pearson method allows usto avoid the extremely long computer run-time needed by pure MC methods.In Fig 2.4 and 2.5, it is shown that the lifetime increases with the number of relayand the amount of initial energy. A more noticeable fluctuation of the simulation withsmall sample can also be seen. This is because the increase in the number of relays orthe amount of initial energy adds to the total number of possible lifetime values. ThePearson method curve agree closely to the simulation curve with large sample for all bothscenarios. Figure 2.4 and 2.5 show that it requires larger sample size for MC method toChapter 2. WRN Lifetime Distribution under Rayleigh Fading 194900 5000 5100 5200 5300 54000123456789x 10−3  Case 1, SimCase 2, SimCase 1, PearsonCase 2, PearsonFigure 2.5: Network lifetime pdf with Rayleigh fading, K = 3, e = [30, 30, 30] J. (a)Case 1: Nsamples = 104. (b) Case 2: Nsamples = 107.obtain the lifetime pdf as the relay number and the initial energy at each relay increaseswhile the Pearson method is not sensitive to this increase for approximating the lifetimepdf.In Fig 2.6, the initial energy at each relay is not equal. Compared to Fig 2.3, thelifetime decreases although the total initial energy is the same. This is because the relayswith low initial energy yield a higher probability that an outage occurs in the network.Thus, the total residual energy is larger when the network becomes inoperable.Chapter 2. WRN Lifetime Distribution under Rayleigh Fading 201200 1300 1400 1500 1600 1700 180001234567x 10−3  Case 1, SimCase 2, SimCase 1, PearsonCase 2, PearsonFigure 2.6: Network lifetime pdf with Rayleigh fading, K = 3, e = [5, 10, 15] J. (a) Case1: Nsamples = 104. (b) Case 2: Nsamples = 107.2.5 Approximating Network Lifetime PDF Usingthe Central Limit TheoremThis section examines the use of the Central Limit Theorem (CLT) to approximatethe network lifetime pdf. Numerical results are given to illustrate the accuracy of theapproach.The Central Limit Theorem (CLT) states that the sample mean of a sufficiently largenumber of independent and identically distributed (i.i.d.) rvs is approximately normallydistributed. More specifically, let {X1, ..., Xn} be a sequence of i.i.d. rvs, each with meanChapter 2. WRN Lifetime Distribution under Rayleigh Fading 21µ and variance σ2. Denote the sample average as X¯n ,∑ni=1 Xin and ζn , X¯n−µσ/√n , then as nis sufficiently large, the rv ζn converges in distribution to a standard normal distribution,N(0, 1) [26].Let Xi denote the relay transmission power at the ith transmission. Since {X1, X2, ...}are i.i.d. rv’s, Let each rv have the same distribution as the rv X . It can be shown thatthe pmf of X is given by [8]Pr{X = αl} = Pr{Γαl−1 ≤ γth}K − Pr{Γαl ≤ γth}K , (2.5)where αl, l = 0, 1, ..., L is the transmit power level, αL = Pmax, α0 = 0 and Γαl is theequivalent end-to-end SNR when the relay transmit power is αl. Using (2.5), the pmf ofX can be computed and the mean µ and variance σ of X can be therefore obtained.Applying the CLT, the sample mean X¯n =∑ni=1 Xin converges in distribution toN(µ, σ2/n). The probability that the network lifetime L is larger than an integer value Tis equivalent to the probability that the sum of T relay transmit energies is smaller thanthe energy that is consumed, i.e. E0 − Er, where E0 and Er are the total initial energyand total residual energy of all relays in the network respectively. Therefore, we havePr{L > T} = Pr{ T∑i=1Xi < E0 −Er}= Pr{T × X¯T < E0 −Er}= Pr{X¯T <E0 −ErT}= Φ(E0−ErT − µσ/√T)(2.6)where Φ(x) = 1√2pi∫ x−∞ e−t2/2dt.Chapter 2. WRN Lifetime Distribution under Rayleigh Fading 22Based on (2.6), we can obtain the cdf and pmf of the lifetime distribution, asFL(T ) = Pr{L ≤ T} = 1− Φ(E0−ErT − µσ/√T), (2.7)fL(T ) =FL(T )− FL(T − 1) if T > 0FL(0) if T = 0(2.8)The values of E0, µ and σ are known as the network parameters, while Er is a rv whichdepends on the lifetime definition, relaying scheme, link fading, initial energy E0, and thesystem outage probability.Although we could obtain the distribution of Er by finding all possible final states,where a state is the current residual energies of all the relays, and computing the proba-bility of each state, this approach is extremely time-consuming. Fortunately, the valuesof Er are typically very small compared to E0. In practice, a good estimate of the networklifetime distribution can be obtained by assuming Er = 0 in (2.7). Alternatively, we canalso apply the Monte Carlo method to obtain the mean of Er and use (2.8) to obtain thelifetime distribution.2.5.1 Numerical ResultsIn this part, the value of Er is first evaluated, followed by the lifetime distribution calcu-lated by (2.8). The accuracy of the proposed CLT method is examined and the influenceof Er in the lifetime distribution is illustrated.In our simulations, we assume that the channel gain and the receiver noise to have unitChapter 2. WRN Lifetime Distribution under Rayleigh Fading 23power. The source transmit power is Ps = 12 dBm and the end-to-end SNR thresholdrequirement is γth = 8 dB. The system outage probability threshold is set as η = 10%and transmit power limit Pmax = 80 mW. The relay transmit power is assumed to bediscretized to L power levels 0 = α0 < α1 < α2 < ... < αL = Pmax. The relay transmitpower must satisfy Pk ≤ min{Pmax, ek}, where ek is the residual energy of the kth relay.The simulation results are obtained using 107 iterations. For each iteration, the networkstarts from each relay with initial energy and ends when the network is determined to beinoperable.Fig 2.7 and Fig 2.8 show the ratio, ξ, of total residual energy at all relays when thenetwork becomes inoperable to the total initial energy as a function of the initial energyvalue at each relay using the MTP relaying scheme. It can be seen that ξ decreases asthe initial energy increases from 1 J to 1 kJ. Furthermore, ξ is less than 1% as the initialenergy in each relay is over 10 J for K = 3.In Fig 2.7, we compare ξ with different relay numbers K. We select K = 3, 4, 5, 10and all relays have equal initial energy. It is shown that ξ increases as K. This is becausethere is more total residual energy when the network becomes inoperable since thereare more relay sensors. In Fig 2.8, we show ξ with different initial energy distributionsamong relays. K is set to be three. The initial energies at the relays, e, in four cases,is Case 1: e = [e, e, e]; Case 2: e = [0.9e, e, 1.1e], Case 3: e = [0.8e, e, 1.2e] and Case 4:e = [0.7e, e, 1.3e], where e = E0/3. It can be seen that ξ increases as the difference amonginitial energies among the relays. This is because the relay with low initial energy yieldsChapter 2. WRN Lifetime Distribution under Rayleigh Fading 24a greater probability that an outage occurs in the system.103 104 105 10610−410−310−210−1total initial energy (mJ)ratio of total residual energy to total initial energy  K=3K=4K=5K=10Figure 2.7: Ratio of total residual energy to total initial energy at all relays as a functionof total initial energy at all relays with K = 3, K = 4, K = 5, K = 10. Allthe relays have equal initial energy.Fig 2.9 - 2.12 show the network lifetime pdf’s obtained using the CLT method as wellas the pdf’s obtained using simulation. In each figure, Two CLT curves are shown. Oneuses the estimated residual energy obtained from MC simulation, whereas the other oneassumes that Er = 0. It can be seen that the “CLT with MC method” curve matchesthe simulation curve very closely. The “CLT with Er = 0” curve is similar but with aslightly larger mean value. The reason is because there is some residual energy in thenetwork when it becomes inoperable. As a result, the actual total energy consumed isChapter 2. WRN Lifetime Distribution under Rayleigh Fading 25103 104 105 10610−410−310−210−1total initial energy (mJ)ratio of total residual energy to total initial energy  Case 1Case 2Case 3Case 4Figure 2.8: Ratio of total residual energy to total initial energy at all relays as a functionof total initial energy at all relays with K = 3. (a) Case 1: e = [e, e, e],(b) Case 2: e = [0.9e, e, 1.1e]. (c) Case 3: e = [0.8e, e, 1.2e]. (d) Case 4:e = [0.7e, e, 1.3e], where e = E0/3.less than the initial energy. By assuming the residual energy to be zero, the resultinglifetime is slightly longer so that the “CLT with Er = 0” curve is slightly shifted tothe right of the “CLT with MC method” curve. However, when the initial energy islarger, ξ is quite small and the two “CLT” curves agree closely. The CLT method greatlyreduces the running time since the values of µ and σ can be obtained theoretically. Ifwe don’t assume Er = 0, we can estimate its mean using MC method; according to ourexperiment with simulations, a small sample size of Nsample = 103 is enough to provideChapter 2. WRN Lifetime Distribution under Rayleigh Fading 26good accuracy. Therefore, the running time would be much less in contrast to a largesample size required by pure simulation.In Fig 2.9, K = 3 and initial relay energy e = [50, 50, 50] J; in Fig 2.10, K = 5 andinitial relay energy e = [50, 50, 50, 50, 50] J; in Fig 2.11, K = 3 and initial relay energye = [25, 50, 75] J; in Fig 2.12, K = 3 and initial relay energy e = [9, 9, 9] kJ. Because anAA battery usually has the capacity of 1800 - 2600 mAh under 50 mA constant drain[27], we select e = [9, 9, 9] kJ (2500 mAh for each relay) to simulate the initial energyof an AA battery in practice. Comparing Fig 2.9, 2.10 and 2.12, it can be seen that thelifetime increases with the number of relays and the amount of initial energy. ComparingFig 2.9 and Fig 2.11, it is shown that the network lifetime is larger when the initialenergies in the relays are equal.Fig 2.13 shows the computer time required to obtain the pdf using the Pearsonmethod, the CLT method and pure simulation. It can be seen that running times ofthe Pearson and CLT methods are similar and they greatly reduce the running timecompared to pure simulation.2.6 SummaryIn this chapter, we studied the pdf of the network lifetime for a two-hop Amplify-and-Forward opportunistic WRN in Rayleigh fading. In previous work the average networklifetime has been studied, but the distribution of the lifetime has not received much at-tention. We proposed two different methods the approach this topic: Pearson SystemsChapter 2. WRN Lifetime Distribution under Rayleigh Fading 278300 8400 8500 8600 8700 8800 89000123456x 10−3lifetimelifetime pdf  SimulationCLT with MC methodCLT with Er = 0Figure 2.9: Lifetime distribution for WRN under Rayleigh fading channels with K = 3,initial relay energy e = [50, 50, 50] J.and Central Limit Theorem. Simulation results showed that the accuracy of both pro-posed methods is quite good. The computer time required to obtain the pdf is greatlyreduced compared to pure MC simulation. For future work, we can extend our analysisand consider other kinds of relaying schemes and channel fadings.Chapter 2. WRN Lifetime Distribution under Rayleigh Fading 281.66 1.67 1.68 1.69 1.7 1.71 1.72x 10400.511.522.533.544.5x 10−3lifetimelifetime pdf  SimulationCLT with MC methodCLT with Er=0Figure 2.10: Lifetime distribution for WRN under Rayleigh fading channels with K =5, initial relay energy e = [50, 50, 50, 50, 50] J.Chapter 2. WRN Lifetime Distribution under Rayleigh Fading 297000 7200 7400 7600 7800 8000 82000123456x 10−3lifetimelifetime pdf  SimulationCLT with MC methodCLT with Er=0Figure 2.11: Lifetime distribution for WRN under Rayleigh fading channels with K =3, initial relay energy e = [25, 50, 75] J.Chapter 2. WRN Lifetime Distribution under Rayleigh Fading 301.83 1.832 1.834 1.836 1.838 1.84 1.842 1.844x 10600.511.522.533.544.5x 10−3lifetimelifetime pdf  SimulationCLT with MC methodCLT with Er=0Figure 2.12: Lifetime distribution for WRN under Rayleigh fading channels with K =3, initial relay energy e = [9, 9, 9] kJ.Chapter 2. WRN Lifetime Distribution under Rayleigh Fading 315 10 15 20 25 30 35 40 45 501001021041061081010initial energy at each relay (J)running time (s)  Pearson methodCLT with MC methodpure simulationCLT with Er = 0Figure 2.13: Computer time required to obtain the pdf using the Pearson method, theCLT method and pure simulation. K = 3 and relays have equal initialenergy.32Chapter 3WRN Lifetime Distribution underWeibull and Nakagami FadingIn chapter 2, we studied the lifetime distribution of a WRN in Rayleigh fading using twodifferent methods. In this chapter, we examine the lifetime distributions in Weibull andNakagami fading. We first derive expressions for the system outage probability and thenapply the two methods in Chapter 2 to the network lifetime analysis. Numerical resultsare presented to illustrate the accuracy of the methods.3.1 Motivations and ContributionsIn chapter 2, the network lifetime distribution for an AF opportunistic WRN was studiedunder Rayleigh fading. We now extend our results to Weibull and Nakagami fading byfirst deriving exact expressions for the system outage probability. Then we apply thePearson method and the CLT method to obtain good approximations to the networklifetime distributions.The contributions of this chapter are as follows:Chapter 3. WRN Lifetime Distribution under Weibull and Nakagami Fading 33• We obtain exact expressions for the system outage probability of opportunistic AFrelaying scheme in Weibull fading.• Fast methods, based on the Pearson systems and the CLT are shown to yield goodapproximations to the network lifetime distributions under Weibull and Nakagamifadings.The rest of this chapter is organized as follows. Section 3.2 describes the systemmodel. The system outage probability analysis is described in Section 3.3. Numericalresults are presented in Section 3.4. A summary of this chapter is given in Section 3.5.3.2 System ModelWe use the two-hop system model described in Chap 2 (Fig 2.1). The only difference isthat instead of Rayleigh fading, we assume that all links are subject to Weibull fading orNakagami fading.A Rayleigh distribution is commonly used to model signal fading channel in an urbanenvironment where the received signal consists of large number of components causedby reflection and scattering. However, when the components is small, the Weibull dis-tribution is a more appropriate fading model. Experimental data shows that Weibulldistribution gives a good fit for indoor fading channels. Some evidence also indicatesthat the Weibull distribution can be used to model outdoor multipath fading well insome cases [28]. The Nakagami distribution is commonly used to model a wide variety ofChapter 3. WRN Lifetime Distribution under Weibull and Nakagami Fading 34realistic fading channels in practice [29][30]. An advantage of the Nakagami distributionis that it can be reduced to the Rayleigh distribution and can model fading conditionsmore severe or less severe than those in the Rayleigh case. Another justication for theapplication of the Nakagami fading model is its good fit to empirical fading data.3.3 System Outage Probability AnalysisExpressions for the system outage probability with Weibull and Nakagami fading are firstderived. Simulation results are then used to confirm the theoretical analysis.3.3.1 Weibull Fading Channel AnalysisAccording to the system model, the source transmits the packet to the destination withthe aid of K relays in two hops. We denote the channel gains corresponding to the pathsfrom the source to the kth relay and the kth relay to the destination by hSk and hkD,respectively. For Weibull fading, hSk and hkD are Weibull distributed with parameters(Ω1k, m1k) and (Ω2k, m2k), respectively.Theorem 1 : Let H be Weibull rv with parameters (Ω, m), (i.e., H ∼ Weibull(Ω, m)).Then, H2 ∼ Weibull(Ω2, m/2).Proof : The CDF function of H is [31]FH(h) =1− e−(h/Ω)m h ≥ 00 h < 0(3.1)Chapter 3. WRN Lifetime Distribution under Weibull and Nakagami Fading 35Then,FH2(h) = Pr{H2 ≤ h}= Pr{−√h ≤ H ≤√h}= Pr{H ≤√h}=1− e−(h/Ω2)m/2 h ≥ 00 h < 0(3.2)Thus, H2 ∼ Weibull(Ω2, m/2).Theorem 2 : Let H be Weibull rv with parameters (Ω, m) and I = aH, a > 0. Then,I ∼ Weibull(aΩ, m).Proof : The CDF function of H is [31]FH(h) =1− e−(h/Ω)m h ≥ 00 h < 0(3.3)Then,FI(i) = Pr{aH ≤ i}= Pr{H ≤ i/a}=1− e−(i/aΩ)m i ≥ 00 i < 0(3.4)Thus, I ∼ Weibull(aΩ, m).Assume the channel gains for the two hops, hSk and hkD are Weibull-distributedwith parameters (ω1k, n1k) and (ω2k, n2k) and the corresponding path SNRs, Γ1k = PSh2SkN0and Γ2k = Pkh2SkN0 are Weibull-distributed with parameters (Ω1k, m1k) and (Ω2k, m2k) re-spectively. Assuming AWGN with unit variance, we have Ω1k = PSω21k, m1k = n1k/2Chapter 3. WRN Lifetime Distribution under Weibull and Nakagami Fading 36and Ω2k = Pkω22k, m2k = n2k/2. As in Chapter 2, all channels are assumed to undergoindependent fading.The equivalent end-to-end SNR through the kth path is given by [8]γeq(k) =γ1kγ2kγ1k + γ2k + 1. (3.5)Because Γ1k and Γ2k are independent, we can find the pdf of Γeq(k) asFΓeq(k)(γ) =∫ ∞0Pr{ Γ1kλΓ1k + λ+ 1≤ γ}fΓ2k(λ)dλ=∫ γ0Pr{Γ1k ≥γλ+ γλ− γ}fΓ2k(λ)dλ+∫ ∞γPr{Γ1k ≤γλ + γλ− γ}fΓ2k(λ)dλ= I1k + I2k(3.6)In the second line, the inequality is reversed because λ − γ is negative based on theintegral interval. Then,I1k =∫ γ0Pr{Γ1k ≥γλ+ γλ− γ}fΓ2k(λ)dλ=∫ γ0fΓ2k(λ)dλ= 1− e−(γ/Ω2k)m2k(3.7)andI2k =∫ ∞γPr{Γ1k ≤γλ + γλ− γ}fγ2k(λ)dλ=∫ ∞γ(1− e− 1Ωm1k1k( γλ+γλ−γ )m1k) m2kΩm2k2kλm2k−1e− λm2kΩm2k2k dλ= e−(γ/Ω2k)m2k − m2kΩm2k2k∫ ∞γe−(λm2kΩm2k2k+ 1Ωm1k1k( γλ+γλ−γ )m1k)dλ(3.8)Chapter 3. WRN Lifetime Distribution under Weibull and Nakagami Fading 37From (3.5) and (3.6), we obtainFΓeq(k)(γ) = I1k + I2k= 1− m2kΩm2k2k∫ ∞γe−(λm2kΩm2k2k+ 1Ωm1k1k( γλ+γλ−γ )m1k)dλ= 1− m2kΩm2k2k∫ ∞0e−((λ+γ)m2kΩm2k2k+ 1Ωm1k1k(γ+ γ2+γλ)m1k)dλ= 1− n2k/2(Pkω22k)n2k/2∫ ∞0e−((λ+γ)n2k/2(Pkω22k)n2k/2+ 1(PSω21k)n1k/2(γ+ γ2+γλ)n1k/2)dλ(3.9)The third equality is obtained by changing the integration variable from λ to λ+ γ.The single path outage probabilities, obtained using (3.9) as well as MC simulation,as a function of the relay transmit power are plotted in Fig 3.1. The initial energy ateach relay is 12 dBm. The MC simulation uses Nsamples = 106 for each of the two casesconsidered: Case 1: ω1k = ω2k = 1, n1k = n2k = 2, Case 2: ω1k = ω2k = 1, n1k = n2k = 3.It can be seen that the theoretical curves derived from (3.9) match the MC simulationresults closely. As is expected, the outage probability decreases with the fading parametern. Note that for n = 2, the Weibull fading model reduces to the Rayleigh fading model.As discussed in Chapter 2, the system outage probability is the product of the outageprobabilities for all K paths.3.3.2 Nakagami Fading Channel AnalysisThe system model is similar to that in Section 3.3.1, except that we assume Nakagamifading instead of Weibull fading. We denote the channel gains corresponding to the hopsfrom the source to the kth relay and from the kth relay to the destination by hSk andChapter 3. WRN Lifetime Distribution under Weibull and Nakagami Fading 380 100 200 300 400 5000.20.30.40.50.60.70.80.91transmit energy (mJ)single path outage probability  Theory, case 1Simulation, case 1Theory,case 2Simulation,case 2Figure 3.1: Simulation and theoretical results (3.7) for the single path outage probabil-ity against relay transmit power under Weibull fading (Case 1: ω1k = ω2k =1, n1k = n2k = 2, Case 2: ω1k = ω2k = 1, n1k = n2k = 3).hkD, respectively. The gains hSk and hkD are with parameters (m1k, λ1k) and (m2k, λ2k),respectively. Then, the variables Xk = h21k and Yk = h22k have a Gamma distributionwith parameters (m1k, θ1k) and (m2k, θ2k) where θ1k = m1k/λ1k and θ2k = m2k/λ2k [32].All channels are assumed to undergo independent fading.In [32], an exact expression for the outage probability is derived for the outage per-formance of the opportunistic AF relaying scheme under Nakagami fading. The outageChapter 3. WRN Lifetime Distribution under Weibull and Nakagami Fading 39probability for a single kth path is given asP kout = 1−2e−η0θ1kθm2k2kΓ(m2k)m1k−1∑n=01n! (η0θ1k)n∑j=0nCjn(̟k)j(η0θ1k̟kθ2k)m2k−j2 Km2k−j(2√η0θ1kθ2k̟k),(3.10)where ̟k = (γSλ1k + 1)/γk, γS = PS/σ2n, γk = Pk/σ2n, η0 = 1/γS and PS, Pk and σ2n arethe source transmit power, the relay transmit power and the AWGN power.The single path outage probabilities, obtained using (3.10) as well as MC simulation,as a function of the relay transmit power are plotted in Fig 3.2. The initial energyat each relay is 12 dBm. The MC simulation uses Nsamples = 106 for each of the twocases considered: Case 1: λ1k = λ2k = 1, m1k = m2k = 1, Case 2: λ1k = λ2k = 1,m1k = m2k = 2.It can be seen that the theoretical curves derived from (3.10) match the MC simulationresults closely. As is expected, the outage probability decreases with the fading parameterm. Note that for m = 1, the Nakagami fading model reduces to the Rayleigh fadingmodel.3.4 Lifetime DistributionIn this section, the network lifetime pdf’s under Weibull and Nakagami fading are ap-proximated based on the Pearson and CLT methods discussed in Chapter 2 as well asthe outage probability expressions in (3.9) and (3.10).Chapter 3. WRN Lifetime Distribution under Weibull and Nakagami Fading 400 100 200 300 400 5000.20.30.40.50.60.70.80.91transmit energy (mJ)single path outage probability  Theory, case 1Simulation, case 1Theory, case 2Simulation, case 2Figure 3.2: Simulation and theoretical results for the single path outage probabilityagainst relay transmit energy under Nakagami fading (Case 1: λ1k = λ2k =1, m1k = m2k = 1, Case 2: λ1k = λ2k = 1, m1k = m2k = 2).As in Chapter 2, we assume that the channel gains and the receiver noises have unitpower. The source transmit power is Ps = 12 dBm and the end-to-end SNR thresholdis γth = 8 dB. The system outage probability threshold, η, is set as 10% and the relaytransmit power limit, Pmax, is 80 mW. The relay transmit power is assumed to be discreteand each relay transmits at one of L power levels 0 = α0 < α1 < α2 < ... < αL = Pmax.The transmit power must satisfy Pk ≤ min{Pmax, ek}, where ek is the residual energy ofthe kth relay. The MC simulation results are obtained using at least 107 iterations. Ineach iteration, the network state starts with each relay having initial energy e0 and endsChapter 3. WRN Lifetime Distribution under Weibull and Nakagami Fading 41when the network becomes inoperable.Fig 3.3 - 3.10 show the network lifetime pdf obtained using the CLT method as wellas the pdf obtained using simulation for Weibull or Nakagami fading. Two CLT curvesare shown in each figure. One uses the estimated residual energy obtained from MCsimulation, whereas the other one assumes that Er = 0. It can be seen that the “CLTwith MC method” curve matches the simulation curve very closely. The “CLT withEr = 0” curve is similar but with a slightly larger mean value. The reason is becausethere is some residual energy in the network when it becomes inoperable. As a result,the actual total energy consumed is less than the initial energy. By assuming the residualenergy to be zero, the resulting lifetime is slightly longer so that the “CLT with Er = 0”curve is slightly shifted to the right of the “CLT with MC method” curve. However, whenthe initial energy is larger, ξ is quite small and the two “CLT” curves agree closely. TheCLT method greatly reduces the running time since the values of µ and σ can be obtainedtheoretically. If we don’t assume Er = 0, we can estimate its mean using MC method;according to our experiment using simulations, a small sample size of Nsample = 103 isenough to provide good accuracy. Therefore, the running time would be much less incontrast to a large sample size required by pure simulation.In Fig 3.3 - 3.6, we show the lifetime pdf under Weibull fading using different Kand e0. The channel link gains are Weibull-distributed with parameters Ω1k = Ω2k = 1,m1k = m2k = 3. In Fig 3.3, K = 3, e0 = 50 J; in Fig 3.4, K = 5, e0 = 50 J; in Fig 3.5,K = 7, e0 = 50 J; in Fig 3.6, K = 3, e0 = 10 J. Comparing Fig 3.3, 3.4 and 3.5, it canChapter 3. WRN Lifetime Distribution under Weibull and Nakagami Fading 42be seen that lifetime increases with the relay number K. Comparing Fig 3.3 and 3.6,lifetime increases with the initial energy e0.8300 8400 8500 8600 8700 8800 890001234567x 10−3lifetimelifetime pdf  SimulationPearson MethodCLT with MC MethodCLT with Er=0Figure 3.3: Lifetime distribution for Weibull fading (Ω1k = Ω2k = 1, m1k = m2k = 3)with K = 3 and e0 = 50 J using the Pearson and CLT methodsFig 3.7 - 3.10 show the lifetime pdf under Nakagami fading with different K and e0.The channel link gains Nakagami-distributed with parameters λ1k = λ2k = 1, m1k =m2k = 2. In Fig 3.7, K = 3, e0 = 50 J; in Fig 3.8, K = 5, e0 = 50 J; in Fig 3.9, K = 7,e0 = 50 J; in Fig 3.10, K = 3, e0 = 10 J. Comparing Fig 3.7, 3.8 and 3.9, it can beseen that lifetime increases with the relay number K. Compared to Fig 3.10, lifetimeincreases with the initial energy e0.It is also shown in Fig 3.6 and 3.10, “CLT with Er = 0” curves are more shifted toChapter 3. WRN Lifetime Distribution under Weibull and Nakagami Fading 431.76 1.77 1.78 1.79 1.8 1.81x 1040123456x 10−3lifetimelifetime pdf  SimulationPearson methodCLT with MC methodCLT with Er=0Figure 3.4: Lifetime distribution for Weibull fading (Ω1k = Ω2k = 1, m1k = m2k = 3)with K = 5 and e0 = 50 J using the Pearson and CLT methodsthe right of the other three curves. The reason is because the ratio ξ is larger due to thelower total initial energy. Thus, the CLT approximation with Er = 0 is less close to thepure simulation.3.5 SummaryIn this chapter, we extended the lifetime analysis for Rayleigh fading in chapter 2 toWeibull and Nakagami fading. We derived an expression for the system outage probabilityin Weibull fading. We then applied the Pearson method and the CLT method to obtainChapter 3. WRN Lifetime Distribution under Weibull and Nakagami Fading 442.82 2.83 2.84 2.85 2.86 2.87 2.88x 1040123456x 10−3lifetimelifetime pdf  SimulationPearson methodCLT with MC methodCLT with Er=0Figure 3.5: Lifetime distribution for Weibull fading (Ω1k = Ω2k = 1, m1k = m2k = 3)with K = 7 and e0 = 50 J using the Pearson and CLT methodsthe network lifetime pdf. Simulation results show that both methods can provide a verygood approximation for the pdf at a much lower computational cost compared to MCsimulations.Chapter 3. WRN Lifetime Distribution under Weibull and Nakagami Fading 451600 1650 1700 1750 1800 1850 190000.0020.0040.0060.0080.010.0120.014lifetimelifetime pdf  SimulationPearson methodCLT with MC methodCLT with Er=0Figure 3.6: Lifetime distribution for Weibull fading (Ω1k = Ω2k = 1, m1k = m2k = 3)with K = 3 and e0 = 10 J using the Pearson and CLT methodsChapter 3. WRN Lifetime Distribution under Weibull and Nakagami Fading 469100 9200 9300 9400 9500 960001234567x 10−3lifetimelifetime pdf  SimulationPearson MethodCLT with MC MethodCLT with Er=0Figure 3.7: Lifetime distribution for Nakagami fading (λ1k = λ2k = 1, m1k = m2k = 2)with K = 3 and e0 = 50 J using the Pearson and CLT methodsChapter 3. WRN Lifetime Distribution under Weibull and Nakagami Fading 471.92 1.925 1.93 1.935 1.94 1.945 1.95 1.955 1.96x 10401234567x 10−3lifetimelifetime pdf  SimulationPearson MethodCLT with MC MethodCLT with Er=0Figure 3.8: Lifetime distribution for Nakagami fading (λ1k = λ2k = 1, m1k = m2k = 2)with K = 5 and e0 = 50 J using the Pearson and CLT methodsChapter 3. WRN Lifetime Distribution under Weibull and Nakagami Fading 483.03 3.035 3.04 3.045 3.05 3.055 3.06 3.065 3.07x 10401234567x 10−3lifetimelifetime pdf  SimulationPearson MethodCLT with MC MethodCLT with Er=0Figure 3.9: Lifetime distribution for Nakagami fading (λ1k = λ2k = 1, m1k = m2k = 2)with K = 7 and e0 = 50 J using the Pearson and CLT methodsChapter 3. WRN Lifetime Distribution under Weibull and Nakagami Fading 491750 1800 1850 1900 195000.0020.0040.0060.0080.010.0120.0140.0160.018lifetimelifetime pdf  SimulationPearson MethodCLT with MC MethodCLT with Er=0Figure 3.10: Lifetime distribution for Nakagami fading (λ1k = λ2k = 1, m1k = m2k = 2)with K = 3 and e0 = 10 J using the Pearson and CLT methods50Chapter 4An Adaptive Relay Transmit PowerScheme for Improving LifetimeIn this chapter, an algorithm based on an energy conserving adaptive transmit powerthreshold is proposed for improving the lifetime in an AF WRN. We first describe thesystem model, different relay selection strategies and our proposed lifetime improvementalgorithm, then analyze the optimal relay selection strategy and finally compare it withour proposed algorithm along with other existing lifetime improving algorithms.4.1 Motivations and ContributionsRelays are employed in wireless networks to reduce transmit power, prolong networklifetime and improve performance [5][6]. In Chapter 2 and Chapter 3, we studied thelifetime distribution of an opportunistic AF WRN. In this chapter, we focus on therelaying scheme and transmit power threshold to improve the network lifetime.These relay scheming strategies in [11] are designed to ensure that the system outageprobability is smaller than a certain threshold value when the system is operable. InChapter 4. An Adaptive Relay Transmit Power Scheme for Improving Lifetime 51[33], a soft transmit power limited scheme to improve the lifetime is proposed, in whichthe selected relay transmit power Pk cannot exceed a threshold, Psoft. The value of Psoftis determined from the end-to-end SNR threshold, the link quality and the number ofrelays. If the Pk value required to meet the end-to-end SNR required exceeds Psoft, therelay does not transmit. This algorithm prevents relays from transmitting signals whenchannels experience deep fades and the system conserves energy while still meets thesystem outage probability target.In this chapter, we further improve the scheme in [33] by adapting each relay’s Psoftto the global CSI and REI. An adaptive Psoft is obtained according to residual energydistribution across all the relays. The algorithm encourages relays with high (low) residualenergy to transmit using a high (low) transmit power limit.The main contributions of this chapter are summarized as follows:• We propose an adaptive transmit power limited scheme based on the global CSIand REI to extend the network lifetime.• The optimal relay selection strategy in [13] is clarified and studied for differentnumbers of relays, numbers of transmit power levels and amounts of initial energy.• The proposed algorithm with the the optimal relay selection strategy is shown toprovide a longer network lifetime compared to other existing lifetime improvingalgorithms.The remainder of this chapter is organized as follows. Section 4.2 describes the systemChapter 4. An Adaptive Relay Transmit Power Scheme for Improving Lifetime 52model and the proposed adaptive transmit power limited scheme. The optimal selectionstrategy is described and analyzed under different scenarios in Section 4.3. Section 4.4presents numeriacal results to compare the performance of the proposed scheme to thoseof some other existing schemes. Finally, a summary of this chapter is given in Section4.5.4.2 Adaptive Relay Transmit Power LimitedAlgorithmIn Chapter 2 and 3, the average system outage probabilities of MTP are analyzed. Inthis section, we discuss a new strategy to increase the network lifetime.The average system outage probability, ¯Pout, is given by [33]¯Pout =NoutNout +Ntx(4.1)where Nout is the number of outages which occurred before the network becomes inoper-able and Ntx is the total number of packets successfully transmitted. It is shown in [33]that for the four relaying schemes, the outage probabilities are typically less than the QoSrequirement. This shows the possibility that the lifetime could be improved by applyingopportunistic transmission while satisfying the QoS requirement. If both Pmax and E0are sufficiently large, the relays will transmit even when the channel condition is verypoor. However, with a soft power limit, Psoft, the relay is prevented from transmittingas long as the QoS requirement is satisfied. In [33], the value of Psoft is calculated byChapter 4. An Adaptive Relay Transmit Power Scheme for Improving Lifetime 53Algorithm 1 Adaptive transmit power limited algorithm to determine Psoft for eachrelay k.1: E = [e1, e2, ..., ek, ..., eK ] represent the current residual energy values at the K relaysduring a packet transmission interval, 1 ≤ k ≤ K2: Based on ek and assuming Pmax = ∞, the path outage probability for relay k, Prk =Pr{Γk ≤ γth}, is obtained3: For the kth relay, Psoft = K√η∏Kn=1 PrnPrkassuming all relays contribute equally to the system outage probability. Therefore, forthe kth relay, with an infinite amount of initial energy, the outage probability of its ownpath isPout(Pk|E0k = ∞) = K√η (4.2)where η is the system outage probability threshold (QoS requirement) and K is thenumber of relays. In order to calculate Psoft, we need to find the lowest non-zero powerlevel at which a relay can transmit. For any Psoft value that is smaller than Pmax,the network lifetime will be increased at the cost of a higher system outage probability.Therefore, as long as QoS requirement is satisfied, this strategy contributes to the networkperformance. However, it is assumed in [33] that after Psoft is calculated at the relays, itremains unchanged during the lifetime of the network. To further improve this strategyfor lifetime increase, we propose an adaptive transmit power limited strategy that canyield additional improvements on the network lifetime.We present an adaptive algorithm, Algorithm 1, for computing Psoft in our proposedscheme. As can be seen, Psoft is calculated based on not only the number of relays andQoS requirement, but also on the residual energy at each relay. For a WRN with K relays,at each transmission, we denote the energy values at the relays as e = [e1, e2, ..., ek, ..., eK ].Chapter 4. An Adaptive Relay Transmit Power Scheme for Improving Lifetime 54Each Prk = Pr{Γαl−1 ≤ γth} is computed based on e. Then relay k calculates its Psoft,kas K√η∏Kn=1 PrnPrk.In Fig 3.1 and Fig 3.2, it is observed that the single path outage probability decreasesrapidly from 1 to 0.3 within the first 100 mJ but decreases very slowly with furtherincrease in energy. As previously discussed, the network is considered to be inoperablewhen Pout reaches the end-to-end SNR threshold. Therefore, in our proposed scheme,we calculate a Psoft value for each relay based on its residual energy. The larger therelay energy, the higher the Psoft value. Relays with high residual energies will tend totransmit using higher powers while those with low residual energy will tend to transmitat lower powers. This slows down the increase in Pout and results in a longer networklifetime.4.3 Optimal Selection Algorithm AnalysisIn order to compare the performance of our proposed strategy, a scheduling algorithmfor the optimal average network lifetime, based on global CSI and REI is given by [13].In this section, we apply this algorithm in our system model and analyze it for differentnumbers of relays, initial energies and number of transmit power levels.For a given set of power levels, QoS requirement and initial energy, the optimal net-work lifetime can be obtained using dynamic programming method [13]. In this method,we construct a path for each state that goes from the initial state to a final state throughthe maximum number of transitions, which, according to our lifetime definition, is theChapter 4. An Adaptive Relay Transmit Power Scheme for Improving Lifetime 55maximum network lifetime. Fig 4.1 illustrates the path of a WRN with two relays. Thenetwork initial state is (e10, e20). It transits to (e10 − w10, e20) if relay 1 is chosen and to(e10, e20−w20) is relay 2 is chosen. After that, both states transit to their next states untila state in the set of terminating states is reached. For our system model, the terminatingstates are those for which the system outage probability is greater than or equal to theoutage probability threshold, i.e. the system is inoperable. The objective is to find themaximum transit number from the initial state to a terminating state.Based on Bellman’s equation [34], the maximum average network lifetime is given by[13]E[Loptimal(e)] =11− Pout(e)×(1 +∑u:∃k,s.t.0<uk≤ekPr{w = u} · max0<k≤KE[Loptimal(e− uk1k)]),(4.3)where K is the number of relays, e = (e1, ...eK) are the relay energies at the beginningof a packet transmission, w = (w1, w2, ..., wK) are the transmission energies required tosuccessfully transmit a packet to the destination, u = (u1, u2, ..., uK) is the set of discretetransmission powers with uk ∈ {0, α1, ..., αL}. The optimal network lifetime is obtainedbackwards starting from the final states to the initial state. With the global REI andCSI at all paths, the relaying scheme to maximize the average network lifetime is givenby [13]k∗optimal = arg max0<k≤K⋂ ek≥wkE[Loptimal(e− uk1k)]. (4.4)Although the scheme manages to maximize the average network lifetime, it is difficultto implement in practice because E[Loptimal(e − uk1k)] for each 0 < k ≤ K is requiredChapter 4. An Adaptive Relay Transmit Power Scheme for Improving Lifetime 56	 		 Figure 4.1: Stochastic shortest path of relaying scheme for WRN with two relays.before computing E[Loptimal(e)]. The performance of this scheme serves primarily as anupper bound to other strategies.Fig 4.2 shows the expected network lifetime as a function of the initial energy. Itcan be seen that MOP and MEI perform better than MTP and MRE for large initialrelay energies, with MOP outperforming MEI slightly. Moreover, the MRE scheme out-performs the MTP scheme when the initial energy is small. The reason is that MREChapter 4. An Adaptive Relay Transmit Power Scheme for Improving Lifetime 570 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5x 10401000200030004000500060007000initial energy (mJ)average lifetime  MTPMREOptimalMOPMEIFigure 4.2: Network lifetime with Rayleigh fading channels, two relays, each with equalinitial energy, end-to-end SNR threshold of 0.25 and L = 8 (10, 20,..., 80mW).achieves a better energy balance between the relays and therefore, slows the increase ofsystem outage probability. As the initial energy increases, MTP gradually outperformsMRE because the benefit of less average transmit power of MTP is amplified with a suffi-cient amount of signal transmissions. However, compared with optimal network lifetime,significant performance loss of all schemes can be seen and the loss increases with theinitial energy.Fig 4.3 shows the average lifetime with different initial energy distributions in eachChapter 4. An Adaptive Relay Transmit Power Scheme for Improving Lifetime 580 1000 2000 3000 4000 5000 600005001000150020002500300035004000total initial energy (mJ)average lifetime  Case 1Case 2Case 3Case 4Figure 4.3: Average network lifetime with Rayleigh fading, two relays, end-to-end SNRthreshold of 0.25 and L = 8 (10, 20,..., 80 mW). Case 1: e = [e/6, 5e/6],Case 2: e = [e/4, 3e/4], Case 3: e = [e/3, 2e/3], Case 1: e = [e/2, e/2],where e = E0.relays. It can be seen that the network with equal initial energy has the longest optimalaverage lifetime. The more difference in the initial energies, the smaller the optimalaverage lifetime.We then study the running time of this algorithm with different numbers of relays,initial energies and transmit power levels. Assuming all the relays have the same set oftransmit power levels, for the initial energy set e = (e1, ...eK) and transmit power levelset {α1, ..., αL}, there are O((∏Kk=1 ek)LK) states in total, where E is the initial energyChapter 4. An Adaptive Relay Transmit Power Scheme for Improving Lifetime 59100 200 300 400 500 600 700 800 900 1000020040060080010001200initial energy (mJ)average run−time (s)  Case 1, SimCase 1, theoryCase 2, SimCase 2, theoryFigure 4.4: Running time for optimal selection algorithm as a function of total initialenergy E0 with three relays, end-to-end SNR threshold of 0.25 and L = 8transmit power level L = 8 (10, 20,..., 80 mW) (Case 1: e = [e, e, e], Case2: e = [0.5e, e, 1.5e], where e = E0/3).at each relay, L is the number of transmit power levels, and K is the number of relays.This is because determining the optimal average lifetime requires the table where eachcell contains the optimal average lifetime of the corresponding initial energies. Thus, therunning time depends on the number of cells in the table. For relay k, the number oftotal possible residual energy amounts is ekL/Pmax. Assuming unit Pmax, the runningtime is O((∏Kk=1 ek)LK) or c(∏Kk=1 ek)LK , where c is the coefficient.Fig 4.4, Fig 4.5 and Fig 4.6 verify our theoretical analysis on the algorithm runningChapter 4. An Adaptive Relay Transmit Power Scheme for Improving Lifetime 601 2 3 4 5 6 7 8020406080100120140number of transmit power levelaverage run−time (s)  SimulationAnalysisFigure 4.5: Running time for optimal selection algorithm as a function of transmit powerlevels with three relays, end-to-end SNR threshold of 0.25 and initial energyof 5000 mJ at each relay. For transmit power level L = 1 (80 mW), L = 2(40, 80 mW), L = 4 (20, 40, 60, 80 mW) and L = 8 (10, 20,..., 80 mW).time. The theoretical results are obtained by computing the coefficient c using MCsimulation. Fig 4.4 shows the running time as a function of the initial energy at eachrelay. The system model is a three-relay WSN with QoS requirement of 0.25 and transmitpower level L = 8 (10, 20,..., 80 mW). For a WSN with L = 8, K = 3, in both Case 1and Case 2, the running time T1 and T2 are O(E3). Because e = [e, e, e] mW in Case 1and e = [0.5e, e, 1.5e] mW in Case 2, based on our running time analysis, for the same Land K, T2 = 0.75T1. Fig 4.5 shows the running time using different numbers of transmitChapter 4. An Adaptive Relay Transmit Power Scheme for Improving Lifetime 612 3 410−210−1100101102103104number of relaysaverage run−time (s)  SimulationAnalysisFigure 4.6: Running time for optimal selection algorithm as a function of relay numberswith initial energy of 3000 mJ at each relay, end-to-end SNR threshold of0.25 and transmit power level L = 8 (10, 20,..., 80 mW).power levels with three relays and 500 mJ initial energy at each relay. The running timeis O(L3). Fig 4.6 shows the average running time as a function of number of relays. Weset the initial energy as 3000 mJ and L = 8. Therefore, the running time is O(300K).For all three analysis, the simulation and our theoretical results agree very closely.Chapter 4. An Adaptive Relay Transmit Power Scheme for Improving Lifetime 624.4 Performance ComparisonIn the section, we use simulation results to compare the performance of MTP, MOP, MTPwith our proposed scheme, MTP with fixed Psoft and the optimal selection algorithm.The source power and threshold SNR are selected as Ps = 12 dBm and γth = 8 dB,respectively. We assume Rayleigh fading with unit variance and channel noise withunit variance as well. The number of relays, system outage probability threshold andmaximum relay transmit power are selected as K = 5, η = 0.15, and Pmax = 80 mW,respectively. The results are obtained with Monte Carlo simulation over 105 iterations.Fig 4.7 - 4.9 show the average lifetimes for three other existing relaying schemes (MTP,MOP and MTP with fixed Psoft) as well as the optimal and proposed selection schemesin three different scenarios with different numbers of relays and system outage thresholdvalues. It can be seen that by using our proposed adaptive Psoft scheme, the networklifetime is extended. Its average is also close to optimal. Comparing Fig 4.7 and 4.8, wecan see that the average lifetime increases with the number of relays. Comparing Fig 4.7and 4.9, we can see that the average lifetime increases with the system outage thresholdvalue.Fig 4.10 shows the outage probabilities for MTP, MOP, MTP with fixed Psoft, MTPwith adaptive Psoft and the optimal scheme. It can be seen that introducing Psoft in-creases the outage probability. The simulation results show that when Psoft is applied,the average Poutage increases from 10% to 15%. However, the increased system outageprobability still satisfies the threshold requirement, η = 0.15.Chapter 4. An Adaptive Relay Transmit Power Scheme for Improving Lifetime 630 0.5 1 1.5 2x 10405001000150020002500initial energy (mJ)average lifetime  MTPMOPMTP with fixed PsoftMTP with adaptive PsoftOptimalFigure 4.7: Average lifetime for five relaying schemes (MTP, MOP, Fixed Psoft, Adap-tive Psoft and optimal selection) with K = 3 and η = 0.15.4.5 SummaryIn this chapter, a new adaptive power restriction scheme is proposed. The networklifetime and average outage probability of our proposed scheme are shown in simulationresults compared with other existing relaying strategies. Results show that the proposedscheme increases the average network lifetime compared with some other schemes whilemaintaining the QoS requirements. In addition, we reproduce the optimal selectionstrategy and apply it to our analysis. Simulation results show that our strategy achievesa close-to-optimal performance.Chapter 4. An Adaptive Relay Transmit Power Scheme for Improving Lifetime 640 0.5 1 1.5 2x 10405001000150020002500300035004000initial energy (mJ)average lifetime  MTPMOPMTP with fixed PsoftMTP with adaptive PsoftOptimalFigure 4.8: Average lifetime for five relaying schemes (MTP, MOP, Fixed Psoft, Adap-tive Psoft and optimal selection) with K = 5 and η = 0.15.Chapter 4. An Adaptive Relay Transmit Power Scheme for Improving Lifetime 650 0.5 1 1.5 2x 104050010001500200025003000initial energy (mJ)average lifetime  MTPMOPMTP with fixed PsoftMTP with adaptive PsoftOptimalFigure 4.9: Average lifetime for five relaying schemes (MTP, MOP, Fixed Psoft, Adap-tive Psoft and optimal selection) with K = 3 and η = 0.2.Chapter 4. An Adaptive Relay Transmit Power Scheme for Improving Lifetime 660 0.5 1 1.5 2x 10400.020.040.060.080.10.120.140.16initial energy (mJ)average Pout  MTPMOPMTP with fix PsoftMTP with adaptive PsoftFigure 4.10: Average Poutage of four relaying schemes (MTP, MOP, Fixed Psoft andAdaptive Psoft) with K = 3 and η = 0.15.67Chapter 5Conclusions and Future WorkIn this chapter, we summarize the research results and contributions. We discuss thestrengths and limitations of the research and suggest directions for future work.5.1 ConclusionsIn the thesis, we focused on the lifetime analysis and improvement for a two-hop Amplify-and-Forward opportunistic wireless relay network (WRN). Two different approaches,namely Pearson and CLT were proposed to analyze the lifetime distribution for the op-portunistic AF WRN with Rayleigh fading in Chaper 2. Simulation results showed thatboth methods yield good approximations to lifetime distribution obtained using Monte-Carlo simulation. Moreover, the approximation accuracy can be improved by taking theresidual energy into consideration.In Chapter 3, we extended our work in Chapter 2 to study the network lifetime inWeibull fading and Nakagami fading. We derived an exact expression for the single pathoutage probability for Weibull fading. Applying the approaches from Chapter 2, thenetwork lifetime distributions were obtained for both fadings with Pearson method andChapter 5. Conclusions and Future Work 68CLT method. For both fadings, a comparison with simulation results showing that bothmethods yield very good approximations for the pdf’s.In Chapter 4, we proposed an adaptive relay transmit power limited scheme to improvethe average network lifetime. The scheme uses CSI and REI to compute a maximumtransmit power limit for each relay. This limit restricts relay transmission under poorchannel conditions. For a bench mark to compare the performance of our proposedalgorithm, we employed the optimal selection strategy. The running time of the optimalselection algorithm was also presented against initial energy at each relay, relay numberand relay transmit power level. Simulation results showed that our proposed scheme hasa longer average network lifetime when compared to MTP, MOP and MTP with fixedPsoft.5.2 Future WorkIn the Pearson method based network lifetime distribution analysis, the first momentcan be obtained theoretically but for the second, third and fourth moments, the MonteCarlo simulation was used. It should be useful to obtain these moments theoretically. Forour proposed CLT based analysis, the residual energy is either assumed to be zero or isestimated using Monte Carlo simulation. It would be interesting to derive the pdf of theresidual energy. It is theoretically possible to determine the residual energy distributionby finding all the possible final states, computing the probability of each such state andcombining the probabilities of states with the same residual energy. However, this methodChapter 5. Conclusions and Future Work 69is extremely time-consuming for a WRN with high initial energy and a large number ofrelays. For our analysis on the Weibull and Nakagami channel, there is still room toimprove the single path outage probability result. Although both the results are exact,more compact, closed-form expressions would be preferred.Compared to the fixed Psoft scheme, our adaptive Psoft scheme in Chapter 4 requiresglobal REI to compute the transmit power limit. Obtaining this information consumesenergy. In addition, calculating Psoft requires extra time and thus the running time ofour algorithm is much longer than the other schemes in Chapter 4. Finally, with anadaptive transmit power limit, the transmit power distribution would be different. It isalso interesting to analyze the lifetime distribution for the AF WSN with our proposedalgorithm.70Bibliography[1] G. Joshi and A. 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