On the Performance of Wireless Energy Harvesting Networks in a Boolean-Poisson Model

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1 On the Performance of Wireess Energy Harvesting Networks in a Booean-Poisson Mode Han-Bae Kong, Ian Fint, Dusit Niyato, and Nicoas Privaut Schoo of Computer Engineering, Nanyang Technoogica University, Singapore Schoo of Physica and Mathematica Sciences, Nanyang Technoogica University, Singapore Abstract Wireess radio frequency (RF) energy harvesting has been adopted in wireess networks as a method to suppy energy to wireess nodes, eg, sensors In this paper, we present a new anaysis of the wireess energy harvesting network based on a Booean-Poisson mode This mode considers that the energy sources have a fixed coverage range The energy sources are distributed according to a Poisson point process (PPP) whie their radii of coverage are random and are assumed to foow a given probabiity distribution We derive the performance measures consisting of the energy harvesting probabiity and the transmission success probabiity both in the cases of two nodes and mutipe nodes Our anaysis is vaidated by simuation Index Terms RF energy harvesting, Booean-Poisson mode, mutihop networks I INTRODUCTION Recenty, RF energy harvesting techniques have been deveoped to et mobie and sensor nodes scavenge energy from radiated RF signas from ambient or dedicated RF sources [] Due to its advantages of powering energyconstrained devices and proonging the ifetime of wireess networks, many researchers have studied energy harvesting methods in various wireess network scenarios [] [7] For exampe, in [] [4], transmit beamforming methods which optimize the performance were deveoped for the networks where energy harvesting nodes coect power from the RF signas sent by its dedicated RF energy transmitters To anayze the performance of the network with RF energy harvesting capabiity, anaytica modes based on a Poisson point process (PPP) were introduced [5] [7] In such modes, the ocations of RF energy sources are geographicay distributed according to the PPP, assuming that nodes in the networks harvest energy from the signas transmitted by surrounding energy sources The study in [5] introduced a tradeoff among transmit power, density of base stations, and density of energy sources in an upink ceuar network In [6], the authors derived the outage probabiity of a network overaid with power beacons distributed according to a PPP Aternativey, the authors in [7] investigated the transmission success probabiity in an RF energy harvesting muti-tier upink ceuar network by modeing the eve of stored energy as a Markov chain In wireess energy harvesting networks, one can consider the coverage region of an RF energy source, ie an energy harvesting enabed area formed by the RF signas from the energy source Then, the union of the coverage regions can be understood as the energy harvesting enabed region in the networks In this sense, the probabiity that nodes can harvest energy is reated to not ony the distribution of the ocations of the RF energy sources, but aso characteristics of the coverage region However, most previous works on energy harvesting networks have not taken the features of the coverage regions into account when anayzing the networks In this paper, we assume that the RF energy sources are distributed according to a homogeneous PPP, and the coverage region of a RF source is a disc of random radius This mode is known in the iterature as the Booean mode The Booean mode has been extensivey studied, cf [8] and [9] for a thorough overview, and the coverage properties of the mode were investigated in [] [] The mode has been used in different appications, see [8] More specific exampes are [3] for appications to image anaysis and [4], [5] for appications to wireess networks However, its appication to wireess energy harvesting networks has not been previousy considered The anaysis presented in this paper considers two major situations arising in RF energy harvesting networks, ie, two node harvesting and mutipe node harvesting The performance measures in terms of the energy harvesting probabiity and transmission success probabiity are derived As for the mathematica contributions, Theorem provides the probabiity that two nodes at fixed ocations are covered by the Booean mode This is the situation in which the nodes can harvest energy and are abe to communicate with each other We extend the computation of Theorem to the case of mutipe nodes at aigned fixed points These nodes are part of a mutihop ink, and thus the important performance metric is an end-to-end transmission success probabiity We find that these resuts depend ony on the aw of the radius, and we provide some expicit computations for a pane of common distributions II THE BOOLEAN-POISSON MODEL This section introduces the required background for the study of the Booean-Poisson mode which is aso known as a Booean mode Let us define µ a probabiity measure on [, ) and consider the probabiity space (, F, P) on which we et be a PPP supported on R d [, ) The intensity measure is ` ( µ), where ` indicates the Lebesgue measure on R d We denote by B x (r) the open Eucidean ba of R d centered at x R d with radius r [, ) Moreover, we consider the foowing random set = [ B x(r), () (x,r)

2 Coverage region 5 Y coordinate (m) 5-5 Fig : Description of the Booean-Poisson mode which consists of a points covered by at east one ba The set is caed the Booean-Poisson mode, and is the underying PPP containing the pairs (x, r) of points aong with their corresponding radius We iustrate the mode in Fig In the Booean-Poisson mode, the metric of interest is the probabiity that a fixed point in R d is covered by the union of bas Let us consider a point A in space ocated at x R d and define the set n o C x = (y, r) R d [, ) :y B x(r) () Then, since x / () \C x = ;, the probabiity that the point A fas in can be computed by fairy standard cacuations: P (x ) = P ( \C x = ;) = exp ( (` ( µ)) (C x)) Z = exp ` (B x(r)) µ(dr) Z = exp v d r d µ(dr), (3) where v d, d/ / (d/ + ) represents the voume of the d- dimensiona Eucidean ba, and ( ) is the Gamma function Note that P (x ) = when R r d µ(dr) = Therefore, we can infer that a point in R d is amost surey covered by the Booean-Poisson mode if the d-th moment of µ is infinite III SYSTEM MODEL In this paper, we anayze the wireess energy harvesting networks consisting of a (random) number of ambient RF energy sources In the networks, sensor nodes harvest energy from the RF signas radiated by the ambient RF energy sources, and transmit or receive data by using the harvested energy The homogeneous PPP with density modes the ocations of the ambient RF energy sources Aso, we assume that each energy source has its own coverage region, and the sensor nodes in the region can scavenge energy from one of the energy sources The coverage region of each ambient RF energy source is modeed by the open Eucidean ba of R d centered at the ocation of the energy source with a randomy distributed radius distributed according to µ By construction, the energy harvesting enabed region is modeed as the Booean-Poisson mode as defined in () Fig iustrates a reaization of the network where d =, = 4, and the radius is exponentiay distributed, ie, µ(dr) = exp( r)dr with = X coordinate (m) Fig : A reaization of the network in a Booean-Poisson mode where back dots and bue circes represent the ocations of the ambient RF energy sources and boundaries of the coverage region It is worthwhie to note that expoiting the Booean-Poisson mode to mode the energy harvesting networks is a new approach and different from the conventiona approach in [5] [7] Most previous works on the wireess energy harvesting networks have not considered characteristics of the coverage range and ony focused on the ocations of the energy sources However, the probabiity that a sensor node in the network can harvest energy is strongy dependent on the distribution of the coverage range Therefore, it is important to investigate the performance of the networks where the energy harvesting enabed region is modeed as the Booean-Poisson mode Let us consider n sensor nodes A, A,,A n which are respectivey ocated at x R d,x R d,,x n R d We assume muti-hop networks where node A transmits data to node A n aided by nodes A,,A n In this case, the transmission succeeds when a nodes are ocated in the energy harvesting enabed region, and the received signato-noise-ratio (SNR) of each hop is arger than a certain threshod th which means the minimum SNR required for the successfu data detection Therefore, by defining P H and P T as the probabiities that a nodes can harvest energy and the received SNRs for a hops are higher than th, respectivey, the transmission success probabiity P S can be expressed as where and P S = P H P T, (4) P H = P (8k {,,n}, x k ), (5) P T = P (8k {,,n }, k th) (6) Here, k, Ph k x k+ x k stands for the received SNR for the k-th hop where P is the transmit power at a nodes, h k denotes the fading power for the k-th hop, indicates the pathoss exponent, and accounts for the power of additive white Gaussian noise In this paper, the fading powers

3 3 (a) Non-coverage: ony A is covered Fig 4: An iustration for the case of r / where, kx x k is the distance between A and A Here, I z (a, b) is the reguarized incompete beta function defined by (b) Coverage: both A and A are covered Fig 3: Exampes to iustrate networks with two nodes {h k } are assumed to foow independenty and identicay distributed (iid) exponentia distribution with parameter It is aso assumed that the {h k } are independent of the PPP IV PERFORMANCE ANALYSIS In this section, we first derive an anaytica expression for the transmission success probabiity in the networks where two nodes exist Then, we generaize the anaytica resut to the case of the networks with mutipe nodes A Networks with two nodes Let us consider two nodes A and A which are ocated at x R d and x R d, respectivey Note that both A and A ought to be ocated in in order to scavenge energy As an iustration, the pacement of A in Fig 3a corresponds to a non-coverage situation, whereas that of in Fig 3b corresponds to a coverage situation In the networks with two nodes, the coverage probabiity P H in (5) can be rewritten as P H = P (x ) + P (x ) P ({x }[{x }) Z (a) = exp v d r d µ(dr) P ({x }[{x }), (7) where (a) foows from the resut in (3) and the stationarity of the PPP In the foowing theorem, we provide an anaytica expression for P H Theorem : In the networks with two nodes, the probabiity that both nodes can harvest energy is given by Z P H = exp v d r d µ(dr) Z +exp v d r d µ(dr) Z d + + v d I /(4r ), r d µ(dr), (8) / I z(a, b) = (a + b) R z ua ( u) b du, a,b,z > (a) (b) Proof: Let us define x as a PPP on the same probabiity space as, supported on Cx c with intensity measure ` ( µ) restricted to Cx c, where X c denotes the compement of a set X Reca that C x has been defined in () We define accordingy the Booean-Poisson mode associated to x as x = S (x,r) x B x (r) Then, we have P (x / x )=P({(y, r) x : x B y(r)} = ;) = P ( x \C x = ;) =exp( (` ( µ)) (C x \Cx c )) Z =exp ` (B x (r) \B x (r) c ) µ(dr) Now, we focus on the computation of the voume in the above equation First, if r</, then ` (B x (r) \B x (r) c )= ` (B x (r)) = v d r d Second, if r /, then one has to compute the shaded area in Fig 4 (represented here in dimension d =) The above d-dimensiona voume (known in the iterature as hyperspherica cap) is equa to [6] ` (B x (r)\b x (r) c )=v d r d d + I /(4r ), Hence, P (x / x ) becomes Z P (x / x )=exp v d r d µ(dr) + v d Z I /(4r ) / d +, r d µ(dr) (9) Note that the aw of given \C x = ; coincides with the distribution of x, and therefore P ({x }[{x }) in (7) can be written as P ({x }[{x }) = P (x / ) P (x / x / ) = P (x / ) P (x / \C x = ;) = P (x / ) P (x / x ) () Combining (7), (9) and (), we obtain the resut in (8) In the foowing coroary, we customize Theorem to the case d =, which is the main situation of interest

4 4 We remark that P H can be computed as Fig 5: An exampe to iustrate networks with mutipe nodes Coroary : When d =, the coverage probabiity can be simpified as Z P H = exp r µ(dr) +exp r r Z Z r µ(dr)+ arccos / r r µ(dr), () 4r where arccos( ) denotes the inverse cosine function, and we reca that is the distance between A and A Proof: By using mathematica software such as Mathematica, we may rewrite I /(4r ), 3 as I /(4r ) 3, = = arccos r Z /(4r ) r r r u u du 4r Aso, if d =, v is equa to From these resuts, we derive the expression in () When the networks consists of ony two nodes, P T in (6) is easiy computed as Ph th P T = P th =exp () P From (8), () and (), we deduce the transmission success probabiity in (4) B Networks with mutipe nodes In this subsection, we concentrate on the wireess energy harvesting networks containing mutipe nodes Let us consider A,,A n nodes (here n ) which are ocated at x R d,,x n R d, respectivey, and define the internoda distance as a,b, kx b x a k, a,b {,,n} We assume that the nodes are aigned as shown in Fig 5 In this exampe, as node A 3 is not covered, node A 3 does not have avaiabe power for data decoding and encoding, and therefore outage occurs when node A attempts to communicate with node A 5 through nodes A, A 3 and A 4 P (8k {,,n}, x k ) = P (9k {,,n}, x k / ) (c) X =+ ( ) X P (8k X, x k / ), (3) X {,,n}, X6=; where (c) foows from the incusion-excusion principe and X denotes the cardina of a set X We expoit reation (3) in the foowing theorem, wherein we derive a procedure for computing the probabiity that a n nodes are covered by the Booean-Poisson mode Theorem : Let X = (a,,a n ) {,,n} correspond to one of the terms appearing in (3) Then P (8k X, x k / ) is given inductivey by P (8k {a,,a n}, x k / ) = P (8k {a,,a n }, x k / ) Z exp v d r d µ(dr) + v d Z an,a n I an,a n 4r d +, r d µ(dr), whist noting that the initia term P (x a /, x a / ) has been computed in Theorem Proof: In this setting, one may compute the probabiity in the summation in (3) by induction as foows: P (8k {a,,a n}, x k / ) = P (8k {a,,a n }, x k / ) P x an / \C xa = ;,, \C xan = ; = P (8k {a,,a n }, x k / ) exp (` ( µ)) C xan \ (C xa [ [C xan ) c = P (8k {a,,a n }, x k / ) Z exp ` B xan (r) \B xa (r) c \ \B xan (r) c )µ(dr) From the assumption that the n nodes are aigned, we have B xan (r)\b xa (r) c \\B xan (r) c =B xan (r)\b xan (r) c Thus, in a simiar manner to Theorem, the voume can be computed as ` B xan (r)\b xan (r) c =v d r d d + I, an,a n 4r This concudes the proof In muti-hop networks, P T in (6) becomes P T = P 8k {,,n }, Ph k =exp P X n th k+,k k= k+,k th (4) We may then derive the transmission success probabiity P S in (4) using Theorem, ie, (3) and (4)

5 5 C Distribution of the radius In this subsection, we consider some specific distributions for the radius, ie, discrete distribution, continuous uniform distribution and Gamma distribution Then, we introduce more simpified expressions for the integras in the coverage probabiity For the simpicity of presentation, we assume d =and define the two integras appearing in Coroary as so that,, Z Z / r µ(dr), arccos r r r r µ(dr), 4r P H = exp( )+exp( + ) The Energy Harvesting Probabiity = 5 Networks with Two Nodes 6 = 5 =, Anaysis =, Simuation = 3, Anaysis = 3, Simuation = 5, Anaysis = 5, Simuation ; Fig 6: Comparison of the energy harvesting probabiity for the networks with two nodes and exponentiay distributed radius ) First, assume that the radius can take ony a finite number of vaues, denoted by R,,R m (, ) for some m, and set p i = µ({r i }) for i =,,m, which is the probabiity that the radius is equa to R i This corresponds to the foowing choice of µ mx µ(dr) = Ri (dr), i= where r denotes the Dirac measure at point r [, ) Then, we obtain = P m i= p iri and mx = i= p i {Ri /} arccos R i p i R i s 4R i R i ) Next, et us consider the case where the radius is uniformy distributed on [,R] for a fixed R [, ), ie, µ(dr) = R [,R] dr Then, can be easiy obtained as = R /3 Now, we focus on the computation of for R / as = for R < / Note that an antiderivative of r 7 r (arccos(/(r)) /(r) p /(4r )) is r '(r), r3 r 3 arccos r 3 4r r + 3 n r + 4 4r Hence, we can compute as = '(R) ' R 3) Last, assume that the radius is Gamma distributed, ie, µ(dr) = (m) m rm e r/ dr, where m> and > stand for the shape and scae parameters, respectivey Then, we can readiy cacuate as = m( + m) We remark that to the best of our knowedge, when the radius is Gamma distributed, the integra does not have a cosed form in terms of standard mathematica functions V SIMULATION RESULTS In this section, we provide numerica resuts to vaidate our anaytica resuts We assume that d =and use the ines and symbos to denote the anaytica and simuated resuts, respectivey Figs 6 and 7 iustrate the energy harvesting probabiity of the networks when the radius is exponentiay distributed with parameter, ie, is an inverse of the mean Note that this distribution is same as the Gamma distribution with parameters m =and =/ In Fig 7, the density is fixed as = 5 From the figures, it is shown that the anaytica resuts match we with the simuated resuts Since the mean of the radius decreases as grows, the energy harvesting probabiity decreases when increases When the nodes in the network are cose together, the probabiity that the nodes are in the coverage region becomes high Therefore, in Fig 6, we can see that the energy harvesting probabiity increases as the distance between two nodes becomes sma Aso, in Fig 7, it is shown that the energy harvesting probabiity decays as the number of sensor nodes increases In Figs 8 and 9, we evauate the transmission success probabiity of the networks where the radius is uniformy distributed on [,R] Here, the SNR is defined as P/, and th =, =and =4 In Fig 8, the distance between nodes is set to =, and in Fig 9, the density is fixed as =5 We can see that the transmission success probabiity increases as R grows This is due to the fact that the coverage region becomes arger as R increases Moreover, in Fig 8, it is shown that the transmission success probabiity grows as the density becomes arger Note that the probabiity P T in (6) is the product of the probabiities that the received SNR for each hop is arger than th Consequenty, in Fig 9, we observe that the transmission success probabiity rapidy decays to zero as the number of hops increases if the SNR is ow

6 6 The Energy Harvesting Probabiity Networks with Mutipe Nodes n =,, = 5, Anaysis n =,, = 5, Simuation n = 3, (, ) = (5,3), Anaysis n = 3, (, ) = (5,3), Simuation n = 4, (, ) = (5,3,), Anaysis n = 4, (, ) = (5,3,), Simuation n = 5, (,, 5,4 ) = (5,3,,4), Anaysis n = 5, (,, 5,4 ) = (5,3,,4), Simuation ; Fig 7: Comparison of the energy harvesting probabiity for the networks with mutipe nodes and exponentiay distributed radius The Transmission Success Probabiity =, Anaysis 6 =, Simuation 6 = 5, Anaysis 6 = 5, Simuation 6 =, Anaysis 6 =, Simuation Networks with Two Nodes SNR = 5 db SNR = 5 db R Fig 8: Comparison of the transmission success probabiity for the networks with two nodes and uniformy distributed radius The Transmission Success Probabiity n =,, =, Anaysis n =,, =, Simuation n = 3, (, ) = (,), Anaysis Networks with Mutipe Nodes n = 3, (, ) = (,), Simuation n = 4, (, ) = (,,), Anaysis n = 4, (, ) = (,,), Simuation n = 5, (,, 5,4 ) = (,,,), Anaysis n = 5, (,, 5,4 ) = (,,,), Simuation SNR = 5 db SNR = 5 db R Fig 9: Comparison of the transmission success probabiity for the networks with mutipe nodes and uniformy distributed radius VI SUMMARY In this paper, we have presented a nove anaytica framework for anayzing the performance of wireess energy harvesting networks The framework is based on the cassica Booean-Poisson mode The probabiity of a singe node being covered by the Booean-Poisson mode is we-known We have extended this computation by considering the cases of two nodes and mutipe nodes The former invoves the transmitter and receiver, whie the atter aso incudes reays In both of these settings, the energy harvesting probabiity and transmission success probabiity have been derived We have conducted an extensive simuation to vaidate the described anaytica framework ACKNOWLEDGEMENTS This work was supported in part by the Nationa Research Foundation of Korea (NRF) grant funded by the Korean government (MSIP) (4RA5A478), Singapore MOE Tier (RG8/3 and RG33/) and MOE Tier (MOE4- T--5 ARC 4/5 and MOE-T--33 ARC 3/3) REFERENCES [] X Lu, P Wang, D Niyato, D I Kim, and Z Han, Wireess Networks With RF Energy Harvesting: A Contemporary Survey, IEEE Commun Surv Tut, vo 7, pp , Second Quarter 5 [] R Zhang and C K Ho, MIMO Broadcasting for Simutaneous Wireess Information and Power Transfer, IEEE Trans Wireess Commun, vo, pp 989, May 3 [3] J Park and B Cerckx, Joint Wireess Information and Energy Transfer in a K-User MIMO Interference Channe, IEEE Trans Wireess Commun, vo 3, pp , Oct 4 [4] H Lee, S-R Lee, K-J Lee, H-B Kong, and I Lee, Optima Beamforming Designs for Wireess Information and Power Transfer in MISO Interference Channes, IEEE Trans Wireess Commun, vo 9, pp 48 48, Sep 5 [5] K Huang and V K N Lau, Enabing Wireess Power Transfer in Ceuar Networks: Architecture, Modeing and Depoyment, IEEE Trans Wireess Commun, vo 3, pp 9 9, Feb 4 [6] J Guo, S Durrani, X Zhou, and H Yanikomerogu, Outage Probabiity of Ad Hoc Networks With Wireess Information and Power Transfer, IEEE Commun Lett, vo 4, pp 49 4, Aug 5 [7] A H Sakr and E Hossain, Anaysis of K-Tier Upink Ceuar Networks With Ambient RF Energy Harvesting, IEEE J Se Areas Commun, vo 33, pp 6 38, Oct 5 [8] S N Chiu, D Stoyan, W S Kenda, and J Mecke, Stochastic Geometry and Its Appications Third Edition, Wiey, 3 [9] R Meester and R Roy, Continuum Percoation Cambridge University Press, 996 [] P Ha, On Continuum Percoation, Ann Probab, vo 3, pp 5 66, 985 [] P Ha, On the Coverage of k-dimensiona Space by k-dimensiona Spheres, Ann Probab, vo 3, pp 99, 985 [] H Biermé and A Estrade, Covering the Whoe Space With Poisson Random Bas, ALEA Lat Am J Probab Math Stat, vo 9, pp 3 9, [3] J Serra, Image Anaysis and Mathematica Morphoogy Academic Press, 984 [4] F Baccei, Bartłomiej, and Błaszczyszyn, On a Coverage Process Ranging from the Booean Mode to the Poisson Voronoi Tesseation With Appications to Wireess Communications, Adv in App Probab, vo 33, pp 93 33, [5] M Haenggi, J G Andrews, F Baccei, O Dousse, and M Franceschetti, Stochastic Geometry and Random Graphs for the Anaysis and Design of Wireess Networks, IEEE J Se Areas Commun, vo 7, pp 9 46, Sep 9 [6] S Li, Concise Formuas for the Area and Voume of a Hyperspherica Cap, Asian J Math Stat, vo 4, pp 66 7,