WIRELESS sensor networks (WSNs), which are capable

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1 IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. XX, NO. XX, XXX Lifetime and Enegy Hole Evolution Analysis in Data-Gatheing Wieless Senso Netwoks Ju Ren, Student Membe, IEEE, Yaoxue Zhang, Kuan Zhang, Student Membe, IEEE, Anfeng Liu, Jiane Chen, and Xuemin (Sheman) Shen Fellow, IEEE Abstact Netwok lifetime is a cucial pefomance metic to evaluate data-gatheing wieless senso netwoks (WSNs) whee battey-poweed senso nodes peiodically sense the envionment and fowad collected samples to a sink node. In this pape, we popose an analytic model to estimate the entie netwok lifetime fom netwok initialization until it is completely disabled, and detemine the bounday of enegy hole in a data-gatheing WSN. Specifically, we theoetically estimate the taffic load, enegy consumption, and lifetime of senso nodes duing the entie netwok lifetime. Futhemoe, we investigate the tempoal and spatial evolution of enegy hole, and apply ou analytical esults to WSN outing in ode to balance the enegy consumption and impove the netwok lifetime. Extensive simulation esults ae povided to demonstate the validity of the poposed analytic model in estimating the netwok lifetime and enegy hole evolution pocess. Index Tems wieless senso netwok, netwok lifetime, enegy hole, enegy efficiency, outing. I. INTRODUCTION WIRELESS senso netwoks (WSNs), which ae capable of sensing, computing, and wieless communication [1] [3], ae widely applied to many applications such as militay suveillance, envionmental monitoing, infastuctue and facility diagnosis, and othe industy applications [4]. A data-gatheing WSN consists of a lage numbe of batteypoweed senso nodes that sense the monitoed aea and peiodically send the sensing esults to the sink. Since the batteypoweed senso nodes ae constained in enegy esouce and geneally deployed in unattended hostile envionment, it is cucial to polong the netwok lifetime of WSN. Meanwhile, as enegy consumption is exponentially inceased with the communication distance accoding to the enegy consumption model [5], multi-hop communication is beneficial to data gatheing fo enegy consevation. Howeve, since the nodes close to the sink should fowad the data packets fom othe nodes, they exhaust thei enegy quickly, leading to an enegy hole aound the sink. As a esult, the entie netwok is subect to pematue death because it is sepaated by the enegy hole. Thee have been seveal existing woks studying the enegy consumption and netwok lifetime analysis fo WSNs. Most of Ju Ren, Yaoxue Zhang, Anfeng Liu and Jiane Chen ae with the College of Infomation Science and Engineeing, Cental South Univesity, Changsha, China, Ju Ren is also a visiting PhD student at the Univesity of Wateloo, Canada. {en u, zyx, anfengliu, iane}@csu.edu.cn. Kuan Zhang and Xuemin (Sheman) Shen ae with the Depatment of Electical and Compute Engineeing, Univesity of Wateloo, Wateloo, ON, Canada, N2L 3G1. {k52zhang, xshen}@bbc.uwateloo.ca. Copyight (c) 29 IEEE. Pesonal use of this mateial is pemitted. Howeve, pemission to use this mateial fo any othe puposes must be obtained fom the IEEE by sending a equest to pubs-pemissions@ieee.og. them [6] [8] focus on the duation fom netwok initialization to the time when the fist node dies (i.e., Fist Node Died Time, FNDT), aiming to impove the netwok pefomances and optimize the FNDT. Chen et al. [6] popose an analytic model fo estimating the taffic load of senso nodes and FNDT in a multi-hop WSN. Geneal netwok lifetime and cost models ae also discussed in [9] to evaluate node deployment stategies. Since netwok lifetime is limited by unbalanced enegy consumption, Ok et al. [1] popose a distibuted enegy balanced outing (DEBR) algoithm to balance the data taffic of senso netwoks and consequently polong the FNDT. As hieachical outing has been poved to be beneficial fo netwok pefomance [11], especially fo the scalability and enegy consumption, eseach woks also study the FNDT of cluste-based WSNs. Lee et al. [11] deive the uppe bound of FNDT in cluste-based netwoks and investigate the effects of the numbe of clustes and spatial coelation on this bound. Liu et al. [12] also discuss the FNDT of cluste-based netwoks, and popose a outing potocol to impove FNDT based on unequal cluste adii. Although most of existing woks ae effective to estimate FNDT, the peiod fom FNDT to the time when all the senso nodes ae dead o the netwok is completely disabled (i.e., All Node Died Time, ANDT) is elatively long [13]. Fo most applications, a small potion of dead nodes may not cause a netwok failue, although they can impact the netwok pefomances [12] [14]. Thus, this peiod is nonnegligible fo the entie netwok lifetime. On the othe hand, pefomance analysis on this peiod is difficult and intactable because the netwok is unstable afte a few nodes die. Once the nodes with heavy load die, some othe nodes should elay the data oiginally fowaded by these dead nodes. It leads to dynamical changes of the outing paths, as well as the taffic load of senso nodes. Theefoe, it is necessay to analyze the pefomance and netwok lifetime afte FNDT. Recently, inceasing attention has been paid to the peiod fom FNDT to ANDT and analyzing the entie netwok lifetime [7], [13] [15]. Ozgovde et al. [13] highlight that FNDT is only an impotant stage of the entie netwok lifetime and ANDT is an impotant facto to evaluate the netwok pefomance. To this end, they popose a utility based lifetime measuement famewok called Weighted Cumulative Opeational Time (WCOT), which calculates a netwok lifetime function based on the complete histoy of the netwok states. Lee et al. [15] analyze the entie aging pocess of the senso netwok in a data-gatheing WSN. Howeve, they mainly focus on the netwok connection time athe than analyzing the

2 IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. XX, NO. XX, XXX enegy consumption and lifetime of the senso nodes. In [16], Li pesents an annuli-based analytic model to analyze the netwok lifetime fo data-gatheing WSNs. The netwok is divided into a numbe of annulus whose widths ae equal to the tansmission ange of the senso nodes, and all the senso nodes in the same annuli ae assumed to die simultaneously. The netwok lifetime is defined as the time elapsing fom netwok initialization until the senso nodes in any one annuli ae dead, and it is finally deived as a function of the numbe of annuli. Based on [16], Liu et al. [17] adopt an impoved annuli-based analytic model to study FNDT and ANDT, and popose a non-unifom node distibution scheme to achieve optimal netwok lifetime. Meanwhile, some eseach woks also investigate the entie netwok lifetime fo event-diven WSNs [14], [18], [19]. Since data taffic in event-diven WSNs is busty and follows a specific distibution, netwok lifetime is geneally analyzed by pobabilistic appoaches. Howeve, few of the existing woks povide a compehensive analysis fo the enegy consumption and lifetime of senso nodes, and consides the negative impacts of enegy hole on the netwok lifetime. Enegy hole is cucial and challenging fo lifetime analysis in WSNs, because it can lead to a pematue death of the netwok [2]. Olaiu et al. [21] fist pove that the enegy hole poblem is inevitable in the WSN unde some specific conditions. Peillo et al. [22] analyze in what condition the enegy holes could appea. Rahim et. al. [23] discuss the load balancing techniques to mitigate enegy hole poblem in lagescale WSNs, and popose a distibuted heuistic solution to balance the enegy consumption of senso nodes by adusting thei tansmission powe. The enegy hole poblem has also been studied in cluste-based WSNs [24], [25]. Most of the existing woks [25] [27] suppose that enegy hole locates aound the sink, and design enegy-efficient outing potocols to mitigate the unbalanced enegy consumption and polong the netwok lifetime. Howeve, ecent investigations [12], [17] point out that enegy hole does not always emege close to the sink and highly depends on some netwok paametes, such as the enegy consumption model and tansmission ange of senso nodes. Howeve, theoetic analysis is not povided in existing woks to estimate the emeging time and location of the enegy hole, as well as its evolution pocess. In this pape, we popose an analytic model to estimate the entie netwok lifetime fom netwok initialization until it is completely disabled, and detemine the bounday of enegy hole in data-gatheing WSNs. To accuately estimate the enegy consumption of senso nodes, we conside the enegy consumption not only fo data tansmitting and eceiving, but also fo idle listening. Specifically, ou contibutions ae theefold. (i) We popose an analytic model to estimate the taffic load, enegy consumption and lifetime of senso nodes duing the entie netwok lifetime. Futhemoe, we estimate the netwok lifetime unde a given pecentage of dead nodes, and the emaining enegy of the netwok based on ou analytical esults. Extensive simulations demonstate that the poposed analytic model can estimate the netwok lifetime within an eo ate smalle than 5%. (ii) Based on the lifetime analysis of senso nodes, we investigate the tempoal and spatial evolution of enegy hole fom emeging to patitioning the netwok, which povides a theoetical basis to mitigate o even avoid enegy hole in WSNs. (iii) To validate the effectiveness of ou analytical esults in guiding the WSN design, we apply them to WSN outing. The impoved outing scheme based on ou analytical esults efficiently balances the enegy consumption and significantly impoves the netwok lifetime, including FNDT and ANDT. The emainde of the pape is oganized as follows. Section II intoduces the system model and fomulates ou poblem mathematically. In Section III, we theoetically analyze the taffic load, enegy consumption and lifetime of senso nodes. We detemine the bounday of enegy hole in Section IV as well as some obsevations on netwok chaacteistics. Section V validates the analytic model by compaing the analytical esults with extensive simulation esults. We apply ou analytical esults to WSN outing in Section VI. Finally, Section VII concludes the pape and outlines the futue wok. II. NETWORK MODEL AND PROBLEM STATEMENT A. Netwok Model Conside a data-gatheing WSN [6], [12], whee n homogeneous sensos ae andomly deployed in a cicula egion with the sink (base station) located at the cente [28], [29]. The netwok adius is R and the tansmission ange of each senso is. The senso nodes ae unifomly distibuted in the netwok with a node density ρ [12], [14], [21]. Each senso monitos a specific aea and peiodically sends the sensed data to the sink in a data peiod (o data ound). Theefoe, netwok lifetime can be measued by the numbe of data peiods (o ounds). All the sensed data ae deliveed to the sink using geedy geogaphic outing [21], [24]. Senso nodes fowad packets to one of thei neighboing nodes, which ae geogaphically closest to the sink among all the neighbos. Geogaphic outing is scalable fo lage WSNs, since it only equies local infomation to make fowading decisions. This outing scheme has been widely adopted in multi-hop wieless senso and ad-hoc netwoks [11], [17]. In addition, ou netwok is based on a collision-fee MAC potocol without data loss ust as the assumptions in [6], [12], [3], then we can focus on the impact on the netwok lifetime caused by the outing potocol, to povide a significant guidance fo outing design on the netwok laye. Sensos opeate in active mode o sleep mode. The atio of the time in active mode to a total data peiod is called duty cycle, denoted by γ. In geneal, sensos consume enegy mainly in data eceiving and tansmitting, and idle listening when they ae in active mode [31], [32]. We do not conside the enegy consumption in sleep mode because it is small enough to be neglected [6], [32], [33]. Accoding to the adio model [11], enegy consumption fo tansmitting and eceiving c bits of data ae shown in Eq. (1) and Eq. (2), espectively. { ce elec + ce fs d 2, d d, E t = ce elec + ce amp d 4 (1), d > d.

3 IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. XX, NO. XX, XXX E = ce elec. (2) whee E elec denotes tansmitting cicuit loss and d is the theshold distance. The fee space channel model and the multi-path fading channel model ae used in Eq. (1), accoding to the distance between the tansmitte and eceive. If the tansmission distance d is lage than the theshold distance d, multi-path fading channel model should be adpoted; othewise, fee space channel model would be used. E fs and E amp denote the enegy fo powe amplification in the two models, espectively. B is the data tansmission ate of each senso node. Fo idle listening, the enegy consumption ate of the senso nodes is denoted by E idle. B. Poblem Statement We define the netwok lifetime as the duation fom the netwok initialization to the time when the netwok is disabled. Fo a data-gatheing WSN, the netwok is geneally disabled unde the following two situations. One is that all the senso nodes exhaust thei enegy and die. The othe is that the sink can not eceive any data in a data peiod due to the enegy hole patitioning the netwok, even if thee ae still a lage numbe of alive nodes in the oute egion of the enegy hole. Fo simplification, we denote the netwok lifetime as ANDT and the time when the fist node dies as FNDT. We descibe the entie pocess of netwok lifetime in Fig. 1. Since the senso nodes peiodically send the sensed data to the sink in a data peiod, the netwok lifetime is slotted into a lage numbe of data peiods. We call the data peiods in which at least a senso node dies as death peiods. Since senso nodes die successively though the netwok lifetime, we can set thee ae k (k n) death peiods [DP, DP 1, DP 2,..., DP k 1 ] in the entie netwok lifetime. Theefoe, the entie netwok lifetime is divided into k + 1 stages [S, S 1, S 2,..., S k 1, S k ] by the k death peiods. S i denotes the i-th netwok stage whee the last data peiod is the i-th death peiod, e.g., the fist senso node dies at the end of the stage S and the netwok is totally disabled at the stage S k. The numbe of alive nodes at each stage befoe the death peiod is denoted by [ S, S 1, S 2,..., S k 1, S k ] (e.g., S = n, S k = ), and [{S }, {S 1 }, {S 2 },..., {S k 1 }, {S k }] denotes the sets of the alive nodes (e.g., {S } is the set of all senso nodes, {S k } = (i.e., empty set)). The duation at each stage, namely the numbe of data peiods at each stage, is denoted by [l, l 1, l 2,..., l k 1 ]. Thus, l () is the netwok lifetime fom netwok initialization until the fist node dies (FNDT). The aveage taffic load[ of node in a data ound ] of each stage is denoted by p (), p (1),...,,..., p(k 1). Obviously, p (k) =. The tansfe function fom the taffic load to enegy consumption is f, which can be detemined accoding to the enegy consumption model. Thus, the aveage enegy consumption of node in a data ound of the i-th stage is f. Similaly, we have e(k) =. The fequently used mathematical notations in this pape ae summaized in Table I. The notations defined above can denote diffeent netwok lifetimes. Fo example, FNDT is l () TABLE I FREQUENTLY USED NOTATIONS Notation Definition R Netwok adius (m) Tansmission ange of senso (m) ρ Density of senso nodes E Initial enegy of a senso node B Data tansmission ate of a senso node d Theshold distance in adio model A x A small egion whee the nodes distances to the sink ae equal o close to x DP i The i-th death peiod S i The i-th netwok stage {S i } The set of alive nodes at the beginning of S i S i The numbe of alive nodes at S i Pe-ound taffic load of node at S i l (i) and ANDT is k 1 Pe-ound enegy consumption of node at S i (nj) The duation of the i-th stage (ounds) i= the senso nodes die is l (i), and the netwok lifetime when half of i l (i) whee S i = n/2. = The obective of this pape is to estimate the nodal taffic load, enegy consumption and netwok lifetime fo a given netwok, so as to povide impotant guidelines fo netwok optimization, such as outing design and node deployment. Specifically, we pesent ou obectives as follows. (1) Fo a given netwok, the aveage taffic load and enegy consumption of the senso nodes at each stage, that is, fo each < i n and i k 1,, e(i), as well as the enegy tansfe function f and the duation vecto of the netwok stages [l, l 1, l 2,..., l k 1 ] should be povided by ou analysis. Then, we can descibe the chaacteistics of the taffic load, enegy consumption and lifetime ove the entie netwok lifetime. (2) The bounday and emeging time of the enegy hole should be detemined to povide a theoetical foundation fo mitigating even avoiding the enegy hole poblem. III. ESTIMATION ON NODAL TRAFFIC LOAD, ENERGY CONSUMPTION, AND NETWORK LIFETIME CHARACTERISTICS In this section, we theoetically estimate the taffic load and enegy consumption of senso nodes, as well as the duation of each netwok stage based on ou system model. The main idea of the analytic model can be descibed as follows. We fist divide the netwok into a numbe of small egions whee the nodes have simila distances to the sink. Since the enegy consumption of the senso nodes in the same egion (i.e., with the simila distances to the sink) should be the same fom a statistical point of view, we use the aveage enegy consumption of this egion as the nodal enegy consumption of this egion. Fig. 2 shows a secto zone of the netwok, whee A x is a egion with the width of ε

4 IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. XX, NO. XX, XXX DP S S 1 S i S k-1 S k DP 1 l l 1 l i l k-1 t FNDT ANDT... DP i... DP k-1 Dead Senso Node Data Peiod Fig. 1. Desciption of the entie pocess of netwok lifetime. Fig. 2. A x Ax Data fowading model. R and θ is the angle fomed by A x and the sink. The nodes distances to the sink in A x equal o ae close to x. Since the nodal tansmission adius is, A x is supposed to fowad the data fom A x+ whose distance to A x is. Likewise, A x+ elays the data fom A x+2. Howeve, if the divided egion is elatively lage (i.e., ε is elatively lage), the enegy consumption of the senso nodes in it cannot be balanced. On the othe hand, if the divided egion is too small, thee might be no node in this egion. Theefoe, we specify the constaint of ε and θ as follows. Accoding to the netwok model, senso nodes ae unifomly distibuted with a node density ρ. Since A x is a elatively small egion, if x ε, the aea of A x can be appoximated as W Ax = xεθ; othewise, W Ax = θε 2. Then, the numbe of nodes in A x is N Ax = W Ax ρ = { xεθρ, if x ε; θε 2 ρ, othewise.. (3) As we discussed above, the numbe of nodes in A x should be lage than 1, which means N Ax 1. That is, Ax 2 ε 1 θρ. (4) Theefoe, we can choose the value of ε and θ unde the constaint of Eq. (4) to keep the egion of A x small enough, which can ensue the balanced enegy consumption of the senso nodes in A x. A. Taffic Load Analysis at S S indicates a stage when no node dies, and hence is the most impotant stage with the best pefomance. We fist analyze the taffic load of senso nodes based on ou analytic model descibed above, by the following theoem. Theoem 1: Assume node is in the small egion of A x with the width of ε. Denote x as the distance between A x and the sink, and θ as the angle fomed by A x and the sink. If each node geneates one data packet pe ound, the aveage data amount sent by in a ound at S is: p () (z 1 + 1) + z 1(1 + z 1 ), if x ε; = 2x 1 2 (z 2 + 2)ε 2 θρ + 1, 2 z 2(z 2 + 1)εθρ, othewise. (5) whee z 1 = (R x)/ and z 2 = (R ε)/. Poof: Since node is in the small egion of A x, its taffic load can be calculated as the aveage taffic load in A x accoding to ou analytic model. Theefoe, we fist calculate the aveage taffic load in A x. As shown in Fig. 2, ε is the width of A x and θ denotes the angle fomed by A x and the sink, thus, we can obtain the numbe of nodes in A x accoding to Eq. (3). As these nodes eceive and fowad the data fom the upsteam egions, the numbe of senso nodes in the upsteam egions A x+i, accoding to Eq. (3), is N Ax+i = { (x + i)εθρ < i z 1, if x ε; (ε/2 + i)εθρ < i z 2, othewise., (6) whee z 1 = (R x)/ and z 2 = (R ε)/. Since each node only geneates a data packet pe ound, the numbe of data packets equals to the numbe of the involved nodes. Thus, the numbe of data packets sent by A x is: D Ax = N Ax + N Ax N Ax+z. (7) Based on Eq. (7), we have the aveage taffic load of A x as D A x N Ax. Since the taffic load of the node appoximately equals the aveage taffic load of the senso nodes in A x, the taffic load of the node at S should be p () some simple calculation, we have p () as Eq. (5). = D A x N Ax. With B. Enegy Consumption Analysis at S and Estimation of l () The taffic load of senso nodes at S can be detemined by Thm. 1. If each data packet contains τ bits, the total amount of tansmitted data is p () x τ. In this pape, enegy consumption fo netwok contol is not consideed since it is almost the same fo each node and elatively small in geedy geogaphic outing [14], [17]. Theefoe, we detemine the enegy consumption of senso nodes at S in the following theoem.

5 IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. XX, NO. XX, XXX Theoem 2: Denote T as the time peiod fo one data gatheing ound. Node is in the egion A x, whee x is the distance between A x and the sink. If the data tansmission ate of the senso node is B bits/second, in a data ound, the aveage enegy consumption of is =, + e(),t + e(),i, whee, = (p() x 1)τE elec,t = p() x τ(e elec + ε κ d α ),i = E idlet () x,i = E idle(t a 2p () x τ/b + τ/b) and if x >, d = ; othewise, d = x, and if d d, ε κ = ε fs and α = 2; othewise, ε κ = ε amp and α = 4. Poof: In a data ound, the enegy consumption of node consists of the following thee pats. (a) Enegy consumption fo data eceiving. Since node is in the egion A x, the eceived data amount 1), accoding to Thm. 1. Thus, the enegy consumption fo eceiving is, = (p() 1)τE elec. (b) Enegy consumption fo data tansmitting. in a ound is (p () Since the data amount sent by in a ound is p () enegy consumption fo data tansmitting is { (8), the,t = p() τ(e elec + ε fs d 2 ), if d d,t = p() τ(e elec + ε amp d 4 ), othewise. (9) If x >, d = ; othewise d = x. That is because the tansmission distance is if x ; othewise, it is x. (c) Enegy consumption fo idle listening. Accoding to the netwok model, the duty cycle is γ [31], [32]. Thus, the active time pe ound is t a = T γ. The enegy consumption fo idle listening is the multiplication of E idle and the duation in idle listening. Since the duation in idle listening, denoted by t (),i, is the active time excluding the time fo data tansmitting and eceiving, we have t (),i = t a (p () 1)τ/B p () τ/b. Theefoe, we deive the enegy consumption fo idle listening as,i = E idlet (),i = E idle(t a 2p () x τ/b + τ/b). To summaize, in a ound, the enegy consumption of node, is =, + e(),t + e(),i. Accoding to Thm. 1 and 2, seveal phenomena can be concluded as follows. (a) Nodal taffic load and enegy consumption have a diect elationship with the tansmission adius, which might cause the location of the hotspot deviating fom the adacent aea of the sink. (b) When is fixed, the total enegy consumption is impacted by the enegy consumption fo idle listening. (c) Fo the nodes nea the sink, since the taffic load is elatively lage, the time peiod fo data tansmitting and eceiving is long, while the time peiod in idle listening is elatively shot. Thus, the popotion of enegy consumption fo idle listening fo these nodes is lowe than the nodes fa fom the sink. Since the fist batch of dead nodes must be the ones with the maximum enegy consumption in the netwok, the FNDT l () is l () E =, (1) max( x ) whee E is the initial enegy of the senso nodes. Theefoe, we summaize the analytical esults at S as follows. [ (1) The pe-ound taffic ] load of senso nodes at S, i.e., p () 1, p() 2,..., p(),..., p () n, can be obtained by Thm. 1. (2) [ The pe-ound enegy consumption ] of each node at S, i.e., 1, e() 2,..., e(),..., n, can be obtained by Thm. 2. (3) The FNDT l () can be detemined by the Eq. (1) and the enegy tansfe function f is the enegy consumption fomula as Eq. (8). C. Estimation on Taffic Load, Enegy Consumption and Netwok Lifetime fom S 1 to S k 1 In the pevious subsections, we have detemined the taffic load and enegy consumption of senso nodes at S. In this subsection, we analyze the taffic load and enegy consumption of the senso nodes afte S, which is complicated because netwok outing paths change dynamically afte S. At fist, we should find out that which pat of senso nodes die fist. Accoding to Eq. (1), the senso nodes with the maximum enegy consumption will die fist. And we can detemine the location of the fist batch of dead nodes by combining Eq. (5), (8) and (1). Accoding to ou analytic model, the enegy consumption of the senso nodes in a egion with width of ε ae the same and the enegy consumption of the egions with the same distance to the sink should be the same too. Theefoe, the netwok can be divided into a numbe of ing egions with the same enegy consumption and the width of ε. Without loss of geneality, we set the fist batch of dead nodes in the ing egion of [u, u + ε], the numbe of dead nodes is (π(u + ε) 2 πu 2 ) ρ. Then, we should detemine that whee ae the i-th (i 2) batch of dead nodes. The i-th (2 i k) batch of dead nodes should die in DP i 1 (i.e., the last data peiod of S i 1 ). Based on the Thm. 1 and 2, we have Coollay 1. Coollay 1: Fo 2 i k 1 and 1 n, given [ the enegy consumption ] of node fom S to S i 1 as, e (1),..., e (i 1), and the duation of the netwok stages befoe S i 1 as [ l (), l (1),..., l (i 2)], we have (( )/ i 2 l (i 1) = E e (w) l (w) e (i 1) min {S i 1} w= ), (11) whee {S i 1 } is the set of alive nodes at the beginning of S i 1. And the set of the i-th batch of dead nodes ae (( )/ ) i 2 n died = ag min E e (w) l (w) e (i 1). (12) {S i 1} w= Poof: Fo each alive node at S[ i 1, the enegy consumption of node fom S to S i 1 is, e (1),..., e (i 1), ] and [ the duation of the netwok stages befoe S i 1 is l (), l (1),..., l (i 2)]. Theefoe, the emaining enegy of node at the beginning of S i 1, denoted by E (i 1),emain, is E (i 1),emain = E i 2 e (w) l (w). (13) w=

6 IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. XX, NO. XX, XXX Since the enegy consumption of node at S i 1 is e (i 1), s lifetime at S i 1 can be detemined as l (i 1) = E (i 1),emain / e (i 1). (14) Theefoe, fo each alive node at S i 1, the duation of S i 1 should be the minimum l (i 1) and the i-th (2 i k) batch of dead nodes n died should be the nodes that has the minimum l (i 1). We have, l (i 1) = min ) and (l (i 1) {S i 1} (l(i 1) n died = ag min ), whee {S i 1 } denotes the set of {S i 1} alive nodes at the beginning of stage S i 1. Coollay 1 detemines the duation of S i based on the enegy consumption of the senso nodes at S i. Next, we should analyze and detemine the taffic load and enegy consumption of the netwok afte the i-th (1 i k 2) batch of nodes die. Accoding to ou analytic model, the enegy consumption of the senso nodes in the egions with the same distances to the sink should be equal. Theefoe, as we discussed above, we can set the i-th batch of dead nodes in the egion [u i, u i +ε]. Then, thee cases should be consideed. (a) All the nodes aound the egion ae alive. (b) The nodes close to the egion, paticulaly on the side nea the sink, ae dead. (c) The nodes close to the egion, paticulaly on the side fa fom the sink, ae dead. We denote the continuous dead egion including [u i, u i +ε] and the oiginal dead egion as [u s, u e ], i.e., all nodes in [u s, u e ] ae dead. In the fist case, [u s, u e ] is equal to [u i, u i +ε]. Thus, fo the dead egion [u s, u e ], the oiginal data fowading should be changed. Specifically, the taffic load of the following fou egions is diffeent fom that at S i 1. (1) The taffic load of the senso nodes in the dead egion [u s, u e ] is. (2) The taffic load inceases in the egion [u e, u e + ε]. The data oiginally tansmitted by the egion [u s, u e ] is fowaded by egion [u e, u e + ε] now. In othe wods, the data of [u s +, u e + ] that is supposed to be fowaded by the egion [u e, u e +ε] leads to the incement of taffic load in [u e, u e +ε]. (3) The taffic loads incease in the egions [u e i, u e ue i + ε] i z 3, whee z 3 =. Due to the inceased taffic load in the egion [u e, u e + ε], the taffic loads in the coesponding downsteam egions [u e i, u e i + ε] < i z 3 incease. (4) If u s >, the taffic loads decease in the egions [u s us i, u e i] < i z 4, whee z 4 =. Because the data of egion [u s, u e ] is oiginally fowaded by the egions [u s i, u e i] < i z 4. Since the nodes in the egion [u s, u e ] ae dead, the taffic load in these downsteam egions should be deceased. Except the egions discussed above, the taffic load in othe egions stays the same. We summaize the taffic load changes in diffeent egions of the netwok in Thm. 3. Theoem 3: Let the i-th (1 i k) batch of dead nodes be in the egion [u i, u i +ε], and [u s, u e ] denote the continuous dead egion including [u i, u i +ε] and the oiginal dead egion. Afte the i-th (1 i k) batch of nodes die at S i 1, fo each alive node in the egion A x whee x is distance bwteen A x A died A hot 1 A 1 hot A died Fig. 3. The change of data fowading afte S. R and the sink, the taffic load at S i changes to =, if x [u s, u e ] = p (i 1) + D died /(((u e + ε) 2 (u e ) 2 )πρ) if x [u e, u e + ε] z 3 = p (i 1) D died /(((u s ) 2 (u e ) 2 )πρ) if x [u s, u e ] < z 4 and u s > = p (i 1), othewise (15) z whee D died = f(ρ) (π(u e + k) 2 π(u s + k) 2 ), z = k=1 R us ue us, z 3 = and z 4 =. Poof: As shown in Fig. 3, denote the egion [u s, u e ] by A died. It is obvious that the taffic load of the dead egion A died is. Afte the senso nodes ofa died die, its coesponding upsteam egion A 1 died is supposed to fowad data to the sink though the egion A hot which is close to A died (A hot is the egion [u e, u e + ε]). Thus, A hot beas not only the data taffic of A 1 hot, but also the data taffic of A1 died which is oiginally undetaken by A died. We fist detemine the taffic load of A 1 died. The taffic load of A 1 died consists of the data taffic in its egion and fom its upsteam egions. The aea of A 1 died is π(u e + ) 2 π(u s + ) 2. And the aea of the upsteam egions of A 1 died ae R π(u e + i) 2 π(u s + i) 2 us 1 i z, whee z =. Theefoe, the taffic load of A 1 died is z D died = (π(u e + k) 2 π(u s + k) 2 ) ρ. (16) k=1 Afte the nodes of A died die, the data oiginally sent to this egion is now tansmitted to the nodes aound A died, leading to the data oiginally fowaded by A died is now sent to A hot. Theefoe, A hot and its coesponding downsteam ue egions [u e, u e + ε] < z 3, whee z 3 =, should fowad exta D died data besides its oiginal data. Since the numbe of nodes in these egions is (π(u e + ε) 2 π(u e ) 2 ) ρ z 3, and the inceased taffic load of each node in these egions is D died /((π(u e + ε) 2 π(u e ) 2 ) ρ z 3. Similaly, if u e >, the data of this egion has to be sent to,

7 IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. XX, NO. XX, XXX Algoithm 1 Detemining the taffic load, enegy consumption and lifetime of senso nodes at each netwok stage. Input: Netwok adius R, tansmission adius, node density of the netwok ρ, and othe paametes. Output: Fo each stage i and each node, etun the nodal taffic load, enegy consumption e(i), as well as the enegy tansfe function f and lifetime vecto l. 1: Detemine the taffic load[ and enegy consumption of ] each node at stage S, i.e., p () 1, p() 2,..., p(),..., p () n and [ ] 1, e() 2,..., e(),..., n, accoding to Thm. 1 and 2; 2: i = 1; 3: while the sink can eceive data in a data peiod do 4: Accoding to Coollay 1, calculate the lifetime l (i 1) at stage S i 1, and the i-th batch of dead nodes egion [u i, u i + ε]; 5: Detemine the taffic load and enegy consumption of [ the senso nodes ] at stage S i, [ 1, e(i) 2 i.e., 1, p(i) 2,..., p(i),..., p(i) n and ], accoding to Thm. 3,..., e(i),..., e(i) n and 4, ; 6: i = i + 1; 7: end while 8: etun The taffic load and enegy consumption and (fo each i and ), and the netwok stage duation vecto l (i) (fo each i). the sink via moe than one hop. Then, the downsteam egions us [u s, u e ] < z 4, whee z 4 =, do not fowad the data oiginally tansmitted by A died. Theefoe, the deceased taffic load of each node in these egions is D died /((π(u s ) 2 π(u e ) 2 ) ρ) < z 4. Meanwhile, the taffic load of othe egions should stay the same as that at S i 1. Based on the analysis above, the theoem can be poved. Accoding to Thm. 3, the enegy consumption of senso nodes at S i changes as the following theoem. Theoem 4: Assume node is in a egion A x, whee x is the distance bwtween A x and the sink. Afte senso nodes die at S i 1, the pe-ound enegy consumption of at S i changes to = x, + x,t + x,i, whee, = (p(i) x 1)τE elec,t = p(i) x τ(e elec + ε κ d α ),i = E idlet (i),i = E idle(t a 2 x τ/b + τ/b) (17) and if x >, d = ; othewise, d = x, and if d d, ε κ = ε fs and α = 2; othewise, ε κ = ε amp and α = 4. Poof: Simila with the poof of Thm. 2. Based on the pevious theoems, Alg. 1 shows how to detemine the taffic load, enegy consumption and lifetime of the senso nodes at each netwok stage. D. Analysis on Netwok Lifetime and Remaining Enegy Alg. 1 can detemine the entie netwok lifetime and the duation of each netwok stage. Howeve, diffeent WSN applications have diffeent lifetime equiements. Intuitively, lifetime equiement can be descibed by the pecentage of dead nodes in the netwok, which is also called death atio. If we use l% i to denote the equied lifetime when the death atio eaches w%, l% means FNDT and l1 % means ANDT. Based on ou analytical esults, we can have Coollay 2. Coollay 2: Given a equied death atio w%, the netwok lifetime l% w is lw % = σ 1 i= l(i), whee n S σ w% and { } n n Si σ = ag min w%. 1 i k n Poof: Since the numbe of the senso nodes in the netwok is n and the numbe of alive nodes at the i-th netwok stage is S i, the pecentage of dead nodes at the i-th netwok stage is n S i. n If the equied death atio is w%, it means the netwok lifetime consists of all the netwok stages whee the pecentage of dead nodes is below w%. Theefoe, we can detemine the netwok stage σ whee the pecentage of dead nodes fistly exceeds{ w%. Accoding to } ou analysis above, we have σ = n Si ag min w%, whee n S σ w%. And the 1 i k n n netwok lifetime l% w is the duation fom S to S σ 1, i.e., l% w = σ 1 i= l(i). Netwok lifetime can be detemined by Coollay 2 unde a given death atio. It indicates the poposed analytic model can estimate the netwok lifetime fo WSN applications with diffeent lifetime equiements. When the pecentage of dead nodes in the netwok exceeds the equied death atio, the netwok is consideed as disabled and the emaining enegy of the netwok becomes useless. Theefoe, emaining enegy can be a pefomance metic to evaluate the enegy efficiency of WSNs, and the emaining enegy of the senso nodes can guide the WSN edeployment. We detemine the emaining enegy of senso nodes in the following coollay. Coollay 3: Fo each netwok stage S i, the emaining enegy of node afte S i is ϕ (i) = E i ( l ), (i) c= and the emaining enegy of the netwok afte S i is ϕ (i) = ne i n ( l ). (i) c= = Poof: Since is the aveage enegy consumption of node in a data ound at S i and l (i) is the duation of S i, we can detemine the enegy consumption of node duing S i as E,i = l (i). Fom S to S i, the total enegy consumption of node is E i, use = i c= of the netwok is E i use = E i, and the total enegy consumption i c= = n E i,. As the initial enegy of the netwok is ne, afte S i, the emaining enegy of node is ϕ (i) = E Euse i, = E i ( ) l (i) and c= the emaining enegy of the netwok is ϕ (i) = ne Euse i = ne i n ( l ). (i) c= =

8 IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. XX, NO. XX, XXX Algoithm 2 Detemining the emeging time and bounday of the enegy hole. Input: Netwok adius R, tansmission adius, node density of the netwok ρ, and othe paametes. Output: The enegy hole bounday [d shole, d ehole ] and emeging time t h. 1: Run Alg. 1 until thee is a continuous dead ing whose width d satisfies d ; 2: The bounday of this dead egion is the equest [d shole, d ehole ]; 3: The lifetime at this netwok stage is the emeging time t h ; 4: etun [d shole, d ehole ] and t h. IV. ENERGY HOLE AND NETWORK CHARACTERISTICS A. Analysis on The Enegy Hole Evolution In this section, we investigate the tempoal and spatial evolution of enegy hole based on ou analytical esult. The taffic load and enegy consumption of the senso nodes and the netwok lifetime can be detemined by Alg. 1, whee the temination condition is that the sink cannot eceive any data in a data peiod, which consists of two cases. One is all nodes die due to enegy exhaustion. The othe is some nodes still have emaining enegy, but the sink is sepaated fom the oute nodes afte the fomation of the enegy hole. Thus, even if the netwok still has emaining enegy, the netwok becomes useless and is also consideed as disabled. We can easily udge the algoithm is teminated in which case by checking if thee ae senso nodes with emaining enegy in the netwok. If it is the second case, the fomation of enegy hole should be analyzed tempoally and spatially. Accoding to ou analytic model, at least one senso node will die afte each netwok stage. Since the location of the dead nodes can be detemined by Alg. 1, we can check if the dead senso nodes fom a continuous dead ing with the width d and d afte each netwok stage. The netwok may be patitioned by the continuous dead ing, which is exactly the enegy hole of the netwok. Alg. 2 shows how to detemine the emeging time and bounday of the enegy hole. B. Obsevations on Netwok Chaacteistics The above analysis povides a compehensive solution to detemine the taffic load, enegy consumption, and netwok lifetime, as well as the enegy hole bounday fo a WSN. Based on these analytical esults, we conclude two obsevations on netwok chaacteistics as follows. (1) If the senso nodes ae unifomly deployed in the netwok, the node density has no impact on the FNDT. Accoding to ou analytical esults, FNDT depends on maximum nodal enegy consumption at S, while nodal enegy consumption is detemined by the taffic load of senso nodes. Accoding to Thm. 1, taffic load is unelated to node density, which poves that node density has little impact on taffic load. It also indicates that it is useless to impove the lifetime by inceasing the node density. TABLE II PARAMETER SETTINGS Paametes Values Initial enegy of a senso node E.5 J Duty cycle γ 1% Duation of a data peiod T 1 s Enegy consumption ate fo.88 mj/s idel listening E idle Data tanmission ate B 512 Kb/s Size of a data packet τ 4 bits (2) Thee exists an optimal tansmission ange to maximize the netwok lifetime. Accoding to Thm. 1, the tansmission ange of the senso nodes diectly impacts the taffic load of senso nodes, which detemines the enegy consumption and lifetime of the netwok. Theefoe, we can set the optimal tansmission ange fo the senso nodes to maximize the netwok lifetime. Since netwok lifetime can be estimated unde a equied death atio i% by Coollay 2, and the options of ae limited, the optimal can be found to maximize l% i with bute-foce testing [12], [17]. V. EXPERIMENTAL RESULTS In this section, we validate ou analytic esults by extensive simulations in OMNET++ [5], [34]. We pefom ou simulations in vaious scenaios whee a lage numbe of sensos ae deployed in a cicula aea with diffeent netwok adii R and tansmission anges. The sink is located at the cente of the netwok. We summaize the main paamete settings in Table II, and the settings of the enegy consumption model ae adopted fom [12]. All of simulations ae based on a collisionfee MAC potocol without data loss to be consistent with ou netwok model [6], [12], [3]. A. Compaison of Theoetical Analysis and simulation Results 1) Taffic Load and Enegy Consumption at S : In Fig. 4(a), we compae the simulation esults with the analytical esults in tems of the taffic load at S. It shows that ou analytic model is quite accuate in estimating the taffic load and the eo ate between the theoetical esults and simulation esults is less than 5%, which might be qualified fo most engineeing applications. Fig. 4(b) shows the enegy consumption compaison at S. As shown in this figue, ou analytic model is also accuate in estimating the enegy consumption of the senso nodes. In addition, the tansmission ange of senso node has a significant impact on enegy consumption Since the senso nodes with maximum enegy consumption die fist in the netwok, it indicates that the fist batch of dead nodes ae not always closest to the sink. 2) Taffic Load, Enegy Consumption and Lifetime Compaison fom S 1 to S k 1 : Fig. 5(a) shows the taffic load compaison between at S and the time when 5% senso nodes die. Since the taffic load of the nodes fa fom the sink hadly changes, Fig. 5(a) shows the compaison among the nodes nea the sink. In Fig. 5(a), the nodes fowad moe data afte a few nodes die, because these nodes should undetake the data

9 IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. XX, NO. XX, XXX Theoetical Value Simulation Value R = 4 m = 85 m Distance to The Sink (m) Theoetical Value Simulation Value R = 6 m = 12 m Distance to The Sink (m) R = 4m = 6m Theoetical Value Simulation Value Distance to The Sink (m) R = 6m = 11 m Theoetical Value Simulation Value Distance to The Sink (m) (a) Fig. 4. (a) Taffic load at S (R = 4m, = 85m); (b) Enegy consumption at S (R = 6m, = 12m). (b) (a) (b) Fig. 6. (a) Nodal lifetime compaison (R = 4m, = 85m); (b) Nodal lifetime compaison (R = 6m, = 11m) Afte 5% nodes die At the fist stage R = 5 m = 85 m Distance to The Sink (m) (a) R = 5 m = 85 m The Numbe of Netwok Stage (b) 2 = 6 m = Simulation Ou Analysis Analysis of [17] Netwok Size R (m) Enegy hole evolution po- Fig. 7. ANDT compaison unde Fig. 8. diffeent netwok size. cess R = 5 m = 85 m The uppe bounday 45 The lowe bounday The Numbe of Rounds Fig. 5. (a) Taffic load compaison at S and afte 5% senso nodes die; (b) Duation compaison of diffeent netwok stages. oiginal fowaded by the dead nodes. Thus, thei nodal enegy consumption ates incease, as known as funneling effect [2]. Fig. 5(b) shows the duation compaison of diffeent stages. It can be seen that, except the lifetime l () at stage S is 1248 ounds, the duation of othe stages is much less than l (). Howeve, since the fomation of enegy hole needs to go though many stages, thee is still a long peiod fom FNDT to ANDT. Fig. 6(a) and 6(b) show the nodal lifetime compaison in diffeent netwok scenaios. The following phenomenon can be found. (a) The simulation esults ae consistent with theoetical analysis, and the maximum eo of theoetical and simulation esults is only 5.7%, which can meet the needs of geneal application. (b) The lifetime of each node in the figue actually shows death ode of nodes and the fomation pocess of the enegy hole. The estimated esults ae consistent with the simulation esults. (c) As shown in Fig. 6(a), the enegy hole egion is the egion whose distance to the sink is less than. Howeve, if we incease the tansmission ange, as shown in Fig. 6(b), the enegy hole egion changes to [38m,155m]. This shows that the enegy hole cannot be simply consideed as nea the sink and Alg. 2 can accuately estimate the location of the enegy hole unde diffeent netwok paametes. Fig. 7 compaes the ANDT estimation esults between ou analytical esults and the analysis fom [17]. [17] adopts an annulibased analytic model to analyze the netwok lifetime, without consideing the enegy consumption fo idle listening. It can be seen that ou analytical esults ae moe accuate than the esults of [17]. Howeve, with the incease of the netwok size, the gap between ou model and [17] becomes small, which indicates the annuli-based analytic model in [17] ae moe applicable fo lage scale WSNs FNDT in a 3% Nodes Die in a ANDT in a FNDT in b ANDT in b Tansmission Radius (m) The Numbe of Senso Nodes Fig. 9. Netwok lifetime compaison Fig. 1. Lifetime compaison unde unde diffeent 1. diffeent node densities R = 5 m FNDT, =7 m FNDT, =9 m FNDT, =12 m ANDT, =7 m ANDT, =9 m ANDT, =12 m 3) Detemination of Enegy Hole Bounday: The location of the enegy hole can be obtained accoding to Alg. 2. Fig. 8 shows the change of the enegy hole bounday duing the netwok opeation. In fact, the enegy hole egion can also be obtained fom Fig. 6(a) and 6(b). B. Netwok Chaacteistic Obsevations This section focuses on evaluating the netwok chaacteistics we have found in Section IV-B. Fig. 9 shows the lifetime of %(FDNT), 3% and 1%(ANDT) dead nodes atio unde diffeent and diffeent enegy consumption models. It is shown that thee indeed exists an optimal to maximize the netwok lifetime. Moeove, Fig. 9 shows that without the consideation of enegy consumption fo idle listening, the netwok ANDT is almost twice as FNDT. It also poves FNDT is only a pat of the netwok life cycle. Fig. 1 shows netwok lifetime compaison unde diffeent node densities. It can be obseved that node density has little impact on the netwok lifetime, including the FNDT and ANDT. VI. FURTHER DISCUSSIONS In this section, we futhe discuss the significance of ou analytical esults and apply them to the WSN outing design 1 Model a denotes the enegy consumption model adopted in this pape, while Model b denotes the model without enegy consumption in idle listening.

10 IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. XX, NO. XX, XXX to mitigate enegy hole poblem and impove netwok lifetime. Ou analytical esults and obsevations ae instuctive and useful fo WSN deployment, design, and optimization. (a) It has been poposed that we can balance the enegy consumption of the netwok by non-unifom node deployment [3], [35]. Since the enegy hole bounday can be estimated, the location of the hotspot can also be deived to guide the non-unifom node deployment. Moeove, afte the netwok deployment, we can e-evaluate the netwok lifetime by the poposed analytical model. (b) The pape pesents a guideline fo selecting optimal netwok paametes to impove netwok lifetime o pefomance. It has been demonstated that diffeent tansmission adii lead to diffeent netwok lifetime, which is useful to select an optimal fo a given netwok [5], [17]. Besides, since the enegy consumption fo idle listening is elatively lage, it motivates us to design an enegy-efficient sleep scheduling algoithm fo the senso nodes to futhe educe the enegy consumption and impove the netwok lifetime [33]. (c) Although geogaphic outing has inheent advantages to be applied into lage scale WSNs, its negative influence on enegy efficiency and netwok lifetime cannot be neglected. Fom ou analysis, thee is moe than 8% enegy left when the netwok is patitioned by the enegy hole. Thus, ou wok should be helpful to povide navigation fo designing an enegy-efficient outing potocol. In the following two sections, we intend to illustate the significance of the poposed analytical esults in guiding the WSN design and optimization. We take the outing design fo instance to discuss how to impove the netwok lifetime, including both of FNDT and ANDT, by designing an enegy efficient outing based on ou analytical esults. A. Enegy Efficient Routing Design based on Lifetime analytical esults Accoding to ou analytical esults, since the nodes nea the sink should fowad the data fom upsteam nodes, the unbalanced enegy consumption and enegy hole poblem cannot be avoided in a unifomly deployed data-gatheing WSN [3]. Howeve, it is still possible to mitigate the unbalanced enegy consumption of the senso nodes and impove the netwok lifetime by designing an enegy-awae outing scheme. The main idea of most existing enegy-awae outing solutions is to select the next hop based on the esidual enegy to avoid pematue death in hotspot [32]. By this means, FNDT can be significantly extended. Howeve, it is obseved fom ou analytical esults that afte the fist node dies, the enegy consumption ate of the substitute node inceases shaply, leading to an acceleated ANDT. Theefoe, enegy consumption balance should be consideed fom two aspects, nodal esidual enegy and enegy consumption ate. Since the cost function based outing has the inheent advantages in scalability and has been extensively studied fo enegy efficiency [34], ou outing scheme concentates on the cost function design. At fist, an optimal enegy cost function should map small changes in nodal esidual enegy to lage changes in the value of the function. Such a function can ise shaply the cost of a outing path whose esidual enegy is low and offset the cost eseving by path length eduction (if any exists), focing nodes to select the oute with moe esidual enegy. Second, the enegy consumption ate of nodes should be taken into consideation in cost function design. As the nodes in hotspots geneally have highe enegy consumption ate than othe nodes, the enegy can be futhe balanced with intoducing this facto into the cost function. Based on the two pinciples, the Double Cost Function based Routing (DCFR) scheme can be designed as follows. Fo the senso node i, its neighboing senso nodes whose distances to the sink ae smalle than i s constitute the candidate set of next hop, denoted by {BN i }. Fo each senso node {BN i }, denote the esidual enegy of node by e, the enegy consumption fo tansmitting a packet between i and by e i,. Then, we define the enegy cost c i, of the single hop between i and as c i, = e i, exp ( 1/sin ( π π 2 e e )). Denote e t x and e t y as the esidual enegy of node at t x and t y espectively. The enegy consumption ate ER of node is ER = e t x e t y t y t x c i, fom i to is c i, = e i, exp. Thus, the enegy consumption ate cost ( ( π 1/sin 2 + ER )), ER max whee ER max = max {BNi}{ER }. If we combine the enegy cost and enegy consumption ate cost, the total cost of node i selecting as the next hop, denoted by T C i,, is T C i, =c i, + c i, = e i, exp ( + e i, exp 1/sin ( ( 1/sin ( π 2 + ER ER max π π 2 e e )) )). (18) Theefoe, node i selects the node P with the smallest cost T C i,p as the next hop, whee P is P = ag min(t C i, ). Note BN i that, the cost function of each node can be calculated based on one hop neighbouing infomation. It indicates that the DCFR scheme is fully distibuted and can be applied to lage scale senso netwoks. B. Simulation Results on DCFR Scheme In this section, we aim to validate the efficiency of the DCFR scheme via Omnet++ simulations. We compae it with two existing algoithms: Geogaphic Geedy Routing (GGR), and Distibuted Enegy Balanced Routing (DEBR) [1]. GGR is the outing scheme adopted in ou pevious analysis of this pape. Both DEBR and DCFR select the next hop based on the value of the cost function, while the fome only consides the nodal esidual enegy. The paamete settings of the simulations ae the same as the settings in Section V. Fig. 11(a) compaes the ANDT with vaious netwok sizes. It illustates that enegy awae outing schemes, including DEBR and DCFR, have longe netwok lifetime than DC and GGR. With the consideation of nodal enegy consumption ate, DCFR can achieve moe balanced nodal enegy consumption and a longe netwok lifetime than DEBR. Fig. 11(b) shows the changes of netwok lifetime with the inceasing numbe of senso nodes, unde diffeent outing algoithms. It

11 IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. XX, NO. XX, XXX = 85 m =.16 GGR DEBR DCFR Netwok Size R (m) (a) GGR DEBR DCFR R = 4 m = 6 m The Numbe of Senso Nodes Fig. 11. (a) ANDT compaison unde diffeent outing schemes.; (b) ANDT compaison unde diffeent node densities. can be seen that node density has little impact on the netwok lifetime unde unifom node distibution, even in diffeent outing algoithms. VII. CONCLUSION In this pape, we have developed an analytic model to estimate the taffic load, enegy consumption and lifetime of senso nodes in a data-gatheing WSN. With the analytic model, we have calculated the netwok lifetime unde a given pecentage of dead nodes, and analyzed the emeging time and location of enegy hole, as well as its evolution pocess. Moeove, two netwok chaacteistics have been found based on ou analytic esults, which can be leveaged to guide the WSN design and optimization. Ou simulation esults demonstate that the poposed analytic model can estimate the netwok lifetime and enegy hole evolution pocess within an eo ate smalle than 5%. Finally, we have applied ou analytic esults to WSN outing. The impoved outing scheme based on ou analytical esults can efficiently balance the enegy consumption and polong the netwok lifetime. In ou futue wok, we will extend the lifetime analysis into enegy havesting WSNs. Since senso nodes ae supplied by stochastic enewable enegy, it is vey challenging to analyze and optimize the netwok lifetime unde the continuous and unstable enegy supply. ACKNOWLEDGMENT This eseach wok is suppoted by the Intenational Science & Technology Coopeation Pogam of China unde Gant Numbe 213DFB17, the China Hunan Povincial Science & Technology Pogam unde Gant Numbe 212GK416, the National Natual Science Foundation of China unde Gant No and , the National Basic Reseach Pogam of China (973 Pogam) unde Gant No. 214CB4635 and Hunan Povincial Innovation Foundation Fo Postgaduate, and NSERC, Canada. Ju Ren is also financially suppoted by the China Scholaship Council. REFERENCES [1] Y. Tung, F. Tsang, T. Chui, C. Tung, R. Chi, P. Hancke, and F. Man, The geneic design of a high-taffic advanced meteing infastuctue using zigbee, IEEE Tans. Indust. Infomatics, vol. 1, no. 1, pp , 214. [2] C. Tung, F. Tsang, L. Lam, Y. Tung, S. Li, F. Yeung, T. Ko, H. Lau, and V. R., A mobility enabled inpatient monitoing system using a zigbee medical senso netwok, Sensos, vol. 14, no. 2, pp , 214. (b) [3] C. Caione, D. Bunelli, and L. 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12 IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. XX, NO. XX, XXX [29] A. Chakaboty, R. Rout, A. Chakabati, and S. Ghosh, On netwok lifetime expectancy with ealistic sensing and taffic geneation model in wieless senso netwoks, IEEE Sensos J., vol. 13, no. 7, pp , 213. [3] H. Zhang and H. Shen, Balancing enegy consumption to maximize netwok lifetime in data-gatheing senso netwoks, IEEE Tans. Paa. Dist. Sys., vol. 2, no. 1, pp , 29. [31] P. Cheng, F. Zhang, J. Chen, Y. Sun, and X. Shen, A distibuted tdma scheduling algoithm fo taget tacking in ultasonic senso netwoks, IEEE Tans. Indust. Electonics, vol. 6, no. 9, pp , 213. [32] E. Ghadimi, O. Landsiedel, P. Soldati, S. Duquennoy, and M. Johansson, Oppotunistic outing in low duty-cycled wieless senso netwoks, ACM Tans. Senso Netw., vol. 1, no. 4, 214. [33] G. Anastasi, M. Conti, and M. Di, Extending the lifetime of wieless senso netwoks though adaptive sleep, IEEE Tans. Indust. Infomatics, vol. 5, no. 3, pp , 29. [34] A. Liu, J. Ren, X. Li, Z. Chen, and X. Shen, Design pinciples and impovement of cost function based enegy awae outing algoithms fo wieless senso netwoks, Comput. Netw., vol. 56, no. 7, pp , 212. [35] J. Ren, Y. Zhang, and X. Lin, Nund: Non-unifom node distibution in cluste-based wieless senso netwoks, KSII Tans. Intenet Info. Syst., vol. 8, no. 7, pp , 214. Ju Ren [S 13] (en u@csu.edu.cn) eceived his B.Sc. and M.Sc. degees in compute science fom Cental South Univesity, China, in 29 and 212, espectively. He is cuently a Ph.D. candidate in the Depatment of Compute Science at Cental South Univesity, China. Since August 213, he has also been a visiting Ph.D. student in the Depatment of Electical and Compute Engineeing, Univesity of Wateloo, Canada. His eseach inteests include wieless senso netwok, mobile sensing/computing, and cloud computing. Yaoxue Zhang (zyx@csu.edu.cn) eceived the B.S. degee fom Nothwest Institute of Telecommunication Engineeing, China, and eceived the Ph.D. degee in compute netwoking fom Tohoku Univesity, Japan, in Cuently, he is a pofesso in the Depatment of Compute Science at Cental South Univesity, China, and also a pofesso in the Depatment of Compute Science and Technology at Tsinghua Univesity, China. His eseach inteests include compute netwoking, opeating systems, ubiquitous/pevasive computing, tanspaent computing, and active sevices. He has published ove 2 technical papes in intenational ounals and confeences, as well as 9 monogaphs and textbooks. He is a fellow of the Chinese Academy of Engineeing and the pesident of the Cental South Univesity, China. Kuan Zhang [S 13] (k52zhang@bbc.uwateloo.ca) eceived his B.Sc. degee in electical and compute engineeing and M.Sc. degee in compute science fom Notheasten Univesity, China, in 29 and 211, espectively. He is cuently woking towad a Ph.D. degee in the Depatment of Electical and Compute Engineeing, Univesity of Wateloo. His eseach inteests include packet fowading, and secuity and pivacy fo mobile social netwoks. Anfeng Liu (anfengliu@csu.edu.cn) eceived the M.Sc. and Ph.D. degees in compute science fom Cental South Univesity, Changsha, China, 22 and 25, espectively. He is cuently a Pofesso with the School of Infomation Science and Engineeing, Cental South Univesity, China. His mao eseach inteest is wieless senso netwoks. Jiane Chen (iane@csu.edu.cn) eceived the PhD degee in compute science fom New Yok Univesity in 1987 and the PhD degee in mathematics fom Columbia Univesity in 199. He is cuently a pofesso of compute science at Texas A&M Univesity, and Cental South Univesity in Changsha, China. His eseach inteests include compute algoithms, compute gaphics, compute netwoks and bioinfomatics. He is an associate edito fo IEEE Tansactions on Computes and Jounal of Compute and System Sciences (Elsevie). Xuemin (Sheman) Shen [M 97, SM 2, F 9] (xshen@bbc.uwateloo.ca) eceived his B.Sc.(1982) degee fom Dalian Maitime Univesity, China, and his M.Sc. (1987) and Ph.D. (199) degees fom Rutges Univesity, New Jesey, all in electical engineeing. He is a pofesso and univesity eseach chai, Depatment of Electical and Compute Engineeing, Univesity of Wateloo. He was the associate chai fo gaduate studies fom 24 to 28. His eseach focuses on esouce management in inteconnected wieless/wied netwoks, wieless netwok secuity, wieless body aea netwoks, and vehicula ad hoc and senso netwoks. He is a co-autho/edito of six books, and has published moe than 6 papes and book chaptes in wieless communications and netwoks, contol and filteing. He has seved as the Technical Pogam Committee Chai fo IEEE VTC 1 Fall, Symposia Chai fo IEEE ICC 1, Tutoial Chai fo IEEE VTC 11 Sping and IEEE ICC 8, Technical Pogam Committee Chai fo IEEE GLOBECOM 7, IEEE INFOCOM 14, Geneal Co-Chai fo Chinacom 7, QShine 6 and ACM MobiHoc 15, Chai fo IEEE Communications Societys Technical Committee on Wieless Communications, and P2P Communications and Netwoking. He also seves/seved as Editoin-Chief fo IEEE Netwok, Pee-to-Pee Netwoking and Application, and IET Communications; a Founding Aea Edito fo IEEE Tansactions on Wieless Communications; an Associate Edito fo IEEE Tansactions on Vehicula Technology, Compute Netwoks, and ACM/Wieless Netwoks; and as a Guest Edito fo IEEE JSAC, IEEE Wieless Communications, IEEE Communications Magazine, and ACM Mobile Netwoks and Applications. He eceived the Excellent Gaduate Supevision Awad in 26, and the Outstanding Pefomance Awad in 24, 27, and 21 fom the Univesity of Wateloo, the Pemies Reseach Excellence Awad (PREA) in 23 fom the Povince of Ontaio, and the Distinguished Pefomance Awad in 22 and 27 fom the Faculty of Engineeing, Univesity of Wateloo. He is a egisteed Pofessional Enginee of Ontaio, Canada, an Engineeing Institute of Canada Fellow, a Canadian Academy of Engineeing Fellow, and a Distinguished Lectue of the IEEE Vehicula Technology and Communications Societies.