Characterizing Data Deliverability of Greedy Routing in Wireless Sensor Networks

Transcription

1 Chaacteizing Data Deliveability of Geedy Routing in Wieless Senso Netwoks Jinwei Liu, Lei Yu, Haiying Shen, Yangyang He and Jason Hallstom Depatment of Electical and Compute Engineeing, Clemson Univesity, Clemson, SC 29634, USA School of Computing, Clemson Univesity, Clemson, SC 29634, USA Depatment of Compute Science, Geogia State Univesity, Atlanta, GA 332, USA {jinweil, shenh, yyhe, Abstact As a popula outing potocol in wieless senso netwoks (WSNs), geedy outing has eceived geat attention. The pevious woks chaacteize its data deliveability in WSNs by the pobability of all nodes successfully sending thei data to the base station. Thei analysis, howeve, neithe povides the infomation of the quantitative elation between successful data delivey atio and tansmission powe of senso nodes no consides the impact of the netwok congestion o link collision on the data deliveability. To addess these poblems, in this pape, we chaacteize the data deliveability of geedy outing by the atio of successful data tansmissions fom sensos to the base station. We intoduce η-guaanteed delivey which means that the atio of successful data deliveies is not less than η, and study the elationship between the tansmission powe of sensos and the pobability of achieving η-guaanteed delivey. Futhemoe, with consideing the effect of netwok congestion and link collision, we povide a moe pecise and full chaacteization fo the deliveability of geedy outing. Extensive simulation and ealwold expeimental esults show the coectness and tightness of the uppe bound of the smallest tansmission powe fo achieving η-guaanteed delivey. I. INTRODUCTION Wieless senso netwoks (WSNs) have been inceasingly deployed fo envionment monitoing [], [2]. Usually senso nodes (sensos in shot) ae distibuted ove a geogaphic egion of inteest and tansmit the sensed data to a emote base station using multi-hop outing. Thus, data delivey, as a fundamental function of WSNs, has eceived geat attention. Consideable eseach effots have been devoted to studying the eliability [3], timeliness [4] and enegy-efficiency [5], [6] of data delivey. High delivey atio with low enegy consumption is a challenging issue of data delivey in WSNs. Many outing potocols have been poposed to addess this challenge, including data-centic [7], hieachical [8] and location-based [9], [] design. Among these potocols, the location-based geedy outing (geedy outing in shot) potocol [9], [] is paticulaly attactive fo lage-scale senso netwoks due to its simplicity, efficiency and scalability, and thus has been widely exploited. In this potocol, each node makes outing decision with only local knowledge and fowads the packet to its neighbo that has the smallest distance to the destination until the packet eaches the destination. A well-known poblem with geedy outing is that it fails at a node called void node that has no neighbo close to the destination. To handle this poblem, many pevious woks [] [3] theoetically analyzed the elationship between the tansmission adius and the deliveability of geedy outing. Specifically, Wan et al. [] studied the citical tansmission adius (i.e., smallest tansmission adius) fo geedy outing to ensue that packets can be deliveed between any souce-destination pais in andomly deployed wieless ad hoc netwoks. Wang et al. [2] futhe deived highe accuate asymptotic bounds on the citical tansmission adius. Yang et al. [3] studied the elationship between the citical tansmission powe (i.e., smallest tansmission powe) and the pobability of guaanteed data delivey fom all sensos to the cental base station (efeed to as many-to-one). These woks have studied the deliveability of geedy outing in tems of pobability of guaanteeing all deliveies (i.e., pobability of guaanteed delivey) and the tansmission condition (e.g., citical tansmission powe/adius) to eliminate void nodes in the netwok. Howeve, no pevious woks have studied the elationship between the tansmission powe and the packet delivey atio of geedy outing, which is the atio of the nodes that successfully delive thei data to the base station. We call these nodes delivey-success nodes, othewise, delivey-failue nodes. The wok in [4] demonstates that data delivey in WSNs is inheently faulty and unpedictable, and thus the fault toleant potocols ae necessay fo senso applications and the potocols should ensue eliable data delivey while minimizing enegy consumption [5]. Theefoe, the elationship between tansmission powe and packet delivey atio of geedy outing is of geat inteest fo WSN designes in pactice. It helps to infe the numbe of delivey-failue nodes with a given tansmission powe, and povides insights on the impact of void nodes on the numbe of delivey-failue nodes. Accodingly, the designes can detemine whethe it is acceptable to use a elatively lowe tansmission powe fo sensos by estimating the numbe of delivey-failue nodes, since a limited numbe of delivey-failue nodes may be acceptable fo possible easons like edundant node deployment. Thus, η-guaanteed delivey is not tivial [6], [7]. Anothe limitation of these pevious woks is that they neglect the impact of netwok congestion and link collision on the deliveability of geedy outing in theoetical analysis, though these two factos ae also well-known causes fo packet delivey failue in WSNs [8] [2]. Since geedy fowading decisions ae made based on location infomation without the knowledge of taffic flows in the WSN, it could geneate spatial congestion and collision, which may educe packet delivey atio. The impact of netwok congestion and collision on data deliveability poses a challenge to the chaacteization of data deliveability. In this pape, we analyze the geedy outing deliveability fo many-to-one data delivey in WSNs. Unlike the pevious wok [3] that consides the deliveability in tems of the

2 pobability of guaanteeing all sensos to successfully send thei data to the base station, we conside the deliveability in tems of the atio of delivey-success nodes. In paticula, we study the citical tansmission powe equied to ensue that the atio of delivey-failue nodes does not exceed a theshold with a given pobability. We also conside the impact of netwok congestion and link collision on the deliveability in the study. Compaed with the pevious wok [3], ou esults chaacteize the deliveability in geneal sense and is much moe pactical with the additional consideation of the two factos. The main contibutions of this pape ae as follows: We intoduce the concept of η-guaanteed delivey, which guaantees that the atio of delivey-failue nodes is at most η. Based on this concept, we study the elationship between the citical tansmission powe and the atio of delivey-failue nodes, which povides a moe geneal chaacteization of the many-to-one deliveability of geedy outing compaed to the pevious woks. We deive analytical uppe bounds on citical tansmission powe fo the η-guaanteed delivey unde Signal-to- Intefeence-plus-Noise-Ratio (SINR in shot) [2] model. Simulation and eal-wold expeimental esults ae povided to validate ou analysis esults. We futhe conduct ou analysis with the consideation of netwok congestion and link collision to povide a moe accuate chaacteization of the deliveability of geedy outing. The emainde of this pape is oganized as follows. Section II eviews the elated wok. Section III descibes the poblem definition and the system model used in this pape. In Sections IV and V, we deive the uppe bounds on citical tansmission powe with and without netwok congestion and link collision consideations. Section VI pesents the numeical analysis, simulation esults and eal-wold expeimental esults. Section VII concludes ou wok with emaks on ou futue wok. II. RELATED WORK Geedy fowading with geogaphical locations in a WSN may fail at void nodes. The most well-known method to handle the poblem is face outing [9]. It planaizes the netwok gaph and fowads a message along one o a sequence of adjacent faces, which povides pogess towads the destination node. Anothe method is using vitual coodinates. Saka et al. [22] popose to compute a new embedding of the sensos in the plane such that geedy fowading with the vitual coodinates guaantees delivey. To handle this void node poblem, many woks [] [3], [23], [24] theoetically analyzed the deliveability of geogaphic geedy outing in WSNs o wieless ad hoc netwoks. The woks in [], [2], [23], [24] focus on the deliveability between any pai of souce-destination nodes by geedy outing. Howeve, these woks assume packet tansmission with no intefeence, which makes them impossible to accuately chaacteize the data deliveability in pactical scenaios. Yang et al. [3] modeled the elationship between the citical tansmission powe and the pobability of guaanteed delivey in the many-to-one delivey in a 2-D WSN. They showed that the citical tansmission adius fo many-to-one deliveability can be much smalle than that fo any-to-any deliveability. Howeve, they studied the outing deliveability of all nodes in tems of the pobability of guaanteed delivey instead of the packet delivey atio, as indicated in Section I. Also, thei analysis neglect the netwok congestion and link collision, which ae main causes that affect deliveability. Consideing the impotance of many-to-one data collection fo senso netwoks, ou wok tagets at many-toone deliveability of geedy outing and studies the elationship between the citical tansmission powe and the pobability of η-guaanteed delivey. Futhe, ou wok is the fist to analyze the effect of netwok congestion and link collision on the deliveability of geedy outing in the physically ealistic SINR model, which makes ou wok substantially diffeent fom pevious woks and enables ou wok to accuately chaacteize the data deliveability in pactical scenaios. Thus, ou wok is a notable extension compaed to pevious woks. III. SYSTEM MODEL AND PROBLEM DEFINITION A. System Model Fo analytical tactability, we assume that a WSN with N nodes is deployed in a 2-D disk egion with adius R. The base station X bs is located at the cente of the egion. The disk egion is denoted by D(X bs,r). The distibution of the sensos ove the egion follows a homogeneous Poisson point pocess with constant density λ []. Each senso, denoted by X i, has the same tansmission powe. We model the WSN as a gaph G(V,E), in which V epesents the set of nodes in the netwok, and E stands fo the links of the netwok. B. Channel Model In this pape, we use the SINR model to captue channel chaacteistics in WSNs. Many pevious woks [25], [26] on data deliveability assume Unit Disk Gaph (UDG) model fo communication. The UDG model, which assumes that two nodes within cetain distance can communicate diectly, ovesimplifies the channel model [27], because it does not conside intefeence fom othe on-going tansmissions. In SINR, the successful eception of a tansmission depends not only on the eceived signal stength but also the intefeence caused by simultaneously tansmitting nodes and the ambient noise level. Thus, based on SINR, we ae able to povide moe ealistic and accuate analysis on the data deliveability of geedy outing in WSNs. We use v s and v to denote a souce tansmitte and a eceive. Let P ec be the eceived signal powe at the eceive v fom the tansmitte v s. Denote I as the amount of intefeence geneated by othe nodes in the netwok. Let N n be the ambient noise powe level. Then, in the SINR model, eceive v eceives a tansmission iff P ec /(N n + I ) β () whee β is a small constant (depending on the hadwae) and it denotes the minimum signal to intefeence atio equied fo a message to be successfully eceived. The value of the eceived signal powe P ec is a deceasing function of the Euclidean distance d s between the tansmitte v s and the eceive v, epesented by P ec (d s )=P t /d α s (2) whee P t is the tansmission powe of the tansmitte, and the so-called path-loss exponent α is a constant between 2 and 6. α indicates the ate at which the eceived signal powe deceases with the distance between the tansmitte and the eceive. Based on (), the tansmission adius fo successful delivey can be epesented as =sup{d P ec (d) β(n n + I ), <d<+ } (3) whee sup epesents the least uppe bound. In WSNs on 2-D plane, I can be epesented by

3 I = v i V \{v s} P t d i α (4) whee V R 2 is the set of nodes in the 2-D plane. C. Poblem Definition Definition : η-guaanteed delivey: Given a WSN G with N sensos, and a minimum delivey atio equiement (η), a data gatheing of G achieves η-guaanteed delivey if N s /N η, whee N s is the numbe of delivey-success nodes in the data gatheing. η-guaanteed delivey with η<% is usually desied in the applications that can toleate a limited numbe of deliveyfailue nodes, such as statistical infeence to the population with sensed data samples. The detemination of η depends on the numbe of delivey-failue nodes that can be toleated. When η<%, the tansmission powe of sensos to achieve η-guaanteed delivey is much lowe than that equied by %-guaanteed delivey. Based on η-guaanteed delivey, we define the citical tansmission powe and adius and pesent ou poblems below. Definition 2: Citical tansmission powe: The citical tansmission powe Pt ci (η, P th ) denotes the minimal tansmission powe, which ensues that the pobability of achieving η-guaanteed delivey is no less than a theshold P th ( <P th < ), i.e., P{N s /N η} P th (5) Definition 3: Citical tansmission adius: The citical tansmission adius ci (η, P th ), coesponding to the citical tansmission powe Pt ci (η, P th ), denotes the minimal tansmission adius which ensues that the pobability of achieving η-guaanteed delivey is no less than P th. Accoding to the definition, citical tansmission powe Pt ci (η, P th ) is detemined by the delivey atio η and theshold P th. It ensues that the pobability of achieving η- guaanteed delivey fo a WSN is no less than a theshold with minimal enegy consumption. To ensue η-guaanteed delivey with a cetain pobability, we need to find the citical tansmission powe Pt ci (η, P th ). Obviously, a senso using the citical tansmission powe Pt ci (η, P th ) has a coesponding citical tansmission adius ci (η, P th ). Based on the above definitions, we can fomulate ou poblems as follows: Poblem : Given a desied atio of delivey-success n- odes η and a pobability theshold P th, what is the citical tansmission powe Pt ci (η, P th ) to achieve η-guaanteed delivey? Fom the above, we can see the pevious wok [3] is a special case of ou poblem with η = %. Ou poblem povides a moe in-depth and pecise chaacteization on the data deliveability of geedy outing. The study of ou poblem is also vey useful to WSN applications that use appoximate data collection that collects incomplete data fom WSNs, which has been widely studied due to its enegyefficiency [28], [29]. Because netwok congestion and link collision affect geedy outing deliveability, we futhe study Poblem with the consideation of these factos. We pesent this new poblem as Poblem 2 in the following. We conside a continuous data gatheing scenaio, in which all sensos peiodically send sensed data to the base station, and the data is collected ound by ound. In one ound of data gatheing, the atio of deliveysuccess nodes is affected by the cuent status of netwok congestion and link collision. Poblem 2: Given a desied atio of delivey-success n- odes η fo each ound of a continuous data gatheing and a pobability theshold P th, what is the citical tansmission powe Pt ci (η, P th ) to achieve η-guaanteed delivey with the consideation of the impact of netwok congestion and link collision on the atio of delivey-success nodes? IV. CRITICAL TRANSMISSION POWER In this section, we addess Poblem and deive the uppe bounds on citical tansmission powe fo the poblem solution in the SINR model. We fist establish the elationship between the pobability of η-guaanteed delivey and the pobability of a node being a delivey-failue node. Then, we fomulate the elationship between the pobability of a node being a deliveyfailue node and the tansmission powe. As a esult, we can find the uppe bounds on citical tansmission powe. A. The Relationship between η-guaanteed Delivey and Delivey Failue Pobability Fo a senso X i, C(X i ) denotes a Benoulli andom vaiable that equals one iff X i is a delivey-failue n- ode. Fo all nodes V = {X,,X V } in the netwok, C(X ),,C(X n ) ae identically distibuted andom vaiables, whee V is the cadinality of V. As the wok in [3], we assume the distibution of the delivey-failue nodes is statistically independent. Let Z be the numbe of deliveyfailue nodes in the netwok, and we have Z = C(X i) (6) x i V Accoding to Definition 2, fo citical tansmission powe, we have P{Z ( η)n} P th (7) Accoding to Makov s inequality, we have P[Z ( η)n] = P[Z (( η)n +)] E(Z)/(( η)n +) Suppose that C(X i )( i N) ae identically distibuted andom vaiables. Then, the expectation of andom vaiable Z can be computed by + k E[Z] = E[ C(X i)]p( V = k) = k= + k= i= (ke[c(x i)]p( V = k)) = E[C(X i)] + k= k(λπr 2 ) k exp( λπr 2 )/(k!) = λπr 2 E[C(X i)] = λπr 2 P(C(X i)=) whee the distibution of the delivey-failue sensos ove the egion follows a homogeneous Poisson point pocess with constant density λπr 2. Combining Fomulas (7), (8) and (9), we have P(C(X i)=) ( P th )(( η)n +)/(λπr 2 ) () In ode to achieve η-guaanteed delivey, the citical tansmission powe should be chosen to make the delivey failue pobability of any node satisfy (). (8) (9)

4 i Xbs ik Xi+n Xi Xi+ Xbs () u 2 () u u () u S 2 ( u) Xi+k () u Fig. : Routing. Fig. 2: Feasible egion of nodes. B. Uppe Bound on Citical Tansmission Powe Definition 4: Void node: X i is a void node iff it cannot diectly communicate with the cental base station X bs and it is close to X bs than all its neighbos. X i is a delivey-failue node if it cannot diectly communicate with the base station X bs, and also cannot communicate with X bs via multi-hop due to the existence of void nodes on the outing path. To compute the pobability of X i being a delivey-failue node, we fist conside the pobability of X i being a delivey-success node. Suppose that the distance between X i and X bs is ρ and the tansmission adius is, X i is a delivey-success node only if it falls into eithe of the following two cases: Case : ρ is less than o equal to, that is, X i can diectly communicate with X bs. Case 2: ρ is geate than, and thee exists a multi-hop geedy outing path to X bs with no delivey-failue nodes. Suppose X i+,,x i+k,,x i+n ae intemediate n- odes fom X i to the base station, as shown in Fig.. ρ i+k is the distance between the node X i+k (k =,,,n) and the base station X bs. n =if X i can diectly communicate with X bs. Case 2 is satisfied iff X i satisfies both of the following two conditions: Condition E : Thee exists at least one node located in X i s tansmission ange which is close to the base station than X i. Condition E 2 : The next fowading node X i+, one of X i s neighbos who has the smallest distance to the base station among X i and all its neighbos, can successfully fowad the packet to the base station. Next, we fist conside the pobability of E, then deive the pobability of X i being a delivey-success node which is equal to the pobability of both E and E 2 ae satisfied. ) Condition E : We call the aea whee the potential next fowading node X i+ can be located as the feasible egion of node X i. Because X i+ must be in the tansmission ange of X i and also must have smalle distance to the base station than X i, the feasible aea of X i is the intesection aea of the two cicles of adius and ρ centeed at X i and the base station, espectively. We use andom vaiable U to denote the distance between the base station (X bs ) and the next fowading node chosen by the geedy outing algoithm. Conside the feasible egion of X i, whee potential next fowading nodes can be located at some distance u o less fom the base station (shaded egion in Fig. 2). The aea of the feasible egion is denoted by S ρ (u). Accoding to [3], because when ρ is geate than, the pobability of no next fowading nodes existing in the feasible egion of aea is equivalent to the pobability that U is stictly geate than u. The complement of this pobability yields the distibution of U [3] which vaies with u { exp( λsρ(u)), ρ u<ρ F (u) =, u ρ (), u < ρ Xi We can obtain the following pobability density function by diffeentiating the distibution F (u) which is absolutely continuous f(u) =λs ρ(u)exp( λs ρ (u)), ρ u<ρ (2) whee S ρ(u) is the deivative of S ρ (u) with espect to u. We define the angles of the two intesecting sectos as 2α ρ, 2β ρ, as shown in Fig. 2. By the Law of Cosines, we have α ρ(u) = accos( 2 + ρ 2 u 2 ) (3) 2ρ β ρ(u) = accos( u2 + ρ 2 2 ) (4) 2uρ Then, we have S ρ(u) = 2 α ρ(u)+u 2 β ρ(u) uρ sin β ρ(u), ρ u<ρ (5) Based on (3), (4) and (5), we have S ρ(u) 2uβ ρ(u) (6) 2) Being a Delivey-success Node: Consideing that the sensos ae unifomly distibuted on 2-D plan, the nodes which has the same distance to the base station ae equal on the netwok deliveability fo thei packets. Thus, fo a given node X i which has distance ρ to the base station, we let the pobability of X i being a delivey-success node is a function of the distance ρ, denoted by P (ρ). The distance U has pobability density function f(u) given by (2). When U = u, the pobability of X i+ being deliveysuccess node is P (u). Because X i can successfully send a packet to X bs only if it satisfies both Condition E and Condition E 2,wehave P (ρ) = ρ ρ P (u)f(u)du (7) We take the deivative of this equation with espect to ρ fist, and get a diffeential equation. Afte computing this diffeential equation using Mathematica, we get the following analytic solution: P (ρ) =exp( 2exp( λ( 2 accos( 2t ) + accos( 2 +2t 2 )t 2 2 ( 2 +4t 2 ) 2t 2 2 t2 )) t 4 λ accos( 2 +2t 2 ρ )tdt + 2exp( λ( 2 2t 2 accos( 2t ) + accos( 2 +2t 2 )t 2 2t 2 2 t2 2 ( 2 +4t 2 ) ))λ accos( 2 +2t 2 )tdt) t 4 2t 2 Accodingly, the pobability of the node X i being a deliveyfailue node is (P (ρ)) c = P (ρ) (9) whee supescipt c means the complement of P (ρ). 3) Uppe Bound on Citical Tansmission Powe: Consideing all the possible locations of X i, the pobability of a node being a delivey-failue node is P c = 2π whee Hence (P (ρ)) c ρdρdθ = 2 πr 2 R 2 g() = (8) ρ(p (ρ)) c dρ = 2g() R 2 (2) ρ( P (ρ))dρ (2) P(C(X i)=)=(2g())/r 2 (22)

5 Let x =( η)n. Based on (8), (9) and (22), we have P[Z x] (λπr 2 (2g())/R 2 )/(x +) (23) To ensue that P[Z x] P th,wehave g() ( P th )(x +)/(2λπ) (24) g() is stictly deceasing fo, and the fomal poof is povided in ou technical epot [32]. Hence, we can ensue P[Z x] P th as long as the citical tansmission adius ci (η, P th ) satisfies ci (η, P th ) =inf{ g() ( Pth )(x +) } (25) 2λπ whee inf epesents the geatest lowe bound. Letting d s in (2) be, with P ec (d s ) (N n + I )β we can obtain the uppe bound of the citical tansmission powe Pt ci (η, P th ), that is Pt ci (η, P th ) P t = β(n n + I ) α (26) whee I can be computed based on Fomula (4). V. EFFECTS OF NETWORK CONGESTION AND LINK COLLISION In this section, we deive the uppe bound on the citical tansmission powe fo η-guaanteed delivey with consideation of the effects of congestion and collision. The congestion at the eceive node intoduces packet loss due to buffe oveflow. Also, when multiple active senso nodes ty to access the channel simultaneously, collisions could occu and coupt the packet in tansmission. A senso fails in deliveing data to its next hop when the tansmission expeiences a collision o the buffe of its next hop is full. Since the congestion and collision ae well-identified causes of packet loss in WSNs [8], [33], we investigate thei effects on the deliveability of geedy outing to povide ealistic analysis esults. Hee, we assume each senso in the WSN has a buffe size of m packets. To compute the pobability that a given node X i delives data to X bs, we assume the data delivey path fom X i to X bs is X i X i+,, X i+n X bs. n = if X i can diectly communicate with X bs. We fist conside the pobability of successful data tansmission at one hop in the path. A. Delivey Success in One Hop Fo a successful one-hop data tansmission, say X j X j+, the following two conditions must be satisfied. Condition E A : X j is not a void node, i.e., X j has a neighbo whose distance to X bs is smalle than X j s. Condition E B : No link collision occus duing the packet tansmission fom X j to X j+, and when the packet aives at X j+ the buffe queue of X j+ is not full, i.e., no congestion occus to the packet. Hence, we have P(X j X j+) =P(E A)P(E B) (27) ) Condition E A : The pobability that X j is a void node is the pobability that no nodes exist in X j s feasible egion. The aea of X j s feasible egion whee any node has smalle distance fom the base station than X j, denoted by S(ρ j,), can be computed by (5) with u = ρ j whee ρ j, is the distance between X j and X bs, i.e., S(ρ j,)=2ρ 2 jacsin + 2 accos 2ρ j 2ρ j ρ 2 j 2 4 (28) Accoding to spatial Poisson point pocess distibution of nodes, we have P(EA) = exp( λs(ρ j,)) (29) 2) Condition E B : Next, to compute P(E B ), we fist deive the pobability of packet loss caused by netwok congestion and link collision espectively, and then obtain P(E B ). Netwok Congestion: Let P nc be the pobability that a node fails to delive a packet to its next hop due to buffe oveflow. We deive P nc based on M/M//k model. The M/M//k model descibes a stochastic pocess whose state space is the set I = {,, 2,, k} whee the value coesponds to the numbe of packets in the node s buffe. Accoding to [34], steady state pobabilities of the system, denoted by P j (j =,, 2,,k), ae { ϱ, ϱ P = ϱ k+, ϱ = (3) k+ P j = { ϱ j ( ϱ), ϱ k+ ϱ (3), k+ ϱ = Hee ϱ = λ ARR /μ in which μ is the packet tansmission ate and λ ARR is packet aival ate. Since it is a many-to-one model (i.e., all packets go to sink), the aival ate of the senso in the cente (close to the sink) should be highe (moe contending nodes) than that of the senso away fom the sink, and thus we conside the aival ate as a function (invese popotion to the eceive s distance to the base station) of the eceive s distance to the base station so that it can bette eflect the case in eal system [35]. The aival ate of node X j+ is as follows λ ARR (ρ j+ )=(R/2)/ρ j+ λ (32) whee R is the adius of the 2-D disk egion, ρ j+ is the distance between X j+ and X bs, and λ is an expected aival ate. Each senso has a buffe size of m packets. With k = m, the steady state pobability P m is the pobability of a buffe being full which causes packet dop. Obviously, P nc = P m (33) Hence, the pobability that node X j fails to delive a packet to its next hop X j+ due to buffe oveflow is P m with ϱ = λ ARR (ρ j+ )/μ (denoted as P m (ρ j+ )). Link Collision: Since in WSNs wieless channels ae shaed by seveal nodes using CSMA-like (Caie Sense Multiple Access) potocols, we deive the pobability of packet loss due to link collision based on modeling of CSMA/CA in [36]. The binay exponential backoff pocedue is modeled as a Makov chain with the assumption of constant and independent collision pobability of a packet tansmitted by each node. We conside a fix numbe l of contending nodes, each always having a packet available fo tansmission afte the completion of each successful tansmission. Based on [36], we can get the pobability of a packet encounteing collision P lc as P lc = ( τ) l (34) whee τ is the pobability that a node tansmits in a andomly chosen slot time. τ = 2( 2P lc ) ( 2P lc )(W +)+P lc W ( (2P lc ) v ) (35) whee W is the minimum contention window size W = CW min, and the maximum contention window size is CW max =2 v W. v is the maximum backoff stage. In paticula, when v =, i.e., no exponential backoff is consideed, the pobability τ esults to be independent of P lc. Fomula (35) thus simply becomes: τ =2/(W +) (36)

6 Computation of P(E B ): Based on (33) and (34), the pobability of Condition E B is P(E B)=( P nc)( P lc )=( P m)( τ) l (37) B. Delivey Success to the Base Station Fo simplicity, we use the aveage numbe of hops that a packet can tavese fom a node to the base station to appoximately estimate the pobability of successful data delivey fom node X i to base station X bs. ) Aveage Numbe of Hops: If a packet tavels fom a node with distance ρ i to the base station to anothe node with distance ρ i+ to the base station, the distance it advances equals ρ i ρ i+. Pevious wok [37] shows that the pobability density function of pogess in one hop fom X i towads the base station X bs is f c(c X i,x bs = ρ) =σ( 2 π 2 )σ 2(ρ c)[ π 2 acsin( + c2 2 )][(ρ c)2 2ρ(ρ c) acsin( + c2 2 2ρ(ρ c) )+ (38) 42 ρ 2 2 (c 2 2 2ρc) 2 2 acsin( c2 2 2ρc ) 2ρ π(ρ c)2 ] σ, c 2 whee ρ is the distance between X i and X bs, c is the maximum fowad pogess in one hop towads the base station X bs, and σ is the numbe of nodes located in the semi-cicle with adius, computed by σ = λ π2 2 whee λ is the constant density. Based on (38), we can get the aveage pogess pe hop towads the base station c fo X i with distance ρ to X bs as follows: c(ρ) = vf c(v)dv (39) Conside all the possible locations of X i,wehave, 2π c(ρ) c = πr ρdρdθ 2 = 2 vf c(v X i,x R 2 bs = ρ)dvρdρ (4) Since the adius of the geogaphic egion D(X bs,r) is R, we estimate the maximum numbe of hops fo deliveing a packet to the base station X bs by Ĥ max = R/ c (4) Accoding to [38], the aveage numbe of hops a packet taveses in the netwok equals E(H) = Ĥ max k= {k [e (k )2 λπ 2 e k2 λπ 2 ] ( e λa ) k } whee A is the intesection aea between two tansmission anges, illustated by the shaded aea in Fig. 3, and can be computed by: A = 2 (2 accos( x 2 ) sin(2 accos( x ))) (43) 2 2) The Delivey Success: A node succeeds in deliveing a packet to the base station if evey hop on the outing path achieves successful delivey of the packet. Fo simplicity, we assume that the delivey of <x<2 Xj+ (42) Xj- Xj Fig. 3: Geomety of a two-hop connection. each hop tansmission is independent of othe hop tansmissions along the path. Then, given the pobability of delivey success in one hop P(E A )P(E B ) and the aveage numbe of hops fo deliveing a packet to the base station E(H), the pobability that a node X i succeeds in deliveing a packet to the base station can be deived by combining (29), (37) and (42): P(C(X i)= X i,x bs = ρ i)=(p(e A)P(E B)) E(H) =Π i+e(h) 2 j=i P(E A)P(E B) (44) =Π i+e(h) 2 j=i ( exp( λs(ρ i,)))( P m(ρ j+))( τ) l C. Uppe Bound on Citical Tansmission Powe Based on (2), (2) and (44), we can obtain the pobability of a node being a delivey failue node: P(C(X i)= X i,x bs = ρ i) = P(C(X i)= X i,x bs = ρ i) = Π i+e(h) 2 j=i ( exp( λs(ρ i,)))( P m(ρ j+))( τ) l (45) Consideing all the possible locations of X i,wehave P(C(X i)=) = 2π (ρ i( Π i+e(h) 2 j=i P(E A) P(E B)))/(πR 2 )dρ idθ = 2 ρ i( Π i+e(h) 2 R 2 j=i ( exp( λs(ρ i,))) (46) ( P m(ρ j+))( τ) l )dρ i Because the complexity of computing P(C(X i )=), and P c is the pobability of a node being a delivey-failue node caused by void nodes, and P(E A ) is the pobability that a node X j is not a void node, we use P(C(X i ) = ) = Π i+e(h) 2 j=i ( P c )P(E B ) to appoximately compute P(C(X i )=)then we have P(C(X i ) = ) = h(),whee h() = Π i+e(h) 2 j=i ( P c )P(E B) = ( R2 2g() ) (E(H) ) (47) R 2 ( P m(ρ j+))( τ) l Then, based on (8), we can get P[Z x] (λπr 2 h())/(x +) (48) Π i+e(h) 2 j=i Thus, to ensue that P[Z x] P th, we need to ensue h() ( P th )(x +)/(λπr 2 ) (49) h() stictly deceases with, and the fomal poof is povided in ou technical epot [32]. Theefoe, the citical tansmission adius has ci (η, P th ) =inf{ h() ( Pth )(x +) λπr 2 } (5) Given a WSN G(V,E) deployed ove a geogaphic egion D(X bs,r) and a pobability theshold P th, the citical tansmission powe Pt ci (η, P th ) fo SINR model satisfies Pt ci (η, P th ) P t = β(n n + I ) α (5) VI. EXPERIMENTAL RESULTS In this section, we pesent numeical analysis of ou theoetical esults to investigate the elationships among the tansmission powe, pobability of η-guaanteed delivey and minimum delivey atio η. Then, we pesent simulation esults that evaluate the tightness of ou uppe bounds on the citical tansmission powes. Finally, we povide eal-wold expeimental esults to validate ou model s ability of well appoximating eal life pefomance.

7 % guaanteed delivey 85% guaanteed delivey.2 9% guaanteed delivey 95% guaanteed delivey % guaanteed delivey Radio tansmission powe (mw) 8% guaanteed delivey.4 85% guaanteed delivey 9% guaanteed delivey.2 95% guaanteed delivey % guaanteed delivey Radio tansmission powe (mw) (a) W/o congestion & collision (b) W/ congestion & collision Fig. 4: Relationship between pobability of η-guaanteed delivey and tansmission powe with intefeence. A. Numeical Analysis In ou numeical analysis, we assume that 5 senso nodes ae distibuted ove a disk egion D(X bs, m) following a Poisson distibution. The base station is located at the cente of the disk egion. All the senso nodes have the same tansmission powe. Fo SINR model, we set path-loss exponent α = 3, the minimum signal to intefeence atio β =4, and ambient noise powe level N n =nw [39], [4]. The Fomulas (23)-(26) in Section IV show the uppe bound on citical tansmission powe without consideing congestion and collision, and Fomulas (48)-(5) in Section V conside congestion and collision. Based on these esults, Fig. 4(a) and Fig. 4(b) show the elationship between the pobability of η-guaanteed delivey and tansmission powe when η=8%, 85%, 9%, 95%, and %, without and with the existence of congestion and collision, espectively. Both figues show that the pobability of η-guaanteed delivey inceases as the adio tansmission powe inceases. Compaing Fig. 4(b) to 4(a), we see that with the consideation of congestion and collision, geate tansmission powe is equied to achieve the same pobability of η-guaanteed delivey. The pobability of η-guaanteed delivey in Fig. 4(a) eventually goes to when the tansmission powe is lage enough. Howeve, in Fig. 4(b) it appoaches but cannot be (though it is not obvious in the figue) due to the existence of congestion and collision. Both figues show that with a smalle η, the tansmission powe equied to achieve the same pobability of η-guaanteed delivey is smalle. An inteesting obsevation is that the cuve of η = % is widely sepaated fom the cuves of othe η values. This esult indicates that with toleance to a small pecentage of delivey failue nodes, much less tansmission powe is needed compaed to that needed by %-guaanteed delivey, thus obtaining significant enegy saving. Fig. 5 shows the elationship between the pobability of η- guaanteed delivey and η with diffeent tansmission powes. We see that given a tansmission powe, the pobability of η-guaanteed delivey deceases with the incease of η, and highe tansmission powe esults in highe pobability of η- guaanteed delivey. This is because a highe tansmission powe enables nodes to communicate with nodes futhe away, deceasing the pobability of delivey failue caused by void nodes. Compaing Fig. 5(a) and 5(b), fo the same tansmission powe and the same η, the pobability of η-guaanteed delivey in Fig. 5(b) is lowe than that in Fig. 5(a) because of the congestion and collision effects. Fig. 6 shows the elationship between the uppe bound on citical tansmission powe and the node density. We changed the node density by vaying the numbe of senso nodes ove the disk egion D(X bs, m). Fig. 6(a), 6(b), 6(c) and 6(d) show the uppe bounds on the citical tansmission powe fo η= 8%, 85%, 9%, and 95% guaanteed delivey, espectively. Each figue shows uppe bounds deived with congestion Radio tansmission powe: mw Radio tansmission powe: 5mw.2 Radio tansmission powe: 2mw Radio tansmission powe: 25mw Minimum delivey atio equiement η (%) Radio tansmission powe:mw Radio tansmission powe:5mw.2 Radio tansmission powe:2mw Radio tansmission powe:25mw Minimum delivey atio equiement η (%) (a) W/o congestion & collision (b) W/ congestion & collision Fig. 5: Relationship between pobability of η-guaanteed delivey and the minimum delivey atio equiement η. and collision (denoted as cong-col in figues) as well as without congestion and collision. The uppe bounds fo %- guaanteed delivey ae dawn in evey figue fo compaison. Fom these figues, it can be seen that the uppe bounds on citical tansmission powe decease as the numbe of nodes in the netwok (hence node density) inceases. This is because a highe node density leads to a smalle aveage distance between any pai of nodes, which enables each node to use a smalle tansmission adius fo communication. We also see that the uppe bounds on citical tansmission powe decease slowly with the node density. This is because the incease of node density intoduces moe intefeence, offsetting some effect of deceasing aveage distance of any pais. All of these figues show that the uppe bound deived with the consideation of congestion and collision is lage than that without the consideation. This indicates that highe tansmission powe is equied to counte the effect of congestion and collision. We also find that a smalle η geneates a smalle uppe bound on citical tansmission powe. The uppe bound fo %- guaanteed delivey is consideably lage than that fo smalle η, which indicates that highe delivey atio equies highe tansmission powe egadless of the existence of congestion and collision. B. Simulation Results We used netwok simulato NS2 [4] to conduct simulation expeiments. Constant Bit Rate (CBR) Taffic geneato [4] is used fo each senso to ceate a fixed size packet fo evey fixed inteval. To validate the coectness and tightness of ou uppe bound, we compae ou theoetical esults with simulation esults in vaious scenaios. By default, the numbe of nodes in the netwok was set to 2 in the simulation. The nodes ae distibuted ove a disk egion D(X bs, 3m) following a Poisson distibution. The theshold fo decoding a signal was set to P th = 64dBm. Fo each setting of tansmission powe, we geneated 2 andom netwok topologies and fo each topology we computed the atio of delivey success nodes. The pobability of η-guaanteed delivey is estimated with the 2 delivey atio samples. Fig. 7(a)-7(b) show the theoetical uppe bounds on citical tansmission powe and the simulation esults fo 85% and 95% guaanteed delivey. We see that ou theoetical uppe bounds ae vey close to the simulation esults. By examining Fig. 7(a)-7(b), we see the uppe bound on citical tansmission powe inceases as η inceases, which is consistent with ou numeical esults. To futhe validate ou model, we vaied the netwok density and taffic load of the netwok. In Fig. 8, we deceased the numbe of nodes in the netwok to to decease the netwok density. Fig. 8(a)-8(b) show the theoetical uppe bounds on citical tansmission powe and the simulation esults fo 85% and 95% guaanteed delivey. We see that ou theoetical uppe bounds ae still vey close to the simulation esults. We.8.6.4

8 Uppe bound on citical tansmission powe (mw) % guaanteed delivey % guaanteed delivey with 8% guaanteed delivey 8% guaanteed delivey with Numbe of nodes in the WSNs Uppe bound on citical tansmission powe (mw) % guaanteed delivey % guaanteed delivey with 9 85% guaanteed delivey 85% guaanteed delivey with Numbe of nodes in the WSNs Numbe of nodes in the WSNs (a) 8%-guaanteed delivey (b) 85%-guaanteed delivey (c) 9%-guaanteed delivey (d) 95%-guaanteed delivey Fig. 6: Relationship between uppe bound on citical tansmission powe and node density with intefeence Simulation esult Radio tansmission powe (mw) Radio tansmission powe (mw) Fig. 7: η-guaanteed delivey vs. tansmission powe (path-loss exponent α =3, N = 2, inteval=2) Simulation esult Radio tansmission powe (mw) Simulation esult Radio tansmission powe (mw) Fig. 8: η-guaanteed delivey vs. tansmission powe (path-loss exponent α =3, N =, inteval=2). also find that the uppe bound on citical tansmission powe inceases as η inceases. Compaing Fig. 8 with Fig. 7, we find that the uppe bounds on citical tansmission powe in Fig. 8 ae lage than those in Fig. 7, which indicates that the uppe bound on citical tansmission powe inceases as node density deceases. This is because lage node density shotens the aveage distance between nodes and theeby educes the pobability of delivey failue caused by void nodes Simulation esult Radio tansmission powe (mw) Simulation esult Radio tansmission powe (mw) Fig. 9: η-guaanteed delivey vs. tansmission powe (path-loss exponent α =3, N =, inteval=) Simulation esult Radio tansmission powe (mw) Simulation esult Radio tansmission powe (mw) Fig. : η-guaanteed delivey vs. tansmission powe (path-loss exponent α =3, N =, inteval=.5). We then vaied taffic load by diffeent intevals fo CBR taffic geneato. Fig. 8, 9 and show the elationship between the pobability of η-guaanteed delivey and the tansmission powe with nodes in the netwok, unde diffeent intevals 2, and.5. Smalle inteval means highe taffic load. It is obvious to see that ou theoetical uppe bounds ae vey close Simulation esult Uppe bound on citical tansmission powe (mw) % guaanteed delivey % guaanteed delivey with 9% guaanteed delivey 9% guaanteed delivey with Uppe bound on citical tansmission powe (mw) % guaanteed delivey % guaanteed delivey with cong 95% guaanteed delivey 95% guaanteed delivey with Numbe of nodes in the WSNs to the simulation esults. Compaing Fig. 8, 9 and, we find the uppe bounds on citical tansmission powe follows Fig. >Fig. 9>Fig. 8, which indicates the uppe bound on citical tansmission powe inceases as taffic load inceases. This is because heavie taffic load inceases congestion and collision and theeby inceases the pobability of delivey failue. C. Real-wold Expeimental Results Ou testbed [42] consists of 6 Tmote Sky motes [43] unning TinyOS A compute unning Ubuntu 2.4 was used to configue all senso nodes. Each senso node was configued to peiodically sample and tansmit data. The netwok delivey atio was measued unde diffeent taffic loads, netwok densities, and adio tansmission powe levels. Fig. shows the elationship between the pobability of η-guaanteed delivey and adio powe level fo 85% and 95% guaanteed delivey. In the test, the inteval between two consecutive packet tansmissions was set as second. In Fig. 2, we inceased the inteval between two consecutive packet tansmissions to 2 seconds to decease taffic load. Both Fig. and Fig. 2 indicate that the expeimental esults ae close to the theoetical esults. By compaing Fig. (a) and Fig. (b), Fig. 2(a) and Fig. 2(b), similaly, we see that the uppe bound on citical tansmission powe inceases as η inceases, which is consistent with numeical esults and simulation esults. Compaing Fig. 2 and Fig., we find that the uppe bounds on citical tansmission powe in Fig. ae lage than those in Fig. 2. This esult indicates that the uppe bound on citical tansmission powe inceases as taffic load inceases, which is consistent with ou simulation esults Expeimental esult 2 3 Radio Powe level 2 3 Radio Powe level Fig. : η-guaanteed delivey vs. tansmission powe (path-loss exponent α =3, N =6, inteval= second) Expeimental esult Radio Powe level Expeimental esult 2 3 Radio Powe level Fig. 2: η-guaanteed delivey vs. tansmission powe (path-loss exponent α =3, N =6, inteval=2 seconds). Ou theoetical and eal-wold expeimental esults show that by toleancing to a small pecentage of delivey failue nodes, much enegy can be saved Expeimental esult

9 VII. CONCLUSION In this pape, we study the deliveability of geedy outing in 2-D WSNs. As opposed to pevious woks that only analyze the pobability of guaanteeing all deliveies and neglect netwok congestion and collision, we intoduce η-guaanteed delivey, whee η can be vaied and study its pobability with the consideation of netwok congestion and collision. We adopt a moe ealistic model to analyze uppe bounds on citical tansmission powe. Though theoetical analysis, we deive the uppe bounds on the citical tansmission powe fo achieving η-guaanteed delivey with a given pobability. The extensive numeical analysis, simulation and eal-wold expeimental esults show that ou chaacteization is close to the pactical scenaios and ou deived uppe bounds ae coect and tight. Ou futue wok is to evaluate the deliveability of geedy outing with vaious impovements poposed ecently fo handling void nodes not only caused by shot tansmission ange, but also caused by non-homogeneous density o physical obstacles (e.g., lake), localization eos. ACKNOWLEDGEMENTS This eseach was suppoted in pat by U.S. NSF gants NSF-4498, IIS-35423, CNS-2546, CNS-24963, Micosoft Reseach Faculty Fellowship REFERENCES [] B. O Flynn, R. Matínez-Català, S. Hate, C. O Mathuna, J. Cleay, C. Slate, F. Regan, D. Diamond, and H. Muphy, Smatcoast: A wieless senso netwok fo wate quality monitoing, in LCN, 27. [2] A. Cepa, J. Elson, M. Hamilton, and J. Zhao, Habitat monitoing: application dive fo wieless communications technology, in Poc. of ACM SIGCOMM, 2. [3] F. Ye, G. Zhong, S. Lu, and L. Zhang, Gadient boadcast: a obust data delivey potocol fo lage scale senso netwoks, Wieless Netwok, vol., no. 3, pp , May 25. [4] T. He, J. Stankovic, C. Lu, and T. 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