Mobility Improves Coverage of Sensor Networks

Transcription

1 Mobility Improve Coverage of Senor Networ Benyuan Liu Dept. of Computer Science Univerity of Maachuett-Lowell Lowell, MA 1854 Peter Bra Dept. of Computer Science City College of New Yor New Yor, NY 131 Olivier Doue School of Computer and Communication Science EPFL Lauanne, Switzerland Philippe Nain INRIA 692 Sophia Antipoli France Don Towley Dept. of Computer Science Univerity of Maachuett Amhert, MA 12 ABSTRACT Previou wor on the coverage of mobile enor networ focue on algorithm to repoition enor in order to achieve a tatic configuration with an enlarged covered area. In thi paper, we tudy the dynamic apect of the coverage of a mobile enor networ that depend on the proce of enor movement. A time goe by, a poition i more liely to be covered; target that might never be detected in a tationary enor networ can now be detected by moving enor. We characterize the area coverage at pecific time intant and during time interval, a well a the time it tae to detect a randomly located tationary target. Our reult how that enor mobility can be exploited to compenate for the lac of enor and improve networ coverage. For mobile target, we tae a game theoretic approach and derive optimal mobility trategie for enor and target from their own perpective. Categorie and Subject Decriptor C.2.1 [Computer-Communication Networ]: Networ Architecure and Deign Networ Communication, Wirele Communication. General Term Deign, Performance. Keyword Coverage, Senor Networ, Mobility. Permiion to mae digital or hard copie of all or part of thi wor for peronal or claroom ue i granted without fee provided that copie are not made or ditributed for profit or commercial advantage and that copie bear thi notice and the full citation on the firt page. To copy otherwie, to republih, to pot on erver or to reditribute to lit, require prior pecific permiion and/or a fee. MobiHoc 5, May 25 27, 25, Urbana-Champaign, Illinoi, USA. Copyright 25 ACM /5/5...$ INTRODUCTION Recently, there ha been ubtantial reearch in the area of enor networ coverage. The coverage of a enor networ repreent the quality of urveillance that the networ can provide, for example, how well a region of interet i monitored by enor, and how effectively a enor networ can detect intruder (target). It i important to undertand how the coverage of a enor networ depend on variou networ parameter in order to better deign and ue enor networ for different application cenario. In many application, enor are not mobile; they remain tationary after their initial deployment. The coverage of uch a tationary enor networ i determined by the initial networ configuration. Once the deployment trategy and ening characteritic of the enor are nown, the networ coverage can be computed and remain unchanged over time. Recently, there ha been a trong deire to deploy enor mounted on mobile platform uch a mobile robot. Such mobile enor networ are extremely valuable in ituation where traditional deployment mechanim fail or are not uitable, for example, a hotile environment where enor cannot be manually deployed or air-dropped. Alo, in application cenario uch a atmophere and ocean environment monitoring, enor move with the urrounding air or ocean current. The coverage of a mobile enor networ now depend not only on the initial networ configuration, but alo on the mobility behavior of the enor. While the coverage of a enor networ with immobile enor ha been extenively explored and i relatively well undertood [12, 8, 9, 3, 7], reeacher have only recently tarted to tudy the coverage of mobile enor networ. Mot of thi wor focue on algorithm to repoition enor in deired poition in order to enhance networ coverage [5, 1, 1, 15, 14]. More pecifically, thee propoed algorithm trive to pread enor in the field o a to maximize the covered area. The main difference among thee wor are how exactly the deired poition of enor are computed. Although the algorithm can adapt to changing environment and recompute the enor location accordingly, enor mobility i exploited eentially to obtain a new tationary configuration that improve coverage after the enor move to their deired location. In thi paper, we tudy the coverage of a mobile enor

2 networ from a different perpective. Intead of trying to achieve an improved networ configuration a the end reult of enor movement, we identify and characterize the dynamic apect of networ coverage that depend on the movement of enor. Specifically, we are intereted in the coverage reulting from the continuou movement of enor. Thi coverage i not available if the enor top moving. We now briefly decribe the coverage provided by the enor movement, and the related reearch problem. Firt, previouly uncovered area become covered a enor move through them and covered area become uncovered a enor move away. A a reult, the location covered by enor change over time, and a greater area will be covered over time than in the cae where enor are tationary. Alo, a location i now not alway covered. It alternate between being covered and not being covered. Thi raied the following quetion: what i the area coverage at a given time intant? what i the area coverage over a time interval? what are the duration of time that a location i covered and not covered? Second, note that an initially undetected intruder will never be detected in a tationary enor networ if the intruder remain tationary or move along an uncovered path. In a mobile enor networ, an intruder i more liely to be detected a the moving enor patrol the field. Thu, enor mobility provide a time-varying coverage not available in a enor networ with tationary enor. Thi can ignificantly improve the intruion detection capability of a enor networ. Thi raie the additional quetion: how quicly can the enor detect an intruder? how doe the detection time depend on the earching trategy of the enor? and what are the optimal mobility trategie for the enor and the intruder? The main contribution of our wor are: Firt, we characterize the fraction of the area covered by enor for a randomly-deployed tationary enor networ. Thi characterization how how the covered area depend on the denity and ening characteritic of the enor. It provide a baeline for comparion with a mobile enor networ. We then conider a random mobility model for enor and tudy the effect of enor mobility on variou apect of networ coverage. We how that, while the fraction of the covered area at any given time intant remain unchanged, the area being covered during a time interval i improved a enor move around. A a reult, intruder and event that will never be detected in a tationary enor networ can now be detected in a mobile enor networ. Thi cenario i of great importance for application that do not require imultaneou coverage of all location at pecific time intant. Unlie a tationary enor networ where a location alway remain either covered or not covered, due to enor movement, a location i now only covered part of the time, alternating between being covered and not being covered. We characterize thi coverage and covering time tradeoff by the fraction of time a location i covered, which i determined by the denity and ening range of the enor, and doe not depend on the enor mobility. While thi time average characterization how, to a certain extent, how well a point i covered, it doe not reveal the duration of the time that a point i covered and uncovered. The time cale of uch duration are very important for networ planning; they preent the time granularity of the intruion detection capability that a mobile enor networ can provide. To thi end, we further characterize the time duration that a point i covered and not covered. In ome application, it i important to detect intruder in the field of interet a quicly a poible. To thi end, we tudy the detection time of an intruder, which i defined to be the time elaped before the intruder i firt detected. Intruder that will never be detected in a tationary enor networ can now be detected by moving enor. We obtain the ditribution of the detection time for a randomly located tationary intruder. The reult ugget that enor mobility can be exploited to effectively reduce the detection time of a tationary intruder when the number of enor i limited. For mobile intruder, the detection time depend on both the enor and intruder mobility trategie. We tae a game theoretic approach and tudy the bet wort-cae performance of a mobile enor networ in term of the intruder detection time. For a given enor mobility behavior, we aume that an intruder can chooe it mobility trategy o a to maximize it detection time (it lifetime before being detected). On the other hand, enor chooe a mobility trategy that minimize the maximum detection time reulting from the intruder mobility trategy. We prove that the optimal enor mobility trategy i for each enor to chooe it direction uniformly at random. The correponding intruder mobility trategy i to remain tationary in order to maximize the time before it i detected. The remainder of the paper i tructured a follow. The networ model and coverage meaure are preented in Section 2. In Section 3, we tudy the fraction of the area being covered at pecific time intant and during a time interval. In Section 4, we tudy the detection time for both tationary and mobile intruder. In Section 5, we review related wor on the coverage of enor networ. Finally, we ummarize the paper in Section 6. Throughout the paper, horthand X exp(µ) will tand for P (X < x) exp( µx), namely, the random variable X i exponentially ditributed with parameter µ. Alo, we will ue the word intruder and target interchangeably. 2. NETWORK AND MOBILITY MODEL In thi ection, we decribe the networ and mobility model ued in thi tudy, and define three coverage meaure of a mobile enor networ. 2.1 Sening Model We aume that each enor ha a ening radiu, r. A enor can only ene the environment and detect event within it ening area, which i the di of radiu r centered at the enor. A point i aid to be covered by a enor if it i located in the ening area of the enor. The enor networ i thu partitioned into two region, the covered region, which i the region covered by at leat one enor, and the uncovered region, which i the complement of the covered region. An intruder i aid to be detected if it lie within the covered region. 2.2 Coverage meaure To tudy the coverage of a enor networ, we define the following three coverage meaure.

3 Definition 1. Area coverage: The area coverage of a enor networ at time t, f a(t), i the fraction of the geographical area covered by one or more enor at time t. Definition 2. Area coverage over a time interval: The area coverage of a enor networ during time interval [, t), f i(t), i the fraction of the geographical area covered by at leat one enor at ome point of time within [, t). Definition 3. Detection time: Conider a target located at a random poition outide of the covered area of a enor networ at time t =. The detection time of the target, X, i defined to be the time at which the target firt enter the ening area of a enor, i.e., the target i firt detected by the enor. All three coverage meaure depend not only on tatic propertie of the enor networ (enor denity and ening range), but alo on the enor mobility behavior. The characterization of area coverage at pecific time intant i important for application that require imultaneou coverage of the networ, while the area coverage over a time interval i appropriate for application that do not require imultaneou coverage of all location at pecific time intant, but rather prefer to cover the networ within ome time interval. The detection time meaure how quicly a enor networ can detect a randomly located target that i not initially covered. 2.3 Location and Mobility Model We conider a networ coniting of a large number of enor placed in a vat two-dimenional geographical region. For the initial configuration, we aume that, at time t =, the location of thee enor are uniformly and independently ditributed in the region. Such a random initial deployment i deirable in cenario where prior nowledge of the region of interet i not available. Alo, random deployment can be the reult of certain deployment trategie. For example, enor may be air-dropped or launched via artillery in battlefield or unfriendly environment. Under thi aumption, the enor location can be modeled by a tationary two-dimenional Poion point proce. Denote the denity of the underlying Poion point proce a λ. The number of enor located in a region R, N(R), follow a Poion ditribution of parameter λ R, where R repreent the area of the region. P (N(R) = ) = e λ R (λ R ). (1)! Since each enor cover a di of radiu r, the initial configuration of the enor networ can be decribed by a Poion Boolean model B(λ, r). In a tationary enor networ, enor do not move after being deployed and networ coverage remain the ame a that of the initial configuration. In a mobile enor networ, depending on the mobile platform and application cenario, enor can chooe from a wide variety of mobility trategie, from paive movement to highly coordinated and complicated motion. Senor deployed in the air or ocean move paively according to external force uch a air or ocean current; imple robot may have a limited et of mobility pattern, and advanced robot can navigate in a more complicated fahion. In thi wor, we conider the following imple enor mobility model. We aume enor move independently of each other and with coordination among them. The movement of a enor i characterized by it peed and direction. A enor randomly chooe a direction θ [, ) according to ome ditribution with probability denity function fθ(θ). The peed of the enor, V, i randomly choen from a finite range v [, V max ], according to a ditribution denity function of fv (v). Throughout the ret of thi paper, we will refer to the initial enor networ configuration a a random enor networ B(λ, r), and the above mobility model a random mobility model. The coverage meaure defined above are function of the actual location of enor, which vary for different realization. In thi wor, we will tudy the expected value of the coverage meaure. 3. AREA COVERAGE In thi ection, we tudy and compare the area coverage of both tationary and mobile enor networ. We analytically characterize the area coverage. We then dicu the implication of our reult on networ planning and how that enor mobility can be exploited to compenate for the lac of enor to increae the area being covered during a time period. Finally, we point out, due to the enor mobility, a point i only covered part of the time; we further characterize thi effect by determining the fraction of time that a point i covered. Theorem 1. At any given time intant t >, the area coverage of a tationary enor networ B(λ, r) i f a(t) e λπr2. (2) Proof. Thi i a reult from tochatic geometry [4]. Here we preent the ey argument of the proof. Conider a bounded region R; the vacancy V within R i defined to be the area in R not covered by enor. where V = χ(x)dx. R 1 x i not covered χ(x) = otherwie. Uing Fubini theorem we have E(V ) = E{χ(x)}dx. R Conider an arbitrary point x in region R and denote the number of enor which cover the point a N. Point x i covered by enor located within ditance r. It follow immediately from the Poion point proce aumption that N ha a Poion ditribution with parameter λπr 2. Therefore, we have E{χ(x)} = P (x i not covered) = P (N = ) = e λπr2. and E(V ) = E{χ(x)}dx = R e λπr2. R Note that the above derivation i independent of R. The area coverage can thu be obtained a follow. f a E(V ) R 2 e λπr.

4 A illutrated in Figure 1, during time interval [, t), each enor cover a hape of a racetrac whoe expected area i α = E[πr 2 + 2rV t] = πr 2 + 2rE[V ]t. V where E[V ] = max f V (V )dv repreent the expected enor peed. A pointed out in [4], area coverage depend on the ditribution of the random hape only through it expected area. Therefore, we have f i(t) e αλ e λ(πr2 +2rE[V ]t). time time t Figure 1: Coverage of mobile enor networ: the left figure depict the initial networ configuration at time and the right figure illutrate the effect of enor mobility during time interval [, t). The olid di contitute the area being covered at the given time intant, and the union of the haded region and the olid di repreent the area being covered during the time interval. Thi formula characterize the dependence of area coverage on the networ propertie. It allow u to compute the fraction of the area being covered for a given enor denity and ening range. For example, in order to achieve a deired area coverage f a ( < f a < 1) almot urely, the denity required i given by λ = ln(1 f a)/πr 2. In a tationary enor networ, a location alway remain either covered or not covered. The area coverage doe not change over time. The following theorem characterize the effect of enor mobility on networ coverage. Theorem 2. Conider a enor networ B(λ, r) at time t =, with enor moving according to the random mobility model. 1. At any time intant t, the fraction of area being covered i f a(t) e λπr2, t. (3) 2. The fraction of area that ha been covered at leat once during time interval [, t) i f i(t) e λ(πr2 +2rE[V ]t). (4) 3. The fraction of the time a point i covered i f t e λπr2. (5) Proof. Given the initial node placement and the random mobility model, at any time intant t, the location of the enor till form a two dimenional Poion point proce of the ame denity [11, Theorem 9.14]. Therefore, the fraction of the area covered at time t remain the ame a in the initial configuration, f a(t) e λπr2. While an uncovered location will be covered when a enor move within ditance r of the location, a covered location become uncovered a enor covering it move away. A a reult, a location i only covered part of the time. More pecifically, a location alternate between being covered and not being covered, which can be modeled a an alternating renewal proce. We ue the fraction of time that a location i covered to meaure thi effect. The fraction of time that a location i covered equal the probability that it i covered at any given time intant, f t e λπr2. In the next ection, we further characterize the time duration of a point being covered and not being covered. At any pecific time intant, the fraction of the area being covered in a mobile enor networ model decribed above i the ame a in a tationary enor networ. Thi i becaue at any time intant, the poition of the enor are till decribed by a Poion Boolean model with the ame parameter a in the initial configuration. Unlie in a tationary enor networ, area initially not covered can now be covered a enor move around in a mobile enor networ. Conequently, target in the initially uncovered area can be detected by the moving enor. Figure 1 illutrate the effect of enor mobility on area coverage. The union of the olid di contitute the area coverage at given time intant. The area that ha ever been covered during time interval [, t) i depicted a the union of the haded region and the olid di, occupying a larger portion of the total area. Due to enor mobility, the fraction of the area that ha ever been covered increae and approache one a time goe to infinity. The rate at which the covered area increae over time depend on the expected peed of enor mobility. The fater enor move, the more quicly the area i covered. Therefore, enor mobility can be exploited to compenate for the lac of enor to improve the area coverage over an interval of time. Note that the area coverage during a time interval doe not depend on the ditribution of the enor movement direction. Baed on (4), we can compute the expected enor peed required to enure that a certain fraction of the area (f ) i covered within a time interval of length t. E[V ] = λπr2 + log(1 f ), for f 1 e λπr 2. t In a tationary enor networ, a location i either alway covered or not covered, a determined by it initial configuration. In a mobile enor networ, a a reult of enor mobility, a location i only covered part of the time, alternating between being covered and not being covered. The fraction of time that a location i covered correpond to the probability that it i covered, a hown in (5). Note

5 that thi probability i determined by the tatic propertie of the networ configuration (denity and ening range of the enor), and doe not depend on enor mobility. Thi coverage-delay tradeoff can be exploited by application that do not require imultaneou coverage of all location at pecific time intant. 4. DETECTION TIME In the previou ection, we characterized the fraction of area being covered at any given time and over a time interval. However, thee meaure do not reveal how quicly mobile enor detect target in the field. The time it tae to detect an intruding target i of great importance in many ecurity-related application. In thi ection, we tudy the detection time for both tationary and mobile target. For mobile target, we tae a game theoretic approach and explore the bet wort-cae performance of the enor networ. We will derive optimal mobility trategie for target and enor that maximize or minimize the detection time. To facilitate analyi and illutrate the effect of enor mobility on the detection time, we mae the aumption that all enor move at a contant peed v. More general peed ditribution can be approximated uing the fixed peed cenario. 4.1 Stationary Target We firt conider the cenario where target are tationary and do not initially fall into the coverage area of any enor. Obviouly, a tationary enor networ will never detect thee target; the detection time of thee target are infinite. However, in a mobile enor networ, a target can be detected by any enor paing within ditance r of it, where r i the common ening range of the enor. The detection time of a tationary target characterize how quicly the enor can detect a randomly located target previouly not detected. Theorem 3. Conider a enor networ B(λ, r) at time t =, with enor moving according to the random mobility model at a fixed peed v. Let X be the detection time of a randomly located tationary target, we have X exp(2λrv ). (6) Proof. We divide the pace evenly around point p in direction ( ). We now have enor clae and enor of cla i move in the direction θ i = i. Since each enor independently chooe it moving direction according to the ame ditribution, each enor cla i a thinning of the original point proce. Therefore, enor cla i form a Poion point proce with denity λ i = λfθ(θ i) θ, where θ = /. Point p will be detected when a enor move within ditance r from the point. Now let X i (firt hit time) be the time that point p i firt detected by a enor of cla i, the detection time X i the minimum of all the firt hit time, X = min X i. Since all enor of cla i move in the ame direction θ i at the ame peed v, it i more convenient to conider the framewor where enor are relatively tationary and the target move in the oppoite direction ( θ i) at the enor peed v. The ditance from point p to the perimeter of the firt enor to contact it in direction ( θ i) i called the linear contact ditance of that particular direction, and we denote it a Y i. From [13, page 8], we now that Y i follow an exponential ditribution with parameter 2λ ir, i.e., Y i exp(2λ ir). Since X i = Y i/v, the firt hit time in direction i follow an exponential ditribution with parameter 2λ irv, X i exp(2λ irv ). Now the minimum of thee exponential ditributed firt hit time i again an exponential ditribution, with a parameter equal to the um of the parameter of the individual exponential ditribution: lim i=1 2λ irv = lim i=1 2λf Θ(θ i) θrv = 2λrv fθ(θ)dθ = 2λrv. Thu, we have X exp(2λrv ). Compared to the cae of tationary enor where an undetected target alway remain undetected, the probability that the target i not detected in a mobile enor networ decreae exponentially over time, P (X t) = e 2λrvt. The expected detection time of a randomly located target i E[X] 2λrv, which i inverely proportional to the denity of the enor (λ), the ening range of each enor (r), and the peed of the enor movement (v ). Note that the expected target detection time i independent of the enor movement direction ditribution denity function, f Θ(θ). Therefore, in order to quicly detect a tationary target, one can add more enor, ue enor with larger ening range, or increae the peed of the mobile enor. Auming there i a requirement that the expected time to detect a randomly located tationary target be maller than a pecific value T, we have or equivalently, 1 2λrv T λv 1 2rT. Auming the ening range of each enor i fixed, the above formula preent the tradeoff between enor denity and enor mobility to enure certain target detection time requirement. The product of the enor denity and enor peed hould be larger than a contant. Therefore, enor mobility can be exploited to compenate for the lac of enor, and vice vera. In Theorem 2, we pointed out that a location alternate between being covered and not being covered, and then derived the fraction of time that a point i covered. While the time average characterization how, to a certain extent, how well a point i covered, it doe not reveal the duration of the time that a point i covered and uncovered. The time cale of uch time duration are alo very important for networ planning; they preent the time granularity of the intruion detection capability that a mobile enor networ can provide. Having Theorem 3, we now characterize

6 the time duration of a point being covered and not being covered. Corollary 1. Conider a random enor networ B(λ, r) at time t =, with enor moving according to the random mobility model. A point alternate between being covered and not being covered. Denote the time duration that a point i covered a T c, and the time duration that a point i not covered a T n, we have T n exp(2λrv ) (7) 2 E[T c] = eλπr 1. (8) 2λrv Proof. In the proof of Theorem 3, we have obtained the ditribution of time for a randomly located point before it i firt covered by a enor. Thi time i the ame a the time that a point i not covered. Therefore, we have T n exp(2λrv ). Since a point alternate between being covered and not being covered, the fraction of time a point i covered i f t = E[T c] E[T c] + E[T 2 n] e λπr. The lat equality in the above equation i given in (5). Solving for E[T c], we obtain (8). Let T denote the period of a point being covered and not being covered, i.e., T = T c + T n. The expected value of the period i E[T ] = E[T c] + E[T n] = e λπr2 /2λrv. 4.2 Mobile Target We now conider the cae where a target i alo mobile. The detection time of a mobile target depend not only on the mobility behavior of the enor but alo on the movement of the target itelf. Target can adopt a wide variety of movement pattern. In thi wor, we will not conider pecific target movement pattern. Rather, we approach the problem from a game theoretic tandpoint and determine the optimal mobility trategie of the target and enor. Given the mobility model of the enor, fθ(θ), a target chooe the mobility trategy that maximize it expected detection time. More pecifically, a target chooe it peed v t [, vt max ) and direction θ t [, ) o a to maximize the expected detection time. The expected detection time i a function of the enor direction ditribution denity, target peed, and target moving direction. Denote the reulting expected detection time a max vt,θt E[X(fΘ(θ), θ t, v t)]; the enor then chooe the mobility trategy (over all poible direction ditribution) that minimize the maximum expected detection time. optimization problem: We have the following minimax min max E[X(fΘ(θ), θ t, v t)]. (9) f Θ (θ) θ t,v t In the above problem tatement, each enor and the target move in traight line after they chooe their moving direction. More general mobility trategie would allow them to change direction and peed during their movement. However, we can ee that uch flexibility doe not lead to a y Figure 2: Relative enor peed in the framewor where target i tationary. better reult for either the enor networ or the target. If a enor change direction during it movement, there will be an overlap in it own covered area. Thi overlap caue the field to be covered le efficiently and will not help to detect the target ooner. Alo, in order to peed up the earch, each enor hould move at it maximum peed. Since the target only now the mobility trategy of the enor (enor direction ditribution denity function) and it doe not now the location and direction of the enor, changing direction and peed will not help the target to prolong it detection time. To olve the minimax optimization problem, we firt characterize the detection time of a target moving at a contant peed in a particular direction. Theorem 4. Conider a enor networ B(λ, r) at time t =, with enor moving according to the random mobility model at a fixed peed v. Let X be the detection time of a target moving at peed v t along direction θ t. Denote We have c = v t/v, ĉ + c w(u) = 1 4c co ĉ 2 u 2 2 v = v ĉ w(θ θ t)fθ(θ)dθ. X exp(2λrv ). (1) Proof. Similar to the proof of Theorem 3, we divide the pace evenly around point p into direction ( ). We now have enor clae where cla i enor move in the direction θ i = i. Senor of cla i form a Poion point proce with denity λ i = λfθ(θ i) θ, where θ = /. Let X i be the time point p i firt reached by a enor of cla i; the detection time X i the minimum of all of the firt hit time, X = min X i. Conider the reference framewor where the target i tationary. We can compute the relative peed of cla i enor to the target, v(θ i), a illutrated in Figure 2. Let c = v t/v be the ratio of the target peed to the enor peed. x

7 We have v(θ i) = (v co(θ i θ t) v t) 2 + v 2 in 2 (θ i θ t) = (v + v t) 1 2vvt (1 + co(θi θt)) (v + v t) 2 1 4c (θi θt) co2 ĉ2 2 = v ĉw(θ i θ t). = v ĉ In the framewor where cla i enor are relatively tationary, point p move in direction ( θ i) at peed v(θ i). From [13], the linear contact ditance, Y i, follow an exponential ditribution with parameter 2λ ir. Since X i = Y i/v i, we have X i exp (2λ irv i). The minimum of thee exponentially ditributed firt hit time i exponentially ditributed with a parameter equal to the um of the parameter of the individual exponential ditribution: lim i=1 2λ irv i = lim i=1 2λf Θ(θ i) θv(θ i)r = 2λr fθ(θ)v(θ)dθ = 2λrv where v = v ĉ w(θ θ t)fθ(θ)dθ. In the following, we will refer to v a the effective enor peed in the framewor where the target i tationary. From (6) and (1), it can be noted that the detection time of both moving target and tationary target follow exponential ditribution, and that the parameter are of the ame form, except the enor peed in (6) i now replaced by the effective enor peed in (1) for the mobile target cae. Auming that the enor denity and ening range are fixed, ince the target detection time follow an exponential ditribution of parameter 2λrv, maximizing the expected detection time correpond to minimizing the effective enor peed v. In the following, we derive the optimal target mobility trategie for two pecial enor mobility model. Senor move in the ame direction θ : f Θ(θ) = δ(θ θ ). Uing the fundamental property of the delta function f(x)δ(x a)dx = f(a), we have v = v ĉ w(θ θ t)δ(θ θ )dθ = v ĉw(θ θ t). We need to chooe a proper θ t and v t that minimize the above effective enor peed v. Firt, it i eay to ee that we require ĉ θ t = θ. Now, we have v = v 1 4c = ĉ 2 vt v and v i minimized when v v t = if vt max v v max otherwie. t Figure 3: Normalized effective relative enor peed v /v a a function of c = v t/v The above reult how, quite intuitively, that the target hould move in the ame direction a the enor at a peed cloet matching the enor peed. If the maximum target peed i greater than the enor peed, the target will not be detected ince it chooe to move at the ame peed and in the ame direction a the enor. In thi cae, the detection time i infinity. Otherwie, if the maximum target peed i maller than the enor peed, the target hould move at the maximum peed in the ame direction of the enor. The expected detection time i 1 2λr(v v max t ). The direction of enor movement i uniformly choen within [, ): fθ(θ). Figure 3 plot the normalized effective enor peed v /v a a function of c = v t/v, the ratio of the target peed to the enor peed. The effective enor peed i an increaing function of c, and i minimized when c =, or v t =. Therefore, if each enor uniformly chooe it moving direction from to, the maximum expected detection time i achieved when the target doe not move. The correponding expected 1 detection time i 2λrv. The optimal target mobility trategy in thi cae can be intuitively explained a follow. Since enor move in all direction with equal probability, the movement of the target in any direction will reult in a larger relative peed and thu a maller firt hit time in that particular direction. Conequently, the minimum of the firt hit time in all direction (detection time) will become maller. We now preent the olution to the minimax problem in the following theorem. Theorem 5. Conider a enor networ B(λ, r) at time t =, with enor moving according to the random mobility model at a fixed peed v. The optimal mobility trategy of the enor i for each enor to chooe a direction according to a uniform ditribution, i.e., f Θ(θ). Proof. For enor, minimizing target detection time i equivalent to maximizing the effective enor peed after a

8 target elect the optimal peed and direction. We firt prove for any given target peed v t, that among all poible enor direction ditribution, the minimum effective enor peed reulted from the optimal target direction choice, min θt v, i maximized when enor chooe direction according to a uniform ditribution. The formal tatement i decribed a follow. Denote the uniform ditribution denity a fθ uniform = 1/. From Theorem 4, the effective enor peed i a function of enor direction ditribution denity, target peed and direction, v (f Θ(θ), θ t, v t) = w(θ θ t)f Θ(θ)dθ. We will prove that min ν (fθ(θ), θ t, v t) min ν (fθ uniform, θ t, v t) (11) θ t,v t θ t,v t for all f Θ(θ). Firt, let u conider the right-hand ide of (11). We have ν (f uniform Θ, θ t, v t) θ t θ t w(θ θ t) dθ w(u) du w(u) du for all θ t, ince the mapping u w(u) i periodic with period. Thi how that min ν (fθ uniform, θ t, v t) θ t We now come bac to the proof of (11). We have 1 min θ t ν (f Θ(θ), θ t, v t) ν (f Θ(θ), θ t, v t) dθ t f Θ(θ) w(θ θ t)f Θ(θ) dθdθ t θ θ f Θ(θ) dθ w(u) du w(u) du. (12) w(u) du dθ w(u) du = min ν (fθ uniform, θ t, v t) (13) θ t where the lat three equalitie follow from the fact that w(u) i periodic with period, from the fact that fθ(θ) i a probabily denity function on [, ], and from (12), repectively. The proof of (11) i concluded by taing firt the minimum over v t in the left-hand ide of (13), then by taing the minimum over v t in the right-hand ide of (13). It follow from the proof of Theorem 5 that when enor chooe direction according to a uniform ditribution, the optimal target mobility trategy i to tay tationary, v t = (ince v (fθ uniform, θ t, v t) i maximized when c =, i.e., when v t = ), and θ t i irrelevant in thi cae. 5. RELATED WORK Recently, enor deployment and coverage related topic have become an active reearch area. In thi ection, we preent a brief overview of the previou wor on the coverage of both tationary and mobile enor networ that i mot relevant to our tudy. A more thorough urvey of the enor networ coverage i provided by [2]. Many previou tudie have focued on characterizing coverage for tationary enor networ. In [12], the author conidered a grid-baed enor networ and derived the condition for the ening range and failure rate of enor to enure that an area i fully covered. In [8], the author propoed everal algorithm to find path that are mot or leat liely to be detected by enor in a enor networ. Path expoure of moving object in enor networ wa formally defined and tudied in [9], where the author propoed an algorithm to find minimum expoure path, along which the probability of a moving object being detected i minimized. The bet coverage problem i further explored by Li et al in [6]. In [7], the author defined everal important coverage meaure for a large-cale tationary enor networ, namely, the area coverage, detection coverage, and node coverage. Under the aumption that enor location follow a Poion point proce, the author obtained analytical reult for the coverage meaure under a Boolean ening model and a general ening model. While the coverage of tationary enor networ ha been extenively tudied and relatively well undertood, reearcher have tarted to explore the coverage of mobile enor networ only recently. In [5], the author propoed a potentialfield-baed algorithm in which node are treated a virtual particle ubjected to virtual force. The virtual force repel the node from each other and from obtacle, and enure that the initial configuration of node quicly pread out to maximize coverage area. In [15], the author preented another virtual-force-baed enor movement trategy to enhance networ coverage after an initial random placement of enor. The virtual force of a enor i directly derived uing the ditance between the enor and the other enor and obtacle, and i ued to compute the enor movement that attempt to maximize the enor field coverage. After the execution of the algorithm and once the effective enor poition are identified, a one-time movement i carried out to redeploy the enor. In [14], the author propoed everal algorithm that identify exiting coverage hole in the networ and compute the deired target poition where enor hould move in order to increae the coverage. All of thee propoed algorithm trive to pread enor in the field in order to obtain a tationary configuration uch that the covered area i maximized. The main difference i how exactly the deired poition of enor are computed. Note that in thi wor, we tudy the coverage of a mobile enor networ from a very different perpective. Intead of trying to achieve an improved tationary networ configuration a an end reult of enor movement, we have focued on the coverage capabilitie reulted from the continuou movement of the enor. 6. SUMMARY In thi paper, we tudy the dynamic apect of the coverage of a mobile enor networ. More pecifically, we characterized area coverage at pecific time intant and during a

9 time interval, and the detection time of a randomly located target. Thee coverage meaure depend on the proce of enor movement and are unique attribute of mobile enor networ. For the random initial deployment trategy and the enor mobility model under conideration, we have hown that while the area coverage at any given time intant remain unchanged, more area will be covered during a time interval, alo, target that will be detected in a tationary enor networ can now be detected by moving enor. The tradeoff i that a location i only covered part of the time. Thi cenario i important for application that do not require imultaneou coverage at pecific time intant but rather deire to cover a large portion of the area during a certain time period. We ue detection time to meaure how quicly enor detect a randomly located target. The reult ugget that enor mobility can be exploited to effectively reduce the detection time of a tationary target when the number of enor i limited. For mobile target, the target detection time depend on enor and target mobility trategie. We tae a game theoretic approach and tudy the bet wortcae performance of a mobile enor networ in term of the target detection time. For a given enor mobility behavior, we aume that a target can chooe it mobility trategy o a to maximize it detection time, while enor chooe a mobility trategy that minimize the maximum detection time. We have hown that the optimal enor mobility trategy i that each enor chooe it movement uniformly in all direction. The correponding target mobility trategy i to tay tationary in order to maximize it detection time. 7. REFERENCES [1] M. Batalin and G. Suhatme. Spreading out: A local approach to multi-robot coverage. In 6th International Conference on Ditributed Autonomou Robotic Sytem (DSRS2), 22. [2] M. Cardei and J. Wu. Coverage in Wirele Senor Networ in Handboo of Senor Networ. CRC Pre, 24. [3] T. Clouqueur, V. Phipatanauphorn, P. Ramanathan, and K. K. Saluja. Senor deployment tategy for target detection. In Firt ACM International Worhop on Wirele Senor Networ and Application, 22. [4] P. Hall. Introduction to the Theory of Coverage Procee. John Wiley & Son, [5] A. Howard, M. Mataric, and G. Suhatme. Mobile enor networ deployment uing potential field: A ditributed, calable olution to the area coverage problem. In DARS 2, 22. [6] X.-Y. Li, P.-J. Wan, and O. Frieder. Coverage problem in wirele ad-hoc enor networ. IEEE Tranaction on Computer, 52(6): , June 23. [7] B. Liu and D. Towley. A tudy on the coverage of large-cale enor networ. In The 1t IEEE International Conference on Mobile Ad-hoc and Senor Sytem, 24. [8] S. Meguerdichian, F. Kouhanfar, M. Potonja, and M. B. Srivatava. Coverage problem in wirele ad-hoc enor networ. In Proc. IEEE Infocom, page , 21. [9] S. Meguerdichian, F. Kouhanfar, G. Qu, and M. Potonja. Expoure in wirele ad-hoc enor networ. In Mobile Computing and Networing, page , 21. [1] J. Pearce, P. Rybi, S. Stoeter, and N. Papaniolopoulou. Diperion behavior for a team of multiple miniature robot. In IEEE International Conference on Robotic and Automation, 23. [11] R. Serfozo. Introduction to Stochatic Networ. Springer, [12] S. Shaottai, R. Sriant, and N. Shroff. Unreliable enor grid: Coverage, connectivity and diameter. In Proc. IEEE Infocom, 23. [13] D. Stoyan, W. S. Kendall, and J. Mece. Stochatic Geometry and it Application: Second Edition. John Wiley and Son, [14] G. Wang, G. Cao, and T. L. Porta. Movement-aited enor deployment. In Proc. IEEE Infocom, 24. [15] Y. Zou and K. Charabarty. Senor deployment and target localization baed on virtual force. In Proc. IEEE Infocom, 23.