Dynamic Coverage of Mobile Sensor Networks

1 Dynamic Co v erage of Mobile Senso r Networks Benyuan Liu, Olivier Dousse, Philippe Nain and Don T owsle y Abstract In this paper we study the dynam ic aspects of the coverage of a mob ile senso r network resultin g from continuo us movement of sensors. As senso rs move arou nd, initially uncovered loca tions are likely to be covered at a later time. A larger area is covered as time c ontinues, and intru ders that migh t never be detec ted in a stationary sen sor network can n ow be detected by moving sensors. However , this improvement in coverage is achieved at the cost that a location is covered only par t o f the tim e, alternating between c overed and not covered. W e character ize area coverage at specific time instants and during time intervals, as well as th e time durations that a loc ation is covered and uncovered. W e fu rther characterize the time it takes to detect a random ly loca ted intruder . F or mobile intru ders, we take a game theoretic ap proach and der i ve optimal mobility strategies for b oth sensors an d in truders. Ou r resu lts show that sensor mob ility brings abou t u nique dyn amic coverage proper ties not present in a stationar y sensor network, and that mobility can be exploited to compensate for the lack of sen sors to improve coverage. Index T erms W ireless sensor n etworks, coverage, mobility . I . I N T R O D U C T I O N Recently , there has been sub stantial resea rch in the area o f se nsor ne twork coverage. Th e coverage o f a sensor network represe nts the quality of surveillance that the ne twork ca n provide, for example, how well a region of interest is monitored by sensors, and how effecti vely a s ensor network can d etect intruders. It is important to understand how the coverage of a se nsor network d epends on vari ous network parameters in order to better des ign and use se nsor networks for diff erent ap plication scena rios. In many ap plications, sensors are not mobile and remain s tationary after their initial deployment. The coverage of such a stationary sen sor network is determined by the initial network configu ration. Once the deployment strategy a nd s ensing ch aracteristics of the senso rs are known, network c overage can be computed and rema ins uncha nged ov er time. Recently , there ha s bee n increasing interest on b uilding mobile sen sor networks. Potential app lications abound . Sensors can be mounted on mob ile platforms such as mobile robots and move to desired area s [1], [2], [3], [4]. Su ch mobile sens or networks are extremely valuable in s ituations where traditional deployment mec hanisms fail or a re n ot s uitable, for example, a hostile en vironment whe re sens ors ca nnot be manually de ployed or air -dropped. Mobile s ensor networks can also play a vital role in ho meland security . Sen sors can be mounted on vehicles (e.g., s ubway trains, taxis, police c ars, fire trucks, boa ts, etc) or ca rried by people (e.g. , police men, fire fighters, e tc). These sen sors will mov e with their ca rriers, dynamically patrolling a nd monitoring the en vironment (e.g., che mical, biological, or radiolog ical agents). In other application sc enarios such a s atmosphe re and unde r -water en vironment monitoring, airborne or under-w ater sensors ma y move with the surrounding a ir or water currents [5]. The coverage of a mo bile Benyu an Liu is with t he Department of Computer Science, University of Massachusetts Lo well, email: bliu@cs.uml.edu. Olivier Do usse is with Nokia Research Center, Lausann e, Swit zerland, email: odou sse@mac.com. Phili ppe Nain is with INRIA Sophia Antipolis, F rance, email: Philippe.Nain@sophia.inria.fr . Don T owsle y is wit h the Department of Computer Science, Univ ersity of Massachusetts, Amherst, email: to wsley@cs.umass.ed u. sensor network now depends not only on the initial n etwork configurations , but also on the mob ility behavior of the sens ors. While the coverage of a senso r network with stationary se nsors has been extensively explored and is relati vely well unde rstood, resea rchers have only recently started to study the coverage of mobile senso r networks. Most of this work focuses on algorithms to reposition sensors in des ired positions in o rder to enhanc e network c overage [6], [7], [8], [9], [10]. More specifica lly , these proposed algorithms s tri ve to spread sens ors to desired locations to improve coverage. The main differences among these works are h ow exactly the desired p ositions of se nsors are computed. Althoug h the algorithms ca n adapt to cha nging en vironments and recompute the sens or loc ations acco rdingly , s ensor mobility is exploited essentially to obtain a new stationa ry configura tion that improves coverage after the se nsors move to their desired locations. In this pape r , we study the coverage of a mobile senso r n etwork from a dif ferent persp ectiv e. Instead of trying to achiev e an improved stationary network configu ration as the end resu lt of sensor movement, we are interested in the d ynamic aspec ts of network c overage resulting from the continuous movement of s ensors. In a stationa ry sens or network, the c overed a reas are determined by the initial confi guration and do not change ov er ti me. In a mobile sensor network, p reviously unc overed areas beco me covered a s sensors move through them and covered areas bec ome uncovered as sens ors move away . As a result, the areas covered by sensors change over t ime, a nd more area s will be covered at least once as time con tinues. The coverage status o f a location a lso ch anges with time, a lternating betwee n being covered and not being covered. In this work, we assume that sensors are initially randomly and uniformly dep loyed and move independ ently in randomly c hosen directions. Based on this mod el, we characterize the fraction of area covered at a given time instan t, the fraction of a rea ever covered during a time interval, as well as the time durations that a location is covered an d not covered. Intrusion d etection is an important task in many sensor network a pplications. W e measure the intrusion detection cap ability of a mobile se nsor network by the detec tion time of a ran domly located intruder , which is defined to be the time e lapsed before the intrude r is first detected by a s ensor . In a stationary sensor ne twork, an initially unde tected intruder will never be detected if it remains sta tionary or moves along an uncovered pa th. In a mobile sens or n etwork, however , s uch an intruder may be detected as the mobile sens ors patrol the field. This can significantly improve the intrusion detection capa bility of a sens or network. In this paper we characterize the de tection time of a ran domly located intruder . The results sugg est that sensor mobility can be exploited to eff ectiv ely reduce the detection time of a s tationary intruder when the number of sen sors is limi ted. W e further present a lower bound on the distrib ution of the detection time of a randomly located intruder , and s how that it c an be minimized if sensors move in straight lines. In some applica tions, for example, radiation, che mical, and biology age nts detections , there is a sensing time requirement before an intruder is detected. W e find in this case that too much mobility can be harmful if the sensor speed is above a threshold. Intuitiv ely , if a se nsor moves f aster , it will cover an area more quickly and detec t some intruders sooner , howe ver , at the same time, it will miss some intruders due to the sens ing time requiremen t. T o this en d, we fi nd there is an optimal sens or s peed that minimizes the detection time of a rando mly located intruder . For a mobile intruder , the detection time depend s on t he mobility s trategies of bo th sensors and intruder . W e take a g ame theo retic ap proach and study the optimal mobility strategies of s ensors and intruder . Gi ven the sensor mobility pattern, we assume that an intruder can choose its mobility strategy so as to maximize its detection time (its lifeti me before being d etected). On the other hand, sen sors c hoose a mobility strategy tha t minimizes the ma ximum detection time resulting from the intruder’ s mobility strategy . This can be vie wed as a zero-sum minimax game betwee n the collection of mobile s ensors and 2 the intruder . W e prove that the optimal sens or mobilit y strategy is for sens ors to choos e their directions uniformly at random betwe en [ 0 , 2 π ) . The corresponding intruder mobility strategy is to remain stationary to max imize its detection time. Th is so lution represents a mixed strategy which is a Nash equilibrium of the game between mobile sens ors and intruders. If sensors cho ose to move in any fixed direction (a pure strategy), it can be exploited by an intruder by moving in the same direction as sens ors to max imize its detection time. The optimal sensor strategy is to choose a mixture of avail able pure strategies (move in a fixed direction between [ 0 , 2 π ) ). The proportion of the mix should be such that the intruder cannot exploit the choice by pursuing any particular pure strate gy (mov e in the same direction as sen sors), res ulting in a uniformly random distribution for sens or’ s movement. When sens ors a nd intruders follo w their respe cti ve optimal strategies, neither side can achieve better p erformance by deviating from this b ehavior . The remaind er of the paper is structured as follows. In Section II, we revie w related work on the coverage of senso r ne tworks. The network model an d coverage me asures are de fined in Se ction III. In Section IV, we deri ve the fraction of the a rea being covered at sp ecific time instants an d during a time interval. The detection time for both s tationary a nd mobile intruders are studied in Section V and Section VI, res pectively . In Sec tion VI, we also d eri ve the the o ptimal mob ility strategies for se nsors and intruders from a g ame theoretic persp ecti ve. Fina lly , we summa rize the paper in Section VII. I I . R E L AT E D W O R K Recently , sens or deployment and coverage related top ics hav e become an activ e res earch area. In this section, we present a brief ov erview of the previous work on the c overage o f b oth stationary and mobile sensor networks that is most rele vant to our study . A more thorough survey of t he se nsor network coverage problems can be found in [11 ]. Many pre vious studies have focuse d on characterizing various coverage measures for stationary sens or networks. In [12], the autho rs c onsidered a grid-based se nsor network a nd deriv ed the conditions for the sensing range an d failure rate of sens ors to ensure that an area is fully covered. In [13], the authors proposed several algorithms to find pa ths that are most or lea st likely to b e de tected by se nsors in a sensor network. Path expo sure of moving o bjects in sens or ne tworks was formally d efined and studied in [14], where the authors prop osed an algo rithm to find minimum exposu re paths, along which the probab ility of a moving ob ject being detected is minimized . The path exposure problem is further explored in [15], [16], [17]. In [18], [19], [20], the k -coverage problem where eac h point is covered b y at leas t k s ensors was in vestigated. In [21], the authors defined a nd derived s everal important coverage me asures for a lar ge-scale stationary sens or network, n amely , area coverage, de tection c overage, and no de coverage, under a Boolea n s ensing model and a ge neral sensing model. Other coverage mea sures have also bee n studied. In [22], [23], the authors studied a metric of qua lity of surveillance wh ich is de fined to be the av erage distance that an intruder can move before be ing de tected, and prop osed a virtual pa trol mode l for su rveillance operations in sensor ne tworks. The relationsh ip between area coverage and network connec ti vity is in vestigated in [24], [25], [26 ]. While the coverage of stationa ry sensor networks has been extensiv ely studied and relati vely well understood , researche rs h av e started to explore the coverage of mob ile se nsor networks o nly rec ently . In [6], [9], [27 ], virtual-force b ased algorithms are used to repel no des from ea ch other and obs tacles to maximize coverage area. In [10], algo rithms are proposed to identify existing coverage holes in the network and co mpute the d esired target positions where senso rs should move in order to increase the coverage. In [28], a distrib uted c ontrol and coordination algo rithm is propo sed to co mpute the op timal sensor deployment for a class of utili ty functions which e ncode optimal coverage and sens ing po licies. In [29], mobility is use d for s ensor dens ity control su ch that the resultant de nsor density follows the spatial variati on of a scalar field in the en vironment. In [30], a n autonomous planning process is d ev eloped to 3 compute the deployment positions of sens ors and leader waypoints for navigationally-chanllenged se nsor nodes. The deployment of wireles s sensor networks un der mobility c onstraints and the tradeof f between mobility an d sensor dens ity for coverage are studied in [31], [32]. Many of these propos ed algorithms stri ve to sprea d sensors to desired pos itions in order to obtain a s tationary con figuration suc h that the c overage is o ptimized. Th e main diff erence is how the de sired sensor positions are computed. In this work we study the c overage of a mobile sensor n etwork from a very dif ferent persp ectiv e. Instead of trying to ac hiev e an improved stationary n etwork configu ration as an end re sult of sens or movement, we focus on the dynamic c overage properties resulting from the continuous movement of the sens ors. Intrusion detec tion problem in mobile s ensor networks has been c onsidered in a few recent studies, e.g., [33] , [34], [35], [36], [37]. In o ur work we take a stochastic ge ometry bas ed approa ch to deriv e closed-form expres sions for the de tection time under dif ferent network, mob ility , a nd se nsing mode ls. In [38], Chin et. a l. proposed an d studied a similar game theoretic problem formulation for a d if ferent network and mobility mo del. I I I . N E T W O R K A N D M O B I L I T Y M O D E L S In this se ction, we d escribe the network and mobility mo del, and introduce three coverage meas ures for a mo bile sens or network u sed in this study . A. Sensing Mode l W e ass ume that each s ensor has a sensing rad ius, r . A sensor ca n only sens e the en v ironment and detect intruders within its sensing area, which is the disk of radius r ce ntered at the senso r . A point is said to be cover ed by a sen sor if it is loca ted in the s ensing area of the sen sor . The sens or network is thus partitioned into two regions, the covered regi on, which is the region covered by at least one sensor , and the uncovered region, which is the complement o f the covered region. An intruder is sa id to be detected if it lies within the covered region. In rea lity , the sensing area of a senso r is us ually not of disk s hape due to hardware a nd en vironment factors. Nevertheless, the disk mo del ca n be us ed to a pproximate the real se nsing area and provide bou nds for the real case. For example, the irregular sens ing a rea o f a sensor c an be lower and upper bou nded by its maximum inscribed and minimum circumscribed circles, repe cti vely . B. Location an d Mobility Model W e conside r a sens or network consisting of a large nu mber of se nsors placed in a 2-dimens ional infinite plane. This is us ed to mod el a large two-dimensiona l geogra phical region. For the initial c onfiguration, we a ssume that, at time t = 0, the locations o f these sensors are uniformly and independen tly distrib uted in the region. Such a rand om initial deployment is desirable in sc enarios where prior knowledge of the region of interest is not avail able; it can also result from certain d eployment strategies. Unde r this assumption, the sens or locations can be modeled b y a stationary two-dimensional Poisson po int proces s. Denote the density of the unde rlying Poisso n point proce ss as λ . The number of sensors located in a region R , N ( R ) , follo ws a Poiss on distribution with p arameter λ k R k , where k R k represe nts the area of the region. Since each sen sor covers a dis k of radius r , the initial configuration of the sens or n etwork can b e described by a Poisson Boolean model B ( λ , r ) . In a stationa ry sensor network, sensors do not move after being d eployed and n etwork coverage remains the same as that of the initial co nfiguration. In a mobile sensor ne twork, depend ing on the mobile platform and ap plication sc enario, sens ors can ch oose from a 4 wide vari ety of mobility strategies, from passive movement to highly coordinated and complicated motion. For example, s ensors deployed in the air or water may move passively according to external forces such as air or water currents; simple robots may have a limited set of mobility pa tterns, and advanced robots can navigate in more complicated fashions; s ensors mounted on vehicles and peo ple move w ith their carriers, which ma y move ran domly and inde pendently or perform highly coordinated searc h. In this work, we conside r the follo wing s ensor mo bility model. Sensors follow arbitrary random curves independ ently of e ach other without coordination among the mselves. In some ca ses, whe n it helps to yield closed-form results and provide insights, we will make the model more specific by limiting sen sor movement to straight lines. In this model, the movement of a sensor is c haracterized by its speed and direction. A sensor ran domly chooses a direc tion Θ ∈ [ 0 , 2 π ) according to s ome d istrib ution with a probability d ensity function of f Θ ( θ ) . The spe ed of the sensor , V s , is randomly chosen from a finite range [ 0 , v max s ] , according to a distrib ution den sity function o f f V s ( v ) . The sensor speed and direction are independ ently ch osen from their respective distrib utions. The above models make simplified assu mptions for real network s cenarios. Our purpose is to ob tain analytical results based on the simplied assu mptions and provide insight and guideline to the deployment and performance of mob ile sensor n etworks. Th e Poisson distribution and u nit disk mod el h av e been widely us ed in the studies of wireles s networks (e. g., coverage and capa city problems) to ob tain ana lytical results. The Poisson spatial dis trib ution is a goo d approx imation for large networks where n odes are randomly and un iformly distributed. For the mob ility mode l, we cons ider the scena rios where n odes move indepen dently of each othe r . For example, sen sors can be carried by pe ople or mounted on people’ s vehicles, boats, or animals, etc. The se carriers are likely to move independe ntly according to their own activit y patterns without much coordination. This is similar to the u ncoordinated mo b ility model use d in [34]. Note that in some s cenarios (e.g ., se nsors mounted on robots) mob ile s ensors c an c ommunicate with each othe r and coordinate their moves. In that cas e the s ensors ca n optimize their movement patterns and p rovide more e f ficient c overage than the independe nt mobility case. In this paper we will focus on the indep endent mobility model. Throughou t the rest of this paper , we will refer to the initial se nsor network configuration as random sensor ne two r k B ( λ , r ) , t he first mobility model where sensors mov e in arbitrary curves a s random mo b ility model , and the more specific mobility mode l whe re s ensors a ll move in straight lines as s traight-line mobility model . The sh orthand X ∼ exp ( µ ) stands for P ( X < x ) = 1 − exp ( − µx ) , i.e., random variable X is expone ntially d istrib uted with p arameter µ . C. Coverage measu r es T o stud y the dynamic c overage properties of a mobile s ensor network, we de fine the followi ng three coverage me asures. Definition 1: Area coverage : The area coverage of a senso r n etwork a t time t , f a ( t ) , is the probability that a given point x ∈ I R 2 is covered by one or more sensors at time t . Definition 2: Time interval area cov erage : The area coverage of a sen sor network during t ime interval [ s , t ) with s < t , f i ( s , t ) , is the probab ility that gi ven a point x ∈ I R 2 , there exists u ∈ [ s , t ) suc h that x is covered by at least on e sens or at time u . Definition 3: Detec tion time : Suppose that an intrude r has a trajectory x ( t ) and that x ( 0 ) is un covered at time t = 0. The d etection time of the intruder is the smallest t > 0 such that x ( t ) is covered by at leas t one sen sor at time t . All three c overage measures depend not only on static properties of the senso r network (initial se nsor distrib ution, sens or dens ity and sen sing range), but also on se nsor movements. The cha racterization of area coverage at s pecific time instants is important for app lications that requ ire parts of the whole n etwork 5 time 0 time t Fig. 1. Cov erage of mobile sensor network : the left figure depicts the initial network configuration at time 0 and the right figure illustrates the effect of sensor mobility during time interval [ 0 , t ) . T he solid disks constitutes the area being co vered at the giv en time instant, and the union of the shad ed region and the solid disks represents the area being cov ered during the time interv al. be covered at any gi ven time instant. The time interv al a rea c overage is rele vant for applications tha t do not requ ire or ca nnot afford simultaneous c overage o f a ll loca tions at specific time ins tants, but prefer to cover the network within some time interv al. The detection time is important for intrusion detec tion applications, measu ring how q uickly a sen sor network can detect a randomly located intruder . I V . A R E A C OV E R AG E In this section, we study and comp are the area coverages of both stationary and mobile sens or netw orks. W e first an alytically characterize the area coverage. W e then disc uss the implications of ou r results on network p lanning a nd s how that sensor mo bility can be exploited to compensa te for the lac k of sensors to increase the area being covered during a time interval. However , we point out, due to the sensor mob ility , a point is only c overed part of the time; we further characterize this effect by determining the fraction of time that a po int is covered. Finally , we disc uss the optimal moving strategies that maximize the area coverage d uring a time interval. In a stationary se nsor network, a location always remains e ither covered or not c overed. Th e area coverage does not cha nge over time. The effect of sensor mobility o n area coverage is illustrated in Figure 1. The union of the solid disks cons titutes the area coverage at given time instants. As sens ors move arou nd, exact locations that a re covered at dif ferent time instants chan ge over time. The area that has been covered during time interval [ 0 , t ) is depicted as the union of the sha ded region and the solid disks. As can be o bserved, more a rea is covered during the time interval than the initial covered area. The following theorem characte rizes the ef fect of s ensor mobility on area coverage. Theore m 1: Con s ider a senso r network B ( λ , r ) a t time t = 0 , with se nsors moving acc ording to the random mobility mode l. 1) At a ny time instant t , the fraction of area being covered is f a ( t ) = 1 − e − λπ r 2 , ∀ t ≥ 0 . (1) 2) The fra ction of area tha t ha s been covered at least once dur ing time interval [ s , t ) is f i ( s , t ) = 1 − e − λ E ( α ( s , t ) ) . (2) where E ( α ( s , t ) ) is the expected ar ea c over ed by a sensor d uring time interval [ s , t ) . When sensor s all move in straight lines, we have f i ( s , t ) = 1 − e − λ ( π r 2 + 2 r ¯ v s ( t − s )) . (3) 6 where ¯ v s is the average senso r spe ed. 3) The fra ction of the time a point is covered is f t = 1 − e − λπ r 2 . (4) Proof. Denote the initial s ensors location proce ss as N 0 . By assu mption, N 0 is a two-dimensional Poisson point proc ess of density λ over I R 2 . Th e probab ility that a point x ∈ I R 2 is c overed is equa l to the probability that at least on e senso r is located in the disk of radius r centered on x (denoted hereafter D ( x , r ) ). At time t = 0, the location of the sensors is gi ven by N 0 , a nd we hav e f a ( 0 ) = P ( N 0 ∩ D ( x , r ) 6 = / 0 ) = 1 − exp ( − λπ r 2 ) . As ill ustrated in Figure 1, during a time interv al, each s ensor covers a greater area than its s ensing area at specific time instants. Den ote the covered area during time interval [ s , t ) by α ( s , t ) . The area coverage during the time interv al can be computed as f i ( s , t ) = 1 − e − λ E ( α ( s , t ) ) . If sens ors all move in s traight lines, during time interval [ s , t ) , each se nsor ha s covered a s hape of a racetrack whos e expected area is E ( α ( s , t )) = E  π r 2 + 2 r v s ( t − s )  = π r 2 + 2 r ¯ v s ( t − s ) . where ¯ v s = R v max s 0 f V ( V ) dV rep resents the expe cted sensor speed. In this cas e, the area coverage is gi ven by f i ( s , t ) = 1 − e − λ ( π r 2 + 2 r ¯ v s ( t − s )) . While an un covered location will be covered wh en a se nsor moves within distance r of the location, a covered loc ation becomes uncovered as se nsors covering it move aw ay . As a result, a loca tion is only covered p art of the time. More spec ifically , a location alternates between being covered a nd not being covered, which can be modeled as an alternating renewal process . W e use the fraction of time tha t a location is covered to measure this e f fect. As f a ( t ) = 1 − exp ( − λπ r 2 ) for all t , the expected fraction of time that a location is c overed, f t , is also equal to 1 − exp ( − λπ r 2 ) . The cas e where sensors and intruder are confi ned in a finite a rea can be a ddresse d by c onsidering a finite sample o f the model pre sented in this pa per . The av erage fraction of the area being covered at time t (or respectiv ely during the interval [ s , t ] ) is e qual to f a ( t ) ( f i ( s , t ) respe cti vely). Furthermore, a s the sens or location proces s at time t , N t , is a n ergodic point proce ss, if the sample area grows to infinity , the variance of the covered fraction tends to zero. This means tha t if the network a rea tends to infinity , the covered fraction become s de terministic. ✷ Note tha t this theorem is a generalization of The orem 2 in [33]. At any s pecific time instan t, the fraction o f the area being covered b y the mobile sensor ne twork described a bove is the sa me as in a stationary sens or ne twork. T his is becaus e at a ny time insta nt, the positions of the sens ors still form a Poisson point proce ss with the s ame p arameters as in the initial config uration. Howe ver , un like in a stationary sensor ne twork, covered locations change over time; areas init ially not covered will be cov ered as sen sors move a round. Conseq uently , intruders in the initially unc overed areas can be detected b y the moving sensors. When s ensors all move in straight lines, the fraction of the area that has ever been covered increa ses and approac hes one a s time proceeds . Later in this se ction we will prove that, amo ng all possible curves, straight line movement is an optimal s trategy that maximizes the a rea being covered during a time interval. The rate at which the covered area increas es over time depen ds on the expected sens or spee d. 7 The faster sensors move, the more q uickly the deployed region is covered. Therefore, sensor mob ility c an be exploited to comp ensate for the lack o f sensors to improve the area coverage over an interval of time. This is us eful for app lications that do no t require or c annot afford simultaneous coverage of a ll locations at any gi ven time, but need to cover a region within a giv en time interval. Note that the a rea coverage during a time interv al does n ot depend on the distrib ution of sensors mov ement direction. Ba sed on (3), we can compute the exp ected s ensor sp eed required to have a ce rtain fraction o f the area ( f 0 ) c overed within a time interv al of len gth t 0 . ¯ v s = − λπ r 2 + log ( 1 − f 0 ) 2 λ rt 0 , f or f 0 ≥ 1 − e λπ r 2 . Howe ver , the benefit of a greater area being covered a t lea st on ce during a time interv al come s with a price. In a stationa ry sens or network, a loca tion is either always covered or not c overed, as dete rmined by its initial configuration. In a mobile sens or network, as a resu lt of sensor mobility , a location is only covered part of the time, a lternating betwee n covered and no t covered. The fraction o f time tha t a location is covered c orresponds to the prob ability that it is covered, as s hown in (4). Note that this probability is determined by the static p roperties of the ne twork configuration (density and s ensing range of the sensors ), and doe s no t depe nd on senso r mobility . In the next se ction, we will further characterize the duration of the time intervals that a loca tion is covered a nd uncovered. From the proof of Theorem 1, it is easy to see that area coverage during a time interv al is maximized when sens ors move in straight lines. This is bec ause, a mong a ll possible curves, the are a covered by a sensor during time interv al [ s , t ) , α ( s , t ) , is maximized whe n the s ensor moves in a straight line. Based on (2), we have the follo wing theorem. Theore m 2: In a sensor network B ( λ , r ) with s ensor s moving according to the random mob ility model, the fra ction of area covered during a ny time interval [ s , t ) is ma x imized whe n se nsors a ll move in s traight lines. It is important to point out that straight line movement is not the on ly optimal strategy that maximizes the area coverage during a time interv al. There is a family of optimal movement patterns that maximize the coverage. W e conjec ture that the optimal movement patterns have the following properties: 1) the local radius of curvature is greater than the sensing range r ev erywhere alon g the o riented trajectory; 2) if the euclidean distance between two points of the curve is les s than 2 r , the n the dista nce between them along the curve is les s than π r . When these two properties are satisfied, the sensing disk of a sensor does not overl ap with its previously covered areas, an d a point will n ot b e covered red undantly by the same sensor . Th e covering ef ficiency is thus maximized. V . D E T E C T I O N T I M E O F S T A T I O N A RY I N T RU D E R The time it takes to detect an intruder is of great importance in many military a nd s ecurity-related applications. In this section, we stud y the detection time of a rand omly loc ated stationary intruder . Detection time for a mobile intrude r is in vestigated in the n ext sec tion. T o f acilitate the analys is and illustrate the effect of sensor mobility on de tection time, we c onsider the sce nario where all sen sors move at a cons tant s peed v s . More general sensor sp eed distrib ution scenarios can b e approximated using the resu lts of this ana lysis. W e as sume that intruders do not initially fall into the coverage area of any senso r . Obviously , these intruders will never be detected in a s tationary se nsor network. In a mobile sensor n etwork, howev er , an intruder c an be de tected by s ensors p assing within a distance r of it, wh ere r is the c ommon s ensing range of the sensors. The d etection time characterizes how quickly the mobile sens ors can detect a randomly located intruder previously no t detected . W e will first deriv e the detec tion time whe n sens ors all move in straight lines . W e will then c onsider the cas e when sens ors move acc ording to arbitrary curves. 8 s r θ v (t−s) Fig. 2. The region A ( s , t ) under the straight-line mobility model. Theore m 3: Con s ider a sensor network B ( λ , r ) with sensor s moving accor d ing to the straight-line random mo b ility model a nd a static intrude r . The s equence of times at which a ne w senso r detects the intruder forms a P oisson process of intens ity 2 λ r ¯ v s , whe re ¯ v s denotes the average senso r speed. As a conse quence, the time be f ore first the detection of the intruder is e xponentially dis tributed with the same par ameter . Proof: W e denote by A ( s , t ) the random region covered by a sensor in the interv al [ s , t ] , that was not covered be fore time s . Th e shape of this region is illustrated in Figure 2. W e first prove that the number of sensors hitting the intruder in the time interval [ s , t ] is Poisson distrib uted with parameter 2 λ r ¯ v s ( t − s ) . Su ppose without loss of ge nerality that the intruder is located at the origin. The prob ability that a s ensor initially located at po int x ∈ I R 2 hits the intruder within [ s , t ] is e qual to P ( − x ∈ A ( s , t )) . This proba bility only depe nds on the direction and speed of the sen sors; in particular , it does not de pend on the initial Poisso n process gi ving the pos itions of the sen sors. W e can thus define a thinned Poisso n process Φ ( s , t ) b y selecting at time 0 the se nsors that will hit the intruder during the interval [ s , t ] . This proc ess is non-uniform a nd has de nsity λ ′ ( x ) = λ P ( − x ∈ A ( s , t ) ) . The nu mber of sens ors hitting the intruder d uring [ s , t ] is eq ual to the total number of p oints in the thinned proce ss, which is Po isson distrib uted with mean E ( card ( Φ ( s , t ))) = Z I R 2 λ ′ ( x ) d x = λ Z I R 2 P ( − x ∈ A ( s , t ) ) d x = λ Z I R 2 E  1 {− x ∈ A ( s , t ) }  d x = λ E  Z I R 2 1 {− x ∈ A ( s , t ) } d x  = λ E ( || A ( s , t ) || ) , (5) where 1 {·} denotes the indicator function of t he event {·} . Furthermore, it is easy to see that E ( || A ( s , t ) || ) = 2 r ¯ v s ( t − s ) . Second , we show that the n umber of s ensors hitting the intruder d uring disjoint time intervals are independ ent. This is simply done by observing that if [ s 1 , t 1 ] ∩ [ s 2 , t 2 ] = / 0 , each sensor is either s elected in Φ ( s 1 , t 1 ) or in Φ ( s 2 , t 2 ) or not selected at all. Therefore, Φ ( s 1 , t 1 ) and Φ ( s 2 , t 2 ) are two ind epende nt process es. 9 Combining the two properties, we conclud e that the s equenc e of times at wh ich the intruder gets hit is a Poiss on proces s. ✷ Compared to the cas e of stationary se nsors where an undetec ted intruder a lw ays remains un detected, the probability that the intruder is not detected in a mobile sensor network decreases exponentially over time, P ( X ≥ t ) = e − 2 λ rv s t . where X repres ents the dete ction time of the intruder . The expec ted detection time of a randomly located intruder is E [ X ] = 1 2 λ rv s , which is in versely propor- tional to the density of the sens ors ( λ ), the sensing rang e of each sensor ( r ), and the spee d of s ensors ( v s ). Note that the expe cted intruder detection time is indep endent of the sens or movement d irection distrib ution de nsity function, f s Θ ( θ ) . Therefore, in order to quickly detect a stationary intruder , o ne can add more se nsors, use se nsors with larger sensing ranges , or increase the s peed of the mobile senso rs. T o guarantee that the expected time to detect a randomly located stationary intruder be s maller tha n a specific value T 0 , we hav e 1 2 λ r v s ≤ T 0 or equivalently , λ v s ≥ 1 2 r T 0 . If the s ensing range o f ea ch senso r is fixed, the above formula prese nts the trade off be tween sen sor density an d senso r mobility to ensure given exp ected intrude r de tection time requ irement. The product of the sens or density a nd sens or speed s hould be lar ger than a constant. The refore, sen sor mobility c an be exploited to co mpensate for the lac k of sen sors, and vice versa . In the proo f of Theo rem 1, we p ointed out that a loc ation alternates be tween b eing covered a nd not being covered, a nd then deriv ed the fraction of time that a p oint is covered. Wh ile the time average characterization shows, to a certain extent, h ow well a po int is covered, it does not reveal the d uration of the time tha t a p oint is covered an d unc overed. The time scale s of suc h time du rations are a lso very important for network plan ning; they pres ent the time granularity of the intrusion de tection ca pability that a mobile sensor network can provide. Th eorem 3 now a llo ws us to ch aracterize the time durations of a point being covered a nd not b eing covered. Corollary 1: Co n sider a rand om sensor network B ( λ , r ) at t ime t = 0 , with se n sors moving accord ing to the straight-line random mobility model. A point alternates b etween b eing covered and not be ing covered. De note the time duration that a p oint is covered as T c , and the time duration that a point is no t cover ed a s T n , we have T n ∼ exp ( 2 λ r v s ) (6) E [ T c ] = e λπ r 2 − 1 2 λ r v s . (7) Proof. In the proof of Th eorem 3, we know that the seque nce of times at which a n ew sensor hits a giv en point forms a Poiss on proces s of intensity 2 λ r v s . After ea ch s ensor hits the point, it immediately covers the point until it moves out of range. There is no con straint on the nu mber of sensors tha t covere the p oint. Therefore, the covered/uncovered sequ ence experienced by the p oint ca n be s een as a M / G / ∞ queuing process , wh ere the service time of an sen sor is the time duration that the senso r covers the point 10 Intruder Fig. 3. Mobile sensor network with sensors moving along arbitrary curves . before moving out of range. The idle periods of M / G / ∞ queue c orresponds to the time duration that the point is not c overed. It is known that idle periods in such que ues have expone ntially distributed durations . Therefore, we have T n ∼ exp ( 2 λ r v s ) . Since a point a lternates between b eing covered and not b eing covered, the fraction o f time a point is covered is f t = E [ T c ] E [ T c ] + E [ T n ] = 1 − e − λπ r 2 . The last e quality in the above equ ation is giv en in (4). Solving for E [ T c ] , we obtain (7). Let T deno te the period of a p oint b eing covered an d not being c overed, i.e., T = T c + T n . T he expe cted value of the period is E [ T ] = E [ T c ] + E [ T n ] = e λπ r 2 / 2 λ r v s . ✷ Above we obtain the detection time of a stationary intruder when sensors all mov e in straight lines. In practice, mobile s ensors do not always move in straight lines; they may ma ke turns and move in dif ferent curves, as depicte d in Figure 3. Next, we establish the op timal sensor moving s trategy to minimize the detection time of a stationary intrude r . Theore m 4: Con s ider a senso r network B ( λ , r ) a t time t = 0 , with se nsors moving acc ording to the random mo bility mod el at a fixed spee d v s . The detection time of a r andomly located stationar y intruder , X , is minimized in probability if sens ors all move in stra igh t lines. Proof: From Equation (5), we kn ow that the numbe r of s ensors detecting the intruder du ring the interval [ 0 , t ] is Poiss on distrib uted with me an E ( || A ( 0 , t ) || ) . Thus we have P ( X ≤ t ) = P ( ca rd ( Φ ( 0 , t )) ≥ 1 ) = 1 − exp ( − E ( | | A ( 0 , t ) || )) , which is a increasing function of E ( || A ( 0 , t ) || ) . As E ( || A ( 0 , t ) || ) is ma ximized wh en se nsors move along straight lines, the probability of d etecting the intruder is also maximized . ✷ Similar to the arguments on the optimal strategies for area coverage in Section IV, straight line movement is not the only optimal strategy that minimizes the de tection time. There is a famil y of moving patterns that ca n minimize the de tection time, where s traight line movement is one of the m. In the above analys is, we have ass umed that an intrude r is immed iately detected when it is hit b y the perimeter o f a sensor , regardless of the time duration ( t s ) it stays in the sensing ran ge of the se nsor . In many intrusion detection applications , for examp le, radiation, ch emical, and biology threats, due to the probabilistic nature of the phenome non and the sen sing mechanisms, an intruder will not be immediately detected once it enters the sensing range of a sen sor . Instead, it will take a ce rtain a mount o f time to 11 Sensor trajectory r r eff v t /2 d s O Fig. 4. Effecti v e r adius of a mobile sensor . detect the intruder . If the sensing time is too short, an intrude r may es cape undetected . T o a ccount for this sensing time req uirement, we defi ne t d to be the minimum sens ing time in order for a sensor to detect a n intruder . Obviously , it is only interesting when 0 ≤ t d ≤ 2 r / v s . Otherwise, the sensing time of an intruder by a s ensor will be smaller than the minimum requirement t d , and the intruder will never be detected. In o rder to yield clos ed-form results and provide ins ights, w e will consider the straight-line random mobility model. Theore m 5: Con s ider a senso r network B ( λ , r ) a t time t = 0 , with se nsors moving acc ording to the straight-line r a ndom mobilit y model at a fixe d s peed v s . An intrud e r is detec te d iff the sensing time t s is at lea st t d , i.e., t s ≥ t d . Let Y be the detec tion time of a randomly located stationar y intruder initially not loca ted in the sensing area of any sen sor , we have Y = t d + T (8) where T ∼ exp ( 2 λ r ef f v s ) (9) r ef f = r r 2 − v 2 s t 2 d 4 . (10) Proof: W e a ssume without loss of generality that the intruder is loca ted at the o rigin. W e obse rve first that a s ensor c overs the intruder for a time longer than t d if a nd only if the distance be tween its trajectory and the origin is less than r ef f (see Figure 4). W e ca ll such sens ors valid sensors. Similarly as in Theorem 1, we de fine a thinned Poisson proc ess Φ ef f ( 0 , t ) by se lecting the se nsors that will detec t the intruder du ring the interval [ 0 , t ] . T o do so, w e defi ne the effective covered a rea A ef f ( 0 , t ) of a sens or as the area covered by the disk of radius r ef f centered on it. Then, the probability that a sen sor initially located at x detects the intruder during the interval [ 0 , t ] is P ( − x ∈ A ef f ( 0 , t )) . By (5) we find that the expected numb er of points in Φ ef f ( 0 , t ) is λ E ( || A ef f ( 0 , t ) || ) = 2 λ r ef f v s . Denoting by T the time before a valid sensor covers the intruder , we get P ( T ≤ t ) = P ( card ( Φ ef f ( 0 , t )) ≥ 1 ) = 1 − exp ( − 2 λ r ef f v s ) . Then, the intruder is finally dete cted by the sy stem after a time T + t d . ✷ In (8), the de tection time has two terms, namely , a co nstant term t d and an exponentially d istrib uted random variable with me an E [ T ] = 1 / ( 2 λ r ef f v s ) . The first term t d is a direct conse quence of the minimum sensing time requirement. After the pe rimeter of a sen sor hits an intruder , it takes a minimum sen sing time o f t d to detec t the intruder , and henc e the con stant delay . By The orem 3, the se cond term corres ponds 12 to the detec tion time in the cas e where there is no minimum s ensing time requirement but sensors hav e a redu ced sensing radius of r ef f . This is aga in a con sequen ce of the minimum sensing time req uirement and the e f fect is illustrated in Figure 4. An intruder will o nly be detec ted by a mobile sensor if the trajectory of the sensor falls within r ef f from the intruder . The above two e f fects of minimum s ensing time requirement result in an increas ed exp ected de tection time compared to the case without minimum sensing time requ irement. Since t d > 0 and r ef f < r , we have E [ Y ] = t d + 1 / ( 2 λ r ef f v s ) > 1 / ( 2 λ r v s ) = E [ X ] . Sensor sp eed has two opp osite ef fects on an intruder’ s detection time. • On one hand , as sensors move faster , uncovered area s will be c overed more quickly and this tends to spe ed up the de tection of intruders. • On the other hand, the effecti ve sensing radius r ef f decreas es as s ensors increas e their speed due to the se nsing time requirement, making intruders less likely to be detected . In the following, we presen t the optimal sensor spe ed that minimizes the expec ted detection time. Excess mobility will be harmful when the sen sor speed is larger than the optimal value. Theore m 6: Unde r the scenario in Theorem 5, the optimal sensor speed minimizing the e xpected detection time of a ra ndomly located intruder is v ∗ s = √ 2 r / t d . (11) Proof. Let dY / d v s = 0, we have v ∗ s = √ 2 r / t d , and the secon d order deriv ati ve d 2 Y d v 2 s | v ∗ s < 0 . T he corre- sponding minimum expe cted detection time is E [ Y ∗ ] = ( 1 + 2 λ r 2 ) t d / 2 λ r 2 . ✷ V I . D E T E C T I O N T I M E O F M O B I L E I N T R U D E R In this s ection, we c onsider the de tection time of a mobile intruder , which d epends not on ly on the mobility behavior of the sen sors but also on the movement of the intrude r itself. Intruders can adopt a w ide variety of movement p atterns. In this work, we will not c onsider sp ecific intruder movement patterns. Rather , we approach the problem from a game theoretic sta ndpoint and study the optimal mobility strategies of the intruders and sen sors. From Theorem 4, the de tection time of a stationary senso r intruder is minimized when sensors all move in straight lines. This res ult can be e asily extended to a mo bile intruder using similar arguments in the reference framework wh ere the intruder is stationary . From the pe rspectiv e of an intruder , s ince it only knows the mobility strategy of the sens ors (se nsor direction distrib ution density func tion) and does not kn ow the locations and directions of the sensors, ch anging direction and spee d will n ot help prolong its detection time. In the following, we will o nly consider the ca se where s ensors and intruders move in straight lines. Gi ven the mobilit y model of the sens ors, f Θ ( θ ) , an intrude r chooses the mobility strategy that maximizes its expe cted d etection time. More spe cifically , an intrude r choos es its spe ed v t ∈ [ 0 , v max t ) and d irection θ t ∈ [ 0 , 2 π ) so as to maximize the expec ted detection time. Th e expected detection time is a function o f the sensor direction distribution dens ity , intruder s peed, and intruder moving direction. De note the resulting expected detection time a s max v t , θ t E [ X ( f Θ ( θ ) , θ t , v t )] ; the senso rs the n choose the mobility strategy (over all poss ible d irection distrib utions) that minimizes the maximum expected detec tion time. This can be viewed as a zero-su m minimax game b etween the collection of mobile sens ors an d the intruder , wh ere the pa yoff s for the mobile sensors and intruder are − E [ X ( f Θ , θ t , v t )] and E [ X ( f Θ , θ t , v t )] , resp ectiv ely . 13 T o fi nd the optimal mobility strategies for mo bile sensors and the intruder , we consider the follo wing minimax optimization problem: min f Θ max θ t , v t E [ X ( f Θ , θ t , v t )] . (12) T o s olve the minimax optimization problem, we first characterize the d etection time o f an intruder moving at a con stant speed in a particular direction. Theore m 7: Con s ider a senso r network B ( λ , r ) a t time t = 0 , with se nsors moving acc ording to the stright-line rand o m mobility model at a fixed speed v s . L e t X be the detection time of an intruder moving at speed v t along direction θ t . Denote c = v t / v s , ˆ c = 1 + c w ( u ) = q 1 − 4 c ˆ c 2 cos 2 u 2 v s = v s ˆ c R 2 π 0 w ( θ − θ t ) f Θ ( θ ) d θ . W e have X ∼ exp ( 2 λ r v s ) . (13) Proof: T o prove this the orem, we put ourselves in the frame of reference of the intruder a nd look at the speed s of the sens ors. Thus, if a se nsor has an ab solute speed vector v s , its s peed vector in the new frame of referenc e is simply v s − v t , where v t denotes the intruder’ s ab solute spe ed vector . Let θ s denote the direction of v s and θ t the direction of v t . In the n ew frame of referenc e, the intruder is static. Den ote c = v t / v s , ˆ c = 1 + c , and w ( u ) = q 1 − 4 c ˆ c 2 cos 2 u 2 . Using the Law of Cosines, the relati ve speed of the sen sor can be comp uted as || v s − v t || = q v 2 s + v 2 t − 2 v s v t cos ( θ s − θ t ) = v s ˆ cw ( θ s − θ t ) W e know from Equation (5) that P ( X ≤ t ) = P ( ca rd ( Φ ( 0 , t )) ≥ 1 ) = 1 − exp ( − λ E ( | | A ( 0 , t ) || )) . Therefore, if E ( || A ( 0 , t ) || ) is a linear function of t , then X is exp onentially distrib uted. W e get || A ( 0 , t ) || = 2 r || v s − v t || t = 2 rt w ( θ s − θ t ) , so that E ( || A ( 0 , t ) || ) = 2 rt Z 2 π 0 w ( θ − θ t ) f Θ ( θ ) d θ = 2 r t v s where v s = v s ˆ c R 2 π 0 w ( θ − θ t ) f Θ ( θ ) d θ , which can be viewed as the average effecti ve sensor spee d in the reference framew ork wh ere the intruder is stationary . Therefore, the detection time is expone ntially distrib uted with rate 2 λ r v s . ✷ From Theo rems 3 a nd 7, it can be noted tha t the detec tion times of b oth stationa ry and mobile intruders follo w exp onential distribut ions, and that the parameters are of the same form, except that the sensor speed is n ow rep laced by the e f fectiv e sensor spee d for the mobile intruder case. Assuming that the senso r de nsity and sensing range a re fixed, s ince the intruder de tection time follo ws an expo nential distribution with mea n 1 / ( 2 λ r v s ) , max imizing the expected de tection time c orresponds 14 0 0.5 1 1.5 2 2.5 3 1 1.5 2 2.5 3 3.5 v s =v s v t =v s Fig. 5. Normalized ef fecti ve relative sensor speed v s / v s as a function of c = v t / v s to minimizing the effecti ve s ensor spe ed v s . In the following, we deri ve the optimal intruder mobility strategies for two spec ial sensor mobility models. Sensor s move in the same direction θ s : f Θ ( θ ) = δ ( θ − θ s ) . Using the fun damental prope rty of the delta fun ction R ∞ − ∞ f ( x ) δ ( x − a ) d x = f ( a ) , we hav e v s = v s ˆ c Z 2 π 0 w ( θ − θ t ) δ ( θ − θ s ) d θ = v s ˆ cw ( θ s − θ t ) . W e n eed to choos e a p roper θ t and v t that minimizes the ab ove ef fecti ve se nsor speed v s . First, it is easy to se e that we req uire θ t = θ s . Now , we have v s = v s ˆ c r 1 − 4 c ˆ c 2 = | v t − v s | and v s is minimized when v t =  v s if v max t ≥ v s v max t otherwise. The above res ults show , quite intuiti vely , that the intrude r sh ould move in the sa me direction as the sensors a t a speed closest matching the s ensor speed. If the max imum intruder spee d is larger than the sensor spee d, the intrude r will not be d etected since it ch ooses to move at the same sp eed and in the same direction as the sens ors. In this case, the detection time is infinity . Otherwise, if the ma ximum intruder s peed is smaller tha n the sensor spe ed, the intruder sh ould move at the maximum speed in the same direction of the sen sors. The expected detection time is 1 2 λ r ( v s − v max t ) . Sensor s move in un iformly random directions: f Θ ( θ ) = 1 2 π . Figure 5 plots the normalized effecti ve sensor s peed v s / v s as a fun ction of c = v t / v s , the ra tio o f the intruder spee d to the sen sor sp eed. The ef fectiv e se nsor spee d is an increa sing func tion of c , a nd is minimized when c = 0, or v t = 0. Th erefore, if each sen sor uniformly choo ses its moving direction from 0 to 2 π , the maximum expe cted detection time is a chieved when the intruder does no t move. The correspond ing exp ected detection time is 1 2 λ rv s . Th e optimal intruder mobilit y s trategy in this c ase can be 15 intuiti vely explained as follows. Since sen sors move in all directions with eq ual probability , the movement of the intruder in a ny d irection will result in a lar ger relative s peed an d thus a sma ller first hit time in tha t particular direction. Cons equen tly , the minimum of the first h it times in all directions (detection time) will beco me smaller . W e n ow present the solution to the minimax game between the c ollection of mobile sensors and the intruder in the follo wing theo rem. Theore m 8: Con s ider a senso r network B ( λ , r ) a t time t = 0 , with se nsors moving acc ording to the r andom mob ility mod e l at a fixed speed v s . For the ga me between the collection of mobile sensor s and the intruder , the optimal sensor strategy is for eac h se n sor to choose a dir ection accord ing to a unif orm distribution, i.e., f Θ ( θ ) = 1 2 π . The o p timal mobilit y strategy of the intruder is to stay stationar y . This s olution constitutes a Nash equilibrium of the game. Proof. For sen sors, minimizing intruder detection time is equi valent to maximizing the effecti ve se nsor speed after an intruder s elects the optimal s peed a nd d irection. W e first prove for any given intruder sp eed v t , that among all possible sens or d irection distrib utions, the minimum effecti ve s ensor speed resulted from the op timal intruder direction choice, min θ t v s , is maximize d when sens ors choose directions acco rding to a uniform distrib ution. Th e formal statement is described as follows. Denote the uniform distrib ution density as f uniform Θ = 1 / 2 π . From Theorem 7, the effecti ve sensor s peed is a function o f se nsor direction distrib ution density , intruder s peed a nd direction, v s ( f Θ ( θ ) , θ t , v t ) = R 2 π 0 w ( θ − θ t ) f Θ ( θ ) d θ . W e will prove that min θ t , v t ν s ( f Θ ( θ ) , θ t , v t ) ≤ min θ t , v t ν s ( f uniform Θ , θ t , v t ) (14) for all f Θ ( θ ) . First, let us conside r the right-hand side of (14). W e have ν s ( f uniform Θ , θ t , v t ) = 1 2 π Z 2 π 0 w ( θ − θ t ) d θ = 1 2 π Z 2 π − θ t − θ t w ( u ) d u = 1 2 π Z 2 π 0 w ( u ) d u for all θ t , since the mapping u → w ( u ) is periodic with p eriod 2 π . This s hows that min θ t ν s ( f uniform Θ , θ t , v t ) = 1 2 π Z 2 π 0 w ( u ) d u . (15) 16 W e now come ba ck to the p roof of (14). W e have min θ t ν s ( f Θ ( θ ) , θ t , v t ) ≤ 1 2 π Z 2 π 0 ν s ( f Θ ( θ ) , θ t , v t ) d θ t = 1 2 π Z 2 π 0 Z 2 π 0 w ( θ − θ t ) f Θ ( θ ) d θ d θ t = 1 2 π Z 2 π 0 f Θ ( θ )  Z θ θ − 2 π w ( u ) d u  d θ = 1 2 π  Z 2 π 0 f Θ ( θ ) d θ   Z 2 π 0 w ( u ) d u  = 1 2 π Z 2 π 0 w ( u ) d u = min θ t ν s ( f uniform Θ , θ t , v t ) (16) where the last three equalities follow from the fact that w ( u ) is p eriodic with period 2 π , from the fact that f Θ ( θ ) is a probability density function on [ 0 , 2 π ] , a nd from (15), respectiv ely . The proof of (14) is concluded by taking first the minimum over v t in the left-hand s ide of (16), then by taking the minimum over v t in the right-hand side of (16). It follo ws tha t when sensors c hoose directions acc ording to a uniform d istrib ution, the optimal intr uder mobility strategy is to stay s tationary , v t = 0 (since v s ( f uniform Θ , θ t , v t ) is maximized when c = 0 (an d equ als to 1), i.e . when v t = 0), and θ t is irrelev ant in this ca se. Based on the previous discu ssions on different mobility s trategies of senso rs and intruders, under the derived optimal mob ility strategies, neither side can improve the payoff by changing the strategy unilaterally . Specifically , whe n se nsors c hoose their direction uniformly at random, the movement of the intruder in any direction will res ult in a larger relative spe ed an d thus a smaller first h it time in that particular direction. Cons equently , the minimum o f the first hit times in all d irections (de tection time) will become smaller . When the intruder s tays stationary , the dection time will not improve if s ensors choose a diff erent d istrib ution for the moving direction. Therefore, the solution con stitutes a Nash eq uilibrium of the game. ✷ This resu lt su ggests tha t in order to minimi ze the expected detection time o f an intrude r , sensors sh ould choose their direc tions uniformly at random betwee n [ 0 , 2 π ) . The corres ponding op timal mobility strategy of the intruder is to stay stationary . The uniformly ran dom sensor movement represen ts a mixed strategy wh ich is a Nash equilibrium of the game between mobile s ensors an d intruders. If senso rs choose to move in any fixed d irection (pure strategy), it can be exploited by an intruder by moving in the same direction as sens ors to max imize its detection time. The optimal sensor strategy is to choose a mixture of avail able pure strategies (move in a fixed direction between [ 0 , 2 π ) ). The p roportion of the mix should be such that the intruder cannot exploit the choice by pursuing any particular pure strate gy (mov e in the same direction as sen sors), res ulting in a uniformly random distribution for sens or’ s movement. When sens ors a nd intruders follo w their respe cti ve optimal strategies, neither side can achieve better p erformance by deviating from this b ehavior . 17 V I I . S U M M A RY In this pape r , we study the dynamic as pects o f the coverage of a mobile sen sor n etwork resulting from the co ntinuous movement of se nsors. Specifica lly , we studied the c overage meas ures related to the area coverage a nd intrusion detection c apability of a mobile sensor ne twork. For the random initial deployment and the random sen sor mobility model under conside ration, we showed that while the area coverage at any g i ven time instants remains uncha nged, more area will be covered at least o nce during a time interval. This is important for applications that do not require or cannot af ford simultaneous c overage o f all locations but want to cover the d eployed region within a certain time interval. The cost is that a location is o nly covered part of the time, a lternating b etween covered and n ot covered. T o this end, we charac terized the d urations a nd fraction of time that a loca tion is covered and not covered. As sens ors move around, intruders that will never be detec ted in a stationary se nsor network can be detected by moving sensors. W e cha racterized t he de tection time of a randomly located stationary intruder . The results sugges t that sens or mo bility c an be exploited to ef fectiv ely reduce the detection time of an intruder when the number o f sens ors is limited. W e further considered a more rea listic sensing mode l where a minimum sensing time is required to detect an intrude r . W e find that there is an optimal sen sor speed that minimizes the expected detec tion time. Beyond the optimal speed , excess mob ility will be harmful to the intrusion detection performance. Moreover , we discuss ed the optimal mob ility strategies that maximize the area coverage d uring a time interval an d minimize the detection time of intruders. For mobile intr uders, the intruder detection time depen ds o n the mobilit y strategies of the sens ors as well as the intruders. W e took a ga me theoretic app roach and obtained the op timal mobility strategy for se nsors and intruders. W e s howed tha t the optimal senso r mobility strategy is that each sens or choos es i ts direction uniformly at random in a ll directions. B y maximizing the entropy of the senso r direc tion distributi on, the amount of prior information on sen sor mobility strategy revealed to a n intruder is minimized . The correspond ing intruder mobility strategy is to stay stationary in order to maximize its detection time. This so lution represe nts a Nash e quilibrium of the game be tween mobile sensors and intruders. Neither side can a chieve better performanc e by deviating from their respec ti ve optimal strategies. R E F E R E N C E S [1] L. E. Na va rro-Serment, R. Grabo wski, C. J. Paredis, and P . K. Khosla, “Millibots: The dev elopment of a f rame work and algorithms for a distributed heterogeneo us robot team, ” IEE E Robotics and Automation Mag azine , vol. 9, no. 4, 200 2. [2] S. Bergbreiter and K. P ister , “Cotsbots: An off-the-shelf platform for distributed robotics, ” in Proc . of the IEEE/RSJ International Confer ence on I ntell igent Robots and Systems , 2003. [3] G. T . Sibley , M. H. Rahimi, and G. S. S ukhatme, “Robomote: A tiny mobile robot platform for large-scale sensor network s, ” in Proceed ings of the IEEE International Confer ence on Robotics and Au tomation (ICR A ) , 2002. [4] M. A. Batali n and G. S. S ukhatme, “Cov erage, exploration and deployme nt by a mobile robot and communication network, ” T elecommunication Systems J ournal, Special Issue on W ir eless Sensor Networks , vol. 26, no. 2, pp. 181–19 6, 2004. [5] A. C. S anderson, “Rivern et: Distributed sensor nets for en vironmental monitoring, ” nSF IIS- 032983 7. [6] A. Ho ward, M. Mataric, and G. Sukhatme, “Mobile sensor network deployment using potential fields: A dist ributed, scalable solution to the area cov erage prob lem, ” in D A RS 02 , 2002. [7] M. Batalin and G. Sukhatme, “Spreading out: A local approach to multi-robot cov erage, ” i n 6th International Confer ence on Distribu ted Autonomous R obotic Systems (DSRS02) , 200 2. [8] J. Pearce, P . Rybski, S. St oeter , and N. P apaniko lopoulous, “Dispersion behaviors for a team of multiple miniature robots, ” in IE EE International Confer ence on Robotics and Automation , 200 3. [9] Y . Zou and K. Chakrabarty , “Sensor deplo yment and tar get localization based on virtual forces, ” in Proc. IEE E Infocom , 2003. [10] G. W ang, G. Cao, and T . L. Porta, “Mo vemen t-assisted sensor deplo yment, ” in Pr oc. IEEE I nfocom , 2004. [11] M. Cardei and J. Wu, Cover ag e in W ir eless Senso r Networks in Handbook of Sensor Networks . CRC Press, 2004. [12] S. Shakk ottai, R. Srikant, and N. S hrof f, “Unreliable sensor grids: Cov erage, connectiv ity and diameter, ” in Pr oc. IEEE Infocom , 200 3. 18 [13] S. Megue rdichian, F . Ko ushanfa r , M. Potkonjak, and M. B . Sriv astava, “Cov erage problems in wireless ad-hoc sensor networks, ” in Pr oc. IEEE Infocom , 2001, pp. 138 0–1387. [14] S. Megue rdichian, F . Kou shanfar , G. Qu, and M. Potkonjak, “Exposure in wireless ad-hoc sensor networks, ” in ACM Mobile Compu ting and Networking , 2001, pp. 139–150. [15] X.-Y . Li, P .-J. W an, and O. Frieder , “Cov erage problems in wireless ad-hoc sensor networks, ” IEE E T ra nsactions on Computers , vol. 52, no. 6, pp. 753–7 63, June 200 3. [16] G. V eltri, Q. Huang, G. Qu, and M. Potkonjak, “Minimal and maximal expos ure path algorithms for wireless embedded sensor networks, ” in Pr oc. of ACM Sensys , 2003. [17] T .-L. Chin, P . Ramanathan, K. K. Saluja, and K.-C. W ang, “Exposure for collaborativ e detection using mobile sensor networks, ” in The 1st IEEE Internation al Confer ence on Mobile Ad-hoc and Senso r Systems , 200 5. [18] C. Huang an d Y . Tseng, “T he coverage problem i n a wi reless sensor network, ” in ACM International W orkshop on W i reless Sensor Networks and Applications (WSNA) , 2003, pp. 115–121. [19] Z. Zhou, S. Das, and H. Gupta, “Connected k-coverage problem in sensor networks, ” in I ntl . Conf. on Computer Communications and Networks (ICCCN) , 2004. [20] S. Kumar , T . Lai, and J. Balogh, “On k-co verag e i n a mostly sleeping sensor netwo rk, ” in Proceed ings of ACM Mo bicom , 2004. [21] B. Liu and D. T o wsle y , “ A study on the co v erage of large-scale sensor networks, ” in The 1st IEEE International Confer ence on Mob ile Ad-hoc and Sensor Systems , 2004. [22] C. Gui and P . Mohap atra, “Power conserv ation and quality of surveillance in targe t tracking sensor networks, ” in Pr oc. ACM Mobicom , 2004. [23] ——, “V irtual patrol: A ne w po wer conserv ation design for surv eillance using sensor networks, ” in Pro c. Internationa l W orkshop on I nformation Processin g in Sen sor Networks (IP SN) , 2005. [24] X. W ang, G. Xing, Y . Zhang, C. Lu, R. Pless, and C. Gil l, “In tegrated cov erage and conne ctivity configuration in wireless sensor networks, ” in A CM Confer ence on E mbedded Networked Sensor Systems (SenSys’03) , 2003. [25] H. Zhang and J. Hou, “M aintaining sensing co verag e and connectivity in sensor networks, ” in in vited paper in International W orkshop on T heor etical and Algorithmic Aspects of Senso r , Ad Hoc W ir eless and P eer-to-P eer Networks , 2004. [26] X. Bai, S . Kumar , Z . Y un, D. Xuan, and T .-H. Lai, “Deploying wireless se nsors to ac hiev e both coverage and connectivity , ” in Proc. of the ACM International Symposium on Mobile A d Hoc Networking and Computing (MobiHoc) , 2006. [27] S. Poduri and G. S. Sukhatme, “Constrained coverage for mobile sensor networks, ” in IEEE International Confer ence on Robotics and Automation , 2004. [28] J. Cortes, S . Martinez, T . Karatas, and F . Bullo, “Co verage control for mobile sensing networks, ” IE EE T ra nsactions on Robotics and Automation , vol. 20, no. 2, pp. 243–2 55, 2004. [29] B. Zhang and G. S . Suk hatme, “Controlling sensor density using mobility , ” in Pr oceedings of The Sec ond IEEE W orkshop on Embedd ed Networked Sensors , 2005, pp. 141– 149. [30] Y . T ang, B. Birch, and L. E. Park er , “Planning mobile sensor net deployme nt f or nav igationally-challenged sensor nodes, ” in IE EE International Confer ence on Robotics and Automation , 200 4. [31] S. Chellappan, W . Gu, X. Bai, D. Xuan, B. Ma, and K. Zhan g, “Deploying wireless sen sor networks un der li mited mobility constraints, ” IEEE Tr ans. on Mobile Computing , vo l. 6, no. 19, pp. 1142–115 7, 2007. [32] W . W ang, V . Sr ini va san, and K.-C. Chua, “Trad e-offs between mobility and density for coverage in wi reless sensor networks, ” in Pr oceedin gs of ACM Mob icom , 2007. [33] B. Liu, P . Brass, O. Dousse, P . Nain, and D. T o wsley , “Mobility impro ves coverag e of sensor network s, ” in A CM Mobihoc , 2005. [34] T .-L. Chin, P . Ramanathan, and K. Saluja, “ Analytic modeling of detection latenc y in mobile sensor networks, ” in Proc. International W orkshop on Information Pr ocessing in Senso r Networks (IPSN) , 2006. [35] O. Dousse, C. T avoularis, and P . Thiran, “Delay of intrusion detection in wireless senso r netw orks, ” in P r oc. of The ACM International Sympo sium on Mobile Ad Hoc Networking and Computing (MobiHoc) , 20 06. [36] G. Y . K eung, B. Li, an d Q. Zhang, “The intrusion detection in mobile sensor network , ” in Pr oc. of the ACM Interna tional Symposium on Mobile Ad Hoc Networking and Computing (MobiHoc) , 2010. [37] Y . Peres, A. S inclair , P . Sousi, and A. Stauf fer , “Mobile geometric graphs: Detection, cov erage and percolation, ” arXiv :1008.0075v 2. [38] J.-C. Chin, Y . Dong, W .-K. Hon, C. Y . T . Ma, and D. K. Y . Y au, “Detection of intelli gent mobile tar get in a mobile sensor network, ” IEEE /ACM T ransactions on Networking , vol. 18, no. 1, February 2010. 19