Run to Potential: Sweep Coverage in Wireless Sensor Networks

Transcription

1 Run to Potentia: Sweep Coverage in Wireess Sensor Networks Min Xi,KuiWu,Yong Qi,Jizhong Zhao, Yunhao Liu,MoLi Department of Computer Science, Xi an Jiaotong University, China Department of Computer Science, University of Victoria, British Coumbia, Canada Department of Computer Science and Engineering, Hong Kong University of Science and Technoogy Abstract Wireess sensor networks have become a promising technoogy in monitoring physica word. In many appications with wireess sensor networks, it is essentia to understand how we an interested area is monitored (covered) by sensors. The traditiona way of evauating sensor coverage requires that every point in the fied shoud be monitored and the sensor network shoud be connected to transmit messages to a processing center (sink). Such a requirement is too strong to be financiay practica in many scenarios. In this study, we address another type of coverage probem, sweep coverage, when we utiize mobie nodes as suppementary in a sparse and probaby disconnected sensor network. Different from previous coverage probem, we focus on retrieving data from dynamic Points of Interest (POIs), where a sensor network does not necessariy have fixed data rendezvous points as POIs. Instead, any sensor node within the network coud become a POI. We first anayze the reationship among information access deay, information access probabiity, and the number of required mobie nodes. We then design a distributed agorithm based on a virtua 3D map of oca gradient information to guide the movement of mobie nodes to achieve sweep coverage on dynamic POIs. Using the anaytica resuts as the guideine for setting the system parameters, we examine the performance of our agorithm compared with existing approaches. I. INTRODUCTION Recent progress in Micro-Eectro-Mechanica Systems (MEMS) makes it possibe to embed sensors, microprocessors, memory, wireess transceivers, and power suppy within a sensor node of severa cubic miimeters [1]. Cooperating together, a arge number of tiny sensor nodes can form an autonomous networking system, caed a wireess sensor network. Such a network can monitor specific physica phenomena [] (e.g., temperature, humidity, audio/video of animas) and transfer data in rea-time approach [3]. It has been widey used in numerous appications, ranging from environment surveiance, object tracking, to structure monitoring, scientific observation. An important factor that affects the effectiveness of such systems is evauated by the coverage of the networks. The coverage ratio generay determines how we the interested area is monitored by the sensors. Based on different appication scenarios, the coverage probem can be divided into three categories: fu coverage, barrier coverage, and sweep coverage. Fu coverage requires that each point of the area is continuousy monitored by one or more sensors. Such a requirement is strong and requires dense node depoyment for a critica region, where we do not want to miss any specific event. Barrier coverage requires that the network captures any intruder crossing a pre-defined barrier area, for exampe, the border of two countries. In both fu coverage and barrier coverage, the interested area shoud be covered by sensors continuousy. In many appications, however, fu coverage and barrier coverage incur prohibitive system cost. For instance, in a sensor system for farming monitoring and management, fu coverage requires thousands of sensors to cover a arge fied. To save cost, we might want to depoy sensor nodes sparsey. The entire fied may not be covered a the time and the network may not necessariy be connected. We can use a number of mobie nodes moving within the fied to coect data and deiver the data to the processing center. We ca it sweep coverage. Sweep coverage requires that the interesting event, once detected by some sensor node, be recorded ocay on the node so that the information coud be ater retrieved by the mobie nodes within a deay bound. The support of sweep coverage is chaenging. First, when an event happens, the sensor node detecting it becomes a Points of Interest (POI) and records corresponding data, which shoud be coected as soon as possibe. In many appications, the number and ocations of POIs change dynamicay, and it is usuay difficut to predict when and where an interesting event happens. The dynamics of POIs requires an agorithm to coordinate the movement of mobie nodes frequenty. Second, we shoud not aways assume the connectivity of the wireess sensor network especiay when the monitored fied is arge. In a sparse network, how to schedue the movement of mobie nodes is not trivia since it is difficut for the mobie nodes to obtain compete information for their movement coordination. Third, it is common that the mobie nodes have much more powerfu radio transceivers with a much arger transmission range than that of stationary sensor nodes. Logicay we have two radio networks: the network consisting of stationary sensors and the network consisting of mobie nodes. The former one is caed the stationary sensor network; it is static and may be sparse and disconnected. The atter one is caed the dynamic mobie network. It dynamicay changes over time and may or may not be connected from time to time. The benefits as we as the chaenges of using mobie nodes

2 motivate us to design efficient agorithms for sweep coverage with dynamic POIs. In summary, we make the foowing contributions in this paper: 1) We design a nove sweep coverage agorithm that uses the potentias of POIs as the driving force to attract mobie nodes toward the POIs. The potentia of a sensor node is a ogica vaue periodicay updated. The potentia fied over the entire network provides guidance information for the movement of mobie nodes. Our agorithm is buit on the idea of run to higher potentias: by strategicay cacuating the potentias of sensor nodes, our agorithm forces a mobie node to move aong the direction toward a nearby emergency POI. The agorithm ony requires a node to communicate with its oca neighbors and thus is fuy distributed. ) We anayze the bounds on the deay of information retrieva, which is defined as the deay from the time a POI starts recording a detected event to the time the information is retrieved by a mobie node. Our anaytica resuts discose the reationship among the number of mobie nodes, their moving speed, the deay of information retrieva, and the information retrieva probabiity. 3) Using the anaytica resuts as the guideine to set the system parameters, we conduct comprehensive simuations to evauate this design. The resuts demonstrate the efficiency and effectiveness of our proposed sweeping agorithm. The rest of this paper is organized as foow. Section II discusses the existing work. In Section III we present the system mode and formuate the sweep coverage probem. We anayze reationship between deay bound, probabiity of sweep coverage, and number of mobie nodes in Section IV. We present a nove sweep coverage agorithm that buids up the potentia of a POI and drives mobie nodes toward the POI in Section V. We conduct the experimenta performance evauation in Section VI. We finay concude the paper in Section VII. II. RELATED WORK Sensor coverage can be considered as a measure of the quaity of service of a wireess sensor network. It has been an active and important research topic, evidenced by many research contributions to this fied in recent years. The coverage probem can be cassified into three categories: fu coverage, barrier coverage, and sweep coverage. Fu coverage: Fu coverage means that every target in a certain area must be covered at any time. Based on the different types of targets, fu coverage can be divided into area coverage and point coverage [4]. To achieve fu coverage, peope use deterministic or randomized methods to depoy wireess sensors. The deterministic depoyment methods achieve the best coverage, but with too much overhead. The randomized methods are more fexibe, but cannot guarantee 1% confidence. For deterministic sensor depoyment, the studies mainy focus on how to decrease the number of sensors and increase the ifetime of the sensor network. For exampe, [5] proposed depoyment patterns to achieve fu coverage and three-connectivity, and fu coverage and five-connectivity under different ratios of sensor communication range over sensing range. Regarding randomized depoyment methods, the research topics incude the study of critica number of sensors for coverage, energy efficient coverage, mobie sensors assisted coverage, and so on. Kumar et a. [6] investigate the critica number of sensors to achieve coverage with random depoyment. Li et a. [7] uses disjoint sets to decrease energy consumption. There is aso work uses mobie sensors as we as stationary sensors to cover an area. For exampe, Cheappan et a. introduce an approach using mobie sensors to improve coverage [8], and Wang et a. [9] propose a distributed agorithm to et mobie sensors determine their moving direction ony with oca information. Barrier coverage: The barrier coverage probem comes from boundary detection in some appications such as detecting intruders crossing a border of two countries. Kumar et a. [1] discuss the barrier coverage probem for the first time. They transform the barrier coverage probem into the connectivity probem and present the critica condition for k-barrier weak coverage. They aso prove that this probem cannot be ocay soved. If the assumptions and requirements are reaxd, ocaized agorithms can be obtained [11]. Baister et a. [1] study the reationship between sensor density and coverage in a thin strip. Sweep coverage: The probem of sweep coverage comes from appications that do not require continuous sensor coverage whie the system cost for fu coverage is prohibitive. The main goa is to cover the targets in the area within a given time interva. In most cases, mobie nodes are introduced for sweep coverage. The sweep coverage probem has appeared in many forms. The probem is considered as moving barriers to cover the whoe region in [13]. In [14], Rekeitis et a. propose an approach to sweep a the destination zones. They assume that the mobie nodes can communicate with each other and know their positions. The area is divided into stripes, with each mobie node taking care of one stripe. Wong et a. [15] use topoogica mapping to sweep the destination area. They make ce decomposition and cover each ce by a zigzag pattern. Batain and Sukhatme [16] propose a decentraized method and present the frequency coverage metric to evauate the quaity of sweep coverage. Athough mobie nodes do not exchange information with each other, they need to communication with the static sensors to avoid dupicate coverage. Muhammad et a. [17] provide a verification method for sweep coverage.

3 3 Some work [18], [19] uses information gradient to guide data retrieva. Information gradient can be formed using natura measures, such as temperature. Those methods, however, may encounter the probem of gradient pateau or singe point faiure, which makes oca greedy agorithm invaid. Lin et a. [] propose a gradient creation method using harmonic function to guarantee that greedy navigation never gets stuck. Cheng et a. [1] define a new type of sweep coverage. The probem is to investigate how many sensors are needed to sweep pre-defined fixed Points of Interest (POI) rather than the entire area at a specified time interva. They prove that the probem of cacuating the minimum number of sensors is NP-hard and propose a centraized agorithm together with a distributed sweep agorithm, caed DSweep, to obtain an approximate soution. We extend the sweep coverage probem in [1] by aowing POIs to be dynamic, i.e, the POIs are not pre-defined and can emerge at any time and any position. This assumption is more reaistic because in many rea appications, it is difficut, if not impossibe, to accuratey predict when and where an event might happen. Providing sweep coverage with dynamic POIs is much more chaenging, making our work significant different from existing ones. III. PROBLEM FORMULATION We make the foowing assumptions on the system settings, which can be found true in many reaistic appications: The fied: Without oss of generaity, we assume that the monitored fied is a square area with the edge ength of meters. Sensors: There are n stationary wireess sensor nodes randomy (or strategicay) depoyed in the fied. The network consisting of stationary sensors is caed stationary sensor network, which may be sparse and disconnected. The radio transmission range of a the stationary sensors is the same and is equa to r(r <<). Mobie nodes: There are m mobie nodes moving in the fied to coect and process data. The network consisting of mobie nodes is caed dynamic mobie network. Its topoogy changes over time. The radio transmission range of a the mobie nodes is the same and is equa to R. R is normay much arger than r, because mobie nodes can use a fue-powered engine and thus energy concern of mobie nodes is of second-order importance. Moving speed: The moving speed of the mobie nodes is the same and is equa to v(v <<) meters per second. Generay speaking, the radio transmission speed is much faster than the movement speed of mobie nodes. Moving direction: We assume that a mobie node knows the direction of a sensor node if they can directy communicate with each other. Location information, athough important, may not be required to make this assumption feasibe. Symbo n m r R v T γ(.) β(.) μ p μ s TABLE I NOTATIONS Description the number of sensor nodes the number of mobie nodes the edge ength of the square fied radio range of sensor nodes radio range of mobie nodes moving speed of mobie nodes deay bound on sweep time potentia update function at POIs potentia update function at non-poi sensors potentia update interva at POIs potentia update interva at non-poi sensors Restriction on data retrieva: A mobie node can downoad the information from a stationary sensor node if and ony if the mobie node is within the communication range of the sensor node. To save energy of sensor nodes, we do not assume muti-hop radio transmission for data downoad because the voume of data may be arge. Nevertheess, we do not put such a constraint on the contro messages of our protoco, which have a very sma size compared to rea data. Network connectivity: We assume that with inks from both the stationary sensor network and the dynamic mobie network, a nodes (sensors and mobie nodes) are connected most of the time. For ease of reference, the notation used in this paper is isted in Tabe I. With the above settings, ogicay we have two wireess networks: the stationary sensor network and the dynamic mobie network. The stationary sensor network monitors the fied. When an event happens, the sensor node which senses the event becomes a point of interest (POI). One of the mobie nodes in the mobie network shoud move to the POI and process the event. A POI is swept if its data is retrieved by a mobie node. For effective network design, we need to answer the foowing key questions: 1) At east how many mobie nodes are required so that any sensor node, once becoming a POI, can be swept within a time period of T with a high probabiity 1? ) How can the mobie nodes be guided to POIs without a centraized contro? We answer the first question and the second question in the next section and Section V, respectivey. IV. IDEAL SWEEP COVERAGE In this section, we answer the first question by anayzing the sweep coverage in different scenarios. The anaytica resuts discose the reationship among the deay bound, the information access probabiity, and the required number of 1 We do not assume a deterministic bound on time deay because it requires a perfect mobiity coordination among mobie nodes, which is hard to guarantee without a centraized contro in a system with dynamic POIs.

4 4 mobie nodes. They wi serve as the guideine in parameter seection in our ater experimenta evauation. Lemma 1: System with singe POI: With the system settings in Section III, for a network with singe dynamic POI (i.e., any sensor coud become a POI but there is ony one POI within a time duration of T ), if a sensor node, once becoming a POI, is swept within a time deay of T with a probabiity of p, the minimum number of mobie nodes required shoud be no smaer than { 1 if π (T v + r) m = n(1 p) otherwise n(1 π (T v+r) / ) Proof In the idea case, the event that a sensor node becomes a POI is known to one or more mobie nodes, and with a perfect agorithm at east one mobie node moves directy to the POI to retrieve data. Due to the deay constraint, the distance between the mobie node and the POI must be no arger than T v + r. Since the POI coud be any sensor in the fied and the mobie nodes may not be abe to aways communicate with each other for coordinated movement, the ocations of mobie nodes at any time instant are approximated as random points in the fied. The probabiity that the POI fas within the range of T v + r of at east one mobie node is p =1 (1 f) m 1, (1) π (T v+r) where f = min{, 1} is the probabiity that the POI fas within the range of T v + r of a given mobie node, and m 1 is the number of mobie nodes that have known the event. From Equation (1), the minimum number of mobie nodes, m, shoud not be smaer than { 1 if π (T v + r) m 1 = n(1 p) otherwise n(1 π (t v+r) / ) Lemma 1 represents the idea scenario based on three assumptions: (1) There is a perfect agorithm that can guide a mobie node to move directy to the POI; () the event that a sensor node becomes a POI is known to one or more mobie nodes; and (3) the ocations of mobie nodes coud be approximated as random points in the fied. The first assumption is the major chaenge that our sweep agorithm wi address. Based on the assumption on network connectivity, the second assumption is trivia with certain contro signaing messages. Without the second assumption, there is actuay no way to get a bound on deay. The ast assumption is ony for an approximation of the ocations of mobie nodes. This approximation is reasonabe because the ocations of POIs are random and the mobie nodes are attracted to the POIs. Lemma 1 is usefu to estimate the best resut that a network with singe dynamic POI can achieve. It is necessary to note that Lemma 1 is aso suitabe for a system in which at any time period of T ony one event happens, because mutipe sensors that detect the same event shoud be spatiay cose and coud be roughy considered as a singe POI. Lemma : System with mutipe POIs when the deay bound T is arge: With the system settings in Section III, for a network with k(k <<n) dynamic POIs, if any POI is swept within a time period of T, the minimum number of required mobie nodes in the network shoud be no smaer than k if π (T v + r). Lemma is trivia and obvious because a one-to-one mapping between POIs and mobie nodes can meet the requirement. Lemma 3: System with mutipe POIs when the deay bound T is sma: With the system settings in Section III, for a network with k(k <<n) dynamic POIs that are spatiay separated with a distance of at east (T v+r), if the number of mobie nodes is (n k+c) π (T v+r) where c is a positive constant, then a POIs can be swept within a time period of T with a c kπ (T v+r). probabiity no smaer than e e Proof With a tight deay bound, the fied coud be divided into k non-intersecting circuar areas, denoted as A 1,...,A k, respectivey, pus an irreguar area that is not covered by any A i,i=1,...,k. The size of A i (i =1,...,k) is π (T v + r). We add mobie nodes one by one randomy into the fied unti each A i (i =1,...,k) incudes at east one mobie node, i.e., A i is covered. The probem of cacuating the expected number of mobie nodes is simiar to the coupon coector s probem []. Denote the number of required mobie nodes as a random variabe X. We now determine E[X]. IfX i is the number of required mobie nodes to cover a new circuar area whie we have exacty i 1 different circuar areas covered. Ceary, X = k i=1 X i. Each X i (i = 1,...,k) is a geometric random variabe. When exacty i 1 different circuar areas are covered, the probabiity of covering a new circuar area is π (T v+r) p i =(k i +1) g, where g = is the probabiity that the added mobie node fas within the new circuar area. Hence, E[x i ]= 1 p i. Using the inearity of expectation, we have that E[X] =E[ = 1 g k X i ] () i=1 k i=1 1 k i +1 = (n k + Θ(1)) π (T v + r) (4) The above resut indicates that with a high probabiity, using (n k+c) π (T v+r) (c is a positive constant) mobie nodes shoud be abe to sweep a POIs within the deay bound T. Using Chernoff bound and Poisson approximation [], this probabiity is no ess than e e c gk = e e c kπ (T v+r). (3)

5 5 Fig. 1. A virtua map of potentias The cacuation is simiar to that in Section of [] and is omitted to save space. It is extremey hard, if not impossibe, to anayze a network of mutipe dynamic POIs with a moderate deay bound on sweep coverage, because the k circuar areas, A i,i=1,...,k, are ikey to intersect with each other. The anaysis in this case reies on the ocations of POIs and needs to consider a possibe intersecting scenarios of different circuar areas. Nevertheess, Lemma and Lemma 3 provide the ower and upper bounds on the required number of mobie nodes for such a network, respectivey. V. DRIVING MOBILE NODES WITH POTENTIALS A. Basic Idea Without a centraized contro, each mobie node does not have a cear view of the goba state information and as such we need to find a way to guide the movement of mobie nodes toward POIs with oca information ony. To achieve this, our basic idea is to buid a virtua 3D map in the stationary sensor network, with POIs ocating at the peaks as iustrated in Figure 1. Mobie nodes thus cimb to these peaks with the hep of oca information, e.g., the height of the neighboring sensors in the virtua 3D map. Once the data of a POI is retrieved by a mobie node, the POI becomes a norma sensor node and it drops to the bottom in the 3D map, etting the mobie node move to another POI. The information at a sensor that is used to guide the movement of mobie nodes is caed the potentia of the sensor. In this paper, it is represented as a rea number. We need to answer two questions in the above potentiaguided approach. First, how can we buid up the potentias that can effectivey guide the movement of mobie nodes toward POIs? Second, how can mobie nodes coordinate their movement to avoid coision (i.e., severa mobie nodes move toward the same POI and thus resut in futie movement)? In the foowing sections, we address the above probems and finay ead to a fuy distributed sweep coverage agorithm. B. Buiding Up Potentias Ceary the potentia of a sensor node wi change dynamicay from time to time. The changing process of a sensor s potentia can be divided into three phases: Initiaization phase: When the system starts, a the sensors set their potentias to as the initia vaue, meaning that the virtua 3D map is fat. Potentia buid-up phase: When a sensor detects a phenomenon in interest, it becomes a POI and sets its potentia to a vaue arger than that of any of its oca neighbors, i.e., other sensor nodes that can be reached via one-hop radio. It then periodicay increases its potentia at an interva time of μ p with a monotonicay increasing function, γ(t), where t is the time duration between current time and the time when the phenomenon was detected. In this paper, we adopt γ(t) =αt, where α is a constant vaue. Note that other definition of γ function is aso possibe. For a non-poi sensor, it updates its potentia based on the potentias of its oca neighbors at an interva time of μ s (normay μ s < μ p ). The potentia of a non-poi sensor is cacuated using a potentia update function β, which is defined ater in this section. Regression: When a mobie node reaches a POI, the data at the POI can be retrieved and the potentia of the POI is reset to. The function γ decides how fast a POI raises its potentia. It can be considered as the degree of emergency of the POI. If the POI has not been swept for a ong time, its potentia wi become higher and higher, meaning that it is urgent to retrieve data from this POI. The function β decides how to cacuate the potentia at a non-poi sensor node. In order to buid a virtua 3D map shown in Figure 1, this function needs to foow two rues: A POI shoud have the maximum potentia among a its neighbors. This feature is caed the oca maximum of POI. The design of β function shoud not vioate this feature. A non-poi sensor coser to a POI shoud has a higher potentia than other non-poi sensors far away from the POI, that is, the potentias of a POI and its nearby neighbors shoud form a hi with the POI at the top of the hi. In this paper, we adopt a very simpe way to cacuate the function β: the potentia of a non-poi node is cacuated as the average vaue of its direct neighbors potentias. It is very easy to see that this way of cacuating β function meets the above requirements, based on the fact that the average of a set of vaues cannot be arger than the maximum vaue in the set. Figure iustrates an exampe for the potentia update process. As iustrated in Figure, the potentias continuousy change over time to refect the dynamic changes of POIs, without using a centraized contro or any goba information. Guided with the potentias, a mobie node moving from a sensor of ow potentia toward a sensor of higher potentia wi finay reach a POI. When a tie exists, the mobie node randomy seects a direction where the sensor has a potentia

6 6 no ess than the current sensor. In this way, our approach guarantees that a mobie node moves toward a correct direction, that is, a direction incuding sensors that have potentias no smaer than the current sensor. A mobie node wi not be stuck at a sensor uness this sensor is a POI. Once the POI s data is retrieved, its potentia changes to, and the mobie node starts to move to another POI. In addition, no path oops coud be formed because mobie nodes aways cimb higher in the virtua 3D map. (a) Initia potentia vaues (c) First round update (e) Third round update Fig POI.5 POI 1 (b) A node becomes POI POI POI (d) Second round update (f) After the data in the POI is retrieved An exampe of potentia update process C. Anti-Coision The approach proposed above does not hande the coision of mobie nodes and is caed the naive sweep agorithm. Coision happens when mutipe mobie nodes move to the same POI, resuting in unnecessary moves for some mobie nodes. We sove the probem with an improved agorithm, caed the anti-coision sweep agorithm. The anti-coision sweep agorithm uses the same idea of the naive method. The ony difference is that each mobie node uses an agent to mark its route ahead and foows the marks to POIs. The agent is actuay a contro message transmitted in the stationary sensor network. An agent uses the naive sweep agorithm to guide its movement. Once it reaches a POI, it resets the POI s potentia to. Since an agent moves much faster than its corresponding mobie node, some mobie nodes wi change their direction before they actuay reach the POI because the POI s potentia has been reset. In this way, unnecessary moves of mobie nodes are greaty reduced. It is true that the agents might sti coide. But this is not a probem at a because a mobie node aways foows the newest marks made by its agent. The od marks that ead to the coision are obsoete and wi not be used by the mobie nodes. Since the agents use the naive sweep agorithm to guide their movement, they enjoy the same nice features of the naive method: They are guaranteed to move aong correct directions; they are never stuck at a sensor node uness reaching a POI; and no path oops coud be formed. VI. EXPERIMENTAL EVALUATION A. Simuation Mode We perform simuation studies to evauate our sweep coverage agorithm and compare it with the DSweep agorithm introduced in [1]. we fix the foowing parameters to make the simuation resuts concise. The simuated fied is a square area with edge ength of 3 meters. We depoy 9 stationary sensors randomy in the fied. The mobie nodes moving speed is set to 1 m/s, and their initia ocations are random in the fied. The radio range of both mobie nodes and stationary sensor nodes is set to 13 meters. The potentia update interva for POIs (μ p ) is set to 5 seconds. Other system parameters, however, are changed in the simuation to capture important features of our agorithm in different scenarios. These parameters incude the number of POIs, the number of mobie nodes, the constraint on sweep deay, and the potentia update interva for non-poi sensors (μ s ). To refect the fexibiity of our agorithm, we assume that POIs are not known in advance, but instead their ocations are determined by a set of randomy seected sensors and may change over time. We test two important performance resuts: average sweep deay and average moving distance. Average sweep deay is defined as the tota time required to sweep a POIs divided by the tota number of POIs. The time required to sweep a POI is cacuated as the time when a sensor becomes the POI to the time when a mobie node can communicate directy to the POI. The average moving distance is defined as the tota moving distance of a mobie nodes to the tota number of mobie nodes. For each simuation scenario, we run the simuation 5 times and take the average as the fina resuts. B. Performance Resuts We test the performance of our sweep coverage agorithm with the anti-coision enhancement in different scenarios. System with singe POI: According to Lemma 1, if we require the sweep deay fas within 7 seconds with a probabiity higher than 98%, the number of mobie nodes shoud be arger than 15. Figure 3 shows the sweep deay with different mobie nodes. It can be seen that with 15

7 7 sweep deay with singe POI mobie node count = 15 mobie node count = 1 mobie node count = deay bound sweep deay with 1 POI mobie node = 1 mobie node = 16 mobie node = 6 mobie node = 36 tight deay bound oose deay bound Fig. 3. Sweep deay with singe POI Fig. 4. Sweep deay with 1 POIs mobie nodes, the sweep deay is mosty fa within the bound, meaning that our sweep coverage agorithm can actuay achieve the performance cose to the idea system described in Section IV. For comparison purpose, we aso test the sweep deay with 1 and mobie nodes. Ceary, 1 mobie nodes cannot meet the requirement and mobie nodes are overki because the sweep deay mosty fas within 6 seconds. System with mutipe POIs: We seect 1 POIs, which are randomy chosen from the sensors. When the requirement on sweep deay is oose such as 16 seconds, Lemma indicates that a good sweep agorithm shoud require ony 1 mobie nodes. When the requirement is tight such as 45 seconds, Lemma 3 shows that if the deay requirement is met with a probabiity higher than 99%, we shoud require about 6 mobie nodes, cacuated with the constant c in Lemma 3 equa to.75. Figure 4 shows the sweep deay with different mobie nodes. It can be seen that when the deay requirement is oose, using ony 1 mobie nodes our agorithm can meet the requirement most of the time. For a tight deay bound (e.g., 45 seconds), about 6 mobie nodes are required in theory. The sweep deay with our agorithm, however, is a bit arger than the bound if ony 6 mobie nodes are used. This is mainy because a mobie node does not move to a POI directy, but instead foows a marked path toward the POI, which is not a strict ine. We can aso see that adding more mobie nodes (e.g., using 36 mobie nodes) does not reduce sweep deay significanty due to the same reason. C. Benefit of Anti-Coision To iustrate the benefit of the anti-coision mechanism, we compare the performance of the naive sweep agorithm with that of the anti-coision version. Figure 5 shows the resuts of a system with 1 mobie nodes and 5 dynamic POIs. Figure 5(b) shows that the average moving distance of the anti-coision method is much smaer than that of the naive method. The reason has been discussed in Section V-C. As a resut, the sweep deay of the anti-coision method is smaer than that of the naive method, as demonstrated in Figure 5(a). sweep deay anti coision naive (a) sweep deay Fig. 5. moving distance (m) anti coision naive (b) moving distance Compare with anti-coision and naive method D. Performance Comparison Between Our Agorithm and DSweep The DSweep agorithm [1] is proposed for systems with static POIs. It assumes that a mobie nodes know their instant ocations, and each POI has a gobay unique ID. The ocations and the sweep period of a POIs are preknowedge at each mobie node. With the above assumptions, DSweep adopts store-carry-forward method to make mobie nodes, once meeting each other, exchange their newest sweep information. A mobie node decides its movement toward the nearest and most urgent POI that it has known. Athough DSweep is the cosest to our work, it is proposed for systems with static POIs, and as such it is impossibe to compare DSweep directy to our agorithm. To make them comparabe, we make two modifications on DSweep. In the first modification, aso denoted as DSweep in Figure 6, we aow a mobie node to know any newy-formed POI in a range twice of its communication range. Second, whenever a new POI is formed, we aow every mobie node to know the POI s information immediatey. This modification is denoted as DSweep-goba in Figure 6. Figure 6 shows the performance resuts with the same parameters for mutipe POIs in Section VI-B. Because both DSweep and DSweep-goba do not rey on the static sensor network, their performance is independent of potentia update interva. It can be seen that the performance of our agorithm is better than that of DSweep. This is because DSweep ony

8 8 sweep deay anti coision DSweep DSweep goba deay bound (a) sweep deay Fig. 6. moving distance (m) anti coision DSweep DSweep goba (b) moving distance Compare with anti-coision and DSweep uses the dynamic mobie network to make moving decisions, and the dynamic mobie network is not aways connected and thus newest information cannot be guaranteed to propagate to right mobie nodes. Nevertheess, our agorithm has worse performance than DSweep-goba. This is because DSweepgoba assumes that each mobie node knows the accurate goba information to make better moving decisions. This assumption, however, is very hard to impement in reaity. VII. CONCLUSION Traditiona approaches to addressing the probem of sensor coverage use fu coverage or barrier coverage as the major evauation metric. In many appications, however, we need a totay different coverage measure, caed sweep coverage. Existing work studies sweep coverage ony with static POIs. This mode is not fexibe and not suitabe in appications where interesting events coud happen anywhere and thus POIs are hard to predict. We sove the probem in this paper by presenting a nove sweep coverage agorithm for systems with dynamic POIs, reying on a virtua 3D map of oca gradient information to guide the movement of mobie nodes. Our agorithm is fuy distributed and can achieve performance cose to the optimum (i.e., the idea sweep coverage). We anayze the reationship among the number of mobie nodes, the bound on sweep deay, and the probabiity of sweep coverage. The anaytica resuts provide a good guideine for system design and parameter seection. ACKNOWLEDGMENTS This work is supported by the Nationa Natura Science Foundation of China ( , 6883), the Nationa High Technoogy Research and Deveopment Program of China (863 Program) (7AA1Z18, 6AA1Z11), the Nationa Key Basic Research Program of China (973 Program) (6CB333) and the Science and Technoogy Research and Deveopment Program of Shaanxi Province (8KW-). REFERENCES [1] I. F. Akyidiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci, Wireess sensor networks: a survey, Computer Networks, vo. 38, no. 4, pp ,. [] M. Wu, J. Xu, X. Tang, and W.-C. Lee, Top-k monitoring in wireess sensor networks, IEEE Transactions on Knowedge and Data Engineering, vo. 19, no. 6, pp , 7. [3] X. Liu, Q. Wang, L. Sha, and W. He, Optima qos samping frequency assignment for rea-time wireess sensor networks, 3, in Proceedings of the 4th Rea-Time Systems Symposium. [4] M. Cardei and D. Z. Du, Improving wireess sensor network ifetime through power aware organization, ACM Wireess Networks, no. 11, pp , 5. [5] X. 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