IEEE TRANSACTIONS ON AUTOMATIC CONTROL as well as over a time interval.The results hold for a o(d general number of sensors except for the asymptotic case, and can provide guidelines on designing WSNs of high degree coverage.Plus,the detection time in the sensing process is investigated. The rest of the paper is organized as follows.In Sec- tion II,the system model and several performance measures are defined.In Section III,we present our main results.The asymptotic coverage problem in mobile heterogeneous WSNs is studied in Section IV and we discuss the impact of mobility Fig.1.A Typical Distribution of D. and heterogeneity based on the results.In Section V,we study k-coverage under Poisson deployment model.In Section VI, the detection delay is investigated.Finally,we conclude our that there are u groups G1,G2,...,Gu in this heterogeneous work in Section VII. network,where u is a positive constant.For y =1,2,...,u. group Gy consists of ny =cun sensors,where n is the total II.SYSTEM MODEL AND PERFORMANCE MEASURES number of sensors in the network and c(y 1,2,...,u)is In this section,we describe the system model regarding called the grouping index which is a positive constant invariant sensing,deployment and mobility pattern,respectively and ofn.e,cy=9(1》and∑y-19=l,For a given group present several measures to assess the coverage performance Gy,sensors in this group possess identical sensing radius r. of mobile heterogeneous WSNs. When ny =1,u=n,it is the special case where each sensor has different sensing radius from each other. A.Deployment Scheme Let the operational region of the sensor network A be an C.Mobility Pattern unit square and this square is assumed to be a torus.This is Sensors move according to certain mobility patterns. to eliminate the boundary effect and equalize all points in the region for convenience of analysis.We consider two models .I.I.D.MOBILITY MODEL-The sensing process is parti- according to which the sensors are deployed. tioned into time slots with unit length.At the beginning of UNIFORM DEPLOYMENT MODEL-n sensors are ran- each time slot,each sensor will randomly and uniformly domly and uniformly deployed in the operational region, choose a position within the operational region and remain independent of each other. stationary in the rest of the time slot. POISSON DEPLOYMENT MODEL-Sensors are deployed .1-DIMENSIONAL RANDOM WALK MOBILITY MODEL according to a 2-dimensional Poisson point process with Sensors in each group are classified into two types of density parameter A=n. equal quantity,H-nodes and V-nodes.And sensors of each These random deployment strategies are favored in the type move horizontally and vertically,respectively.The situation that the geographical region to be sensed is hostile sensing process is also divided into time slots with unit length.At the very beginning of each time slot,each sensor and inimical.Under such circumstance,wireless sensors might will randomly and uniformly choose a direction along its be sprinkled from aircraft,delivered by artillery shell,rocket. missile or thrown from a ship,instead of being placed by human moving dimension and travel in the selected direction a certain distance D which follows the distribution function or programmed robots.Specifically,uniform deployment model is commonly employed when the priori knowledge of the target fp(d).To make the model non-trivial,fp(d)satisfies such requirement:fp(d)=0 when d<do or d>1,where do area is unavailable,and as a most simple scheme,it provides is an arbitrary constant and 0<do<1.This requirement insights for exploring more complex deployment strategies. implies that the distance a sensor travel can be neither too Poisson deployment is a widely used method in literature to model the location of randomly-dropped sensors due to its short3 nor too long4.A typical distribution is illustrated in Fig 1.We do not set requirements on the velocity of sensor memoryless and annexable property [23]. during its movement,but sensors must reach destination within the time slot. B.Sensing Strategy 2-DIMENSIONAL RANDOM WALK MOBILITY MODEL- Basically,we employ the binary disc sensing model in this Each sensor in group Gu(y=1,2,...,u)randomly and study,where we assume that a sensor is capable to sense independently chooses a directionθ∈[O,2π)according perfectly within the disc of radius r centered at the sensor. Beyond this sensing area,the sensor cannot sense.Here,r 3Short range travel approximates remaining stationary or i.i.d.model and denotes the sensing radius of a sensor. may fail to gain benefits from movements. 4Long range travel is energy-consuming due the movement.And if the sensor Further,our study takes into account the general case that can travel beyond the dimension of the operational area (i.e.d>1).it can sensors in the network have different sensing radii.We assume always cover the area along its moving dimension.IEEE TRANSACTIONS ON AUTOMATIC CONTROL 3 as well as over a time interval. The results hold for a general number of sensors except for the asymptotic case, and can provide guidelines on designing WSNs of high degree coverage. Plus, the detection time in the sensing process is investigated. The rest of the paper is organized as follows. In Sec￾tion II, the system model and several performance measures are defined. In Section III, we present our main results. The asymptotic coverage problem in mobile heterogeneous WSNs is studied in Section IV and we discuss the impact of mobility and heterogeneity based on the results. In Section V, we study k-coverage under Poisson deployment model. In Section VI, the detection delay is investigated. Finally, we conclude our work in Section VII. II. SYSTEM MODEL AND PERFORMANCE MEASURES In this section, we describe the system model regarding sensing, deployment and mobility pattern, respectively and present several measures to assess the coverage performance of mobile heterogeneous WSNs. A. Deployment Scheme Let the operational region of the sensor network A be an unit square and this square is assumed to be a torus. This is to eliminate the boundary effect and equalize all points in the region for convenience of analysis. We consider two models according to which the sensors are deployed. • UNIFORM DEPLOYMENT MODEL — n sensors are ran￾domly and uniformly deployed in the operational region, independent of each other. • POISSON DEPLOYMENT MODEL — Sensors are deployed according to a 2-dimensional Poisson point process with density parameter λ = n. These random deployment strategies are favored in the situation that the geographical region to be sensed is hostile and inimical. Under such circumstance, wireless sensors might be sprinkled from aircraft, delivered by artillery shell, rocket, missile or thrown from a ship, instead of being placed by human or programmed robots. Specifically, uniform deployment model is commonly employed when the priori knowledge of the target area is unavailable, and as a most simple scheme, it provides insights for exploring more complex deployment strategies. Poisson deployment is a widely used method in literature to model the location of randomly-dropped sensors due to its memoryless and annexable property [23]. B. Sensing Strategy Basically, we employ the binary disc sensing model in this study, where we assume that a sensor is capable to sense perfectly within the disc of radius r centered at the sensor. Beyond this sensing area, the sensor cannot sense. Here, r denotes the sensing radius of a sensor. Further, our study takes into account the general case that sensors in the network have different sensing radii. We assume 0 1 d fD(d) d0 Fig. 1. A Typical Distribution of D. that there are u groups G1, G2, ··· , Gu in this heterogeneous network, where u is a positive constant. For y = 1, 2, ··· , u, group Gy consists of ny = cyn sensors, where n is the total number of sensors in the network and cy(y = 1, 2, ··· , u) is called the grouping index which is a positive constant invariant of n (i.e., cy = Θ(1)) and u y=1 cy = 1. For a given group Gy, sensors in this group possess identical sensing radius ry. When ny = 1, u = n, it is the special case where each sensor has different sensing radius from each other. C. Mobility Pattern Sensors move according to certain mobility patterns. • I.I.D. MOBILITY MODEL — The sensing process is parti￾tioned into time slots with unit length. At the beginning of each time slot, each sensor will randomly and uniformly choose a position within the operational region and remain stationary in the rest of the time slot. • 1-DIMENSIONAL RANDOM WALK MOBILITY MODEL — Sensors in each group are classified into two types of equal quantity, H-nodes and V-nodes. And sensors of each type move horizontally and vertically, respectively. The sensing process is also divided into time slots with unit length. At the very beginning of each time slot, each sensor will randomly and uniformly choose a direction along its moving dimension and travel in the selected direction a certain distance D which follows the distribution function fD(d). To make the model non-trivial, fD(d) satisfies such requirement: fD(d)=0 when d<d0 or d > 1, where d0 is an arbitrary constant and 0 < d0 < 1. This requirement implies that the distance a sensor travel can be neither too short 3 nor too long 4. A typical distribution is illustrated in Fig 1. We do not set requirements on the velocity of sensor during its movement, but sensors must reach destination within the time slot. • 2-DIMENSIONAL RANDOM WALK MOBILITY MODEL — Each sensor in group Gy(y = 1, 2, ··· , u) randomly and independently chooses a direction θ ∈ [0, 2π) according 3Short range travel approximates remaining stationary or i.i.d. model and may fail to gain benefits from movements. 4Long range travel is energy-consuming due the movement. And if the sensor can travel beyond the dimension of the operational area (i.e. d > 1), it can always cover the area along its moving dimension