Event Detection Time for Mobile Sensor Networks Using First Passage Processes Hazer Inaltekin, Christina R. Tavoularis and Stephen B. Wicker School of Electrical and Computer Engineering Cornell University Ithaca, NY 14853 {hi27,crt23, wicker}@ece.cornell.edu Abstract— A new technique based on first passage processes is presented as a means for calculating the detection time of an event by wireless mobile sensors located in R d , d =1, 2 and 3. Our proposed method is valid for any type of mobility model, and is amenable to a variety of extensions. We consider heterogenous networks in which sensor platforms have different mobility patterns. The second extension considers unreliable sensors with randomly distributed operational lifetimes. The second general- ization is of paramount importance if the network is deployed in a hostile environment. Our main results are illustrated for random linear motion and Brownian motion. An approach for applying our theorems when sensors move according to a time- homogenous Markov process is also considered. I. I NTRODUCTION In many applications, the length of time that passes until an event is detected is of critical importance. In static networks, if the amount of area covered by each sensor is equal to A and the sensor locations are Poisson distributed with intensity λ> 0, then the fraction of area covered is equal to 1 -exp(-λ · A). This is also the probability that a point target is detected by one of the sensors. In a static network, this probability remains the same over time. In this paper, we determine event detection time probability for wireless mobile sensor networks located in R d , d =1, 2 or 3. A new approach based on first passage processes is proposed to determine the time required to detect an event. This tech- nique reduces event detection time probability calculations to that of finding the first instance in which a randomly initiated trajectory for sensor platform crosses an event boundary. Our method is valid for any kind of mobility model. We make no assumptions with regard to the type of network boundary. Our network may have no rigid, well-defined boundaries, or it may have a reflecting or an absorbing boundary. Furthermore, this method admits very natural extensions to heterogenous networks where different sensors may have different mobility patterns (e.g., linear motion and Brownian motion), and to networks with unreliable sensors having limited lifetimes. Specifically, if the network is deployed in a hostile environ- ment such as a battlefield or the inside of a burning building, it is expected to observe unforeseen sensor failures. We illustrate the applications of our main theorems to linear motion and Brownian motion. We also outline how one can calculate event detection time probability for time-homogenous Markov processes. A. Related Work Coverage of wireless sensor networks and the detection of an event or an intruder are among the fundamental problems considered in the domain of wireless sensors, and have at- tracted considerable amount of research interest. Some of these work (e.g., [1] and [2]) deal with immobile networks, while the other (e.g., [3], [4], [5], [6] and [7]) study the coverage of mobile sensor networks. In [1], the authors analyze the coverage of networks with immobile nodes by combining Voronoi diagrams and graph theoretical techniques. In [2], connectivity and coverage prop- erties of an unreliable sensor grid are analyzed. Papers [3] and [4] focus on algorithmic aspects. They propose network deployment protocols to enhance the coverage of wireless mobile sensor networks after the initial placement of the sensors. Authors of [5] and [6] studied the coverage of sensor networks under random mobility patterns, which is also what we examine in this work. However, they limit themselves to very specific mobility models such as random linear mobility [5] or Brownian motion [6]. The analysis presented in these papers is based on the theory of coverage processes and only valid for infinite size networks. In contrast to them, we do not assume any specific mobility pattern (see Theorem 1). Our analysis is based on the theory of first passage processes and is valid for all finite size networks. We also show validity for infinite networks by taking the limit as the network size increases without bound. We also analyze heterogenous networks in which different sensors may have different mobility patterns (see Theorem 3), and the networks with unreliable sensors having random lifetimes (see Theorem 4). In [7], the intrusion detection problem was considered. Sensor nodes are static but a malicious agent linearly moves inside the network. They determine the first time this agent detected by the largest connected component so that message can be communicated to a sink in a multihop fashion. In this work, we do not take into account the connectivity of the network because we are looking at mobile sensor networks where each node gets connected for periods of time due to node mobility, usually within a small delay in dense networks