A stochastic process model of the hop count distribution in wireless sensor networks☆
Introduction
Target localization in wireless sensor networks (WSNs) is an active area of research with wide applicability. Due to power and interference constraints, the vast majority of WSNs convey messages via multiple hops from a source to one or several sinks, mobile or stationary. Localization techniques which exploit the information about the Euclidean distance from a sensor node, contained in the hop count of a message originating from that node, are referred to as range-free [1]. Range-free localization is applicable to networks of typically low-cost, low-power wireless sensor nodes without the hardware resources needed to accurately measure node positions, neighbor distances or angles (for example using GPS, time or angle of arrival). It is therefore an attractive approach in situations where a compromise is sought between localization accuracy on one hand, and cost, size and power efficiency on the other.
Various hop-count based localization techniques for WSNs have been proposed; for a survey see e.g. [1]. Relating hop count information to the Euclidean distance between sensor nodes, exemplified by the probability distribution of the hop count conditioned on distance, remains a challenging problem. Except in special cases, such as one-dimensional networks [2], only approximations can be obtained; such approximations are often in the form of recursions, which tend to be difficult to evaluate [3], [4], [5]. Moreover, the hop count depends on the chosen path from the source to the sink and is therefore a function of the routing method employed by the network. An approximate, closed-form hop count distribution is proposed in [6] and evaluated for nearest, furthest and random neighbor routing, in which a forwarding node selects the next node from a neighborhood oriented towards the sink, where it is assumed that this neighborhood is not empty. Some of the existing localization algorithms, such as the DV-hop algorithm [7] and its variants, define the hop count between nodes as that of the shortest path [3], [4], [5]. Other localization algorithms use the hop count of a path established through greedy forward routing [8], [9], [10], [11]. In most cases, the overhead incurred by routing is not negligible. Simpler alternatives may be needed when sensor nodes impose more severe complexity constraints.
This paper is motivated by the range-free target localization problem in networks of position-agnostic wireless sensor nodes, which broadcast messages using flooding under the assumption that node-to-node message delays can be characterized by independent, exponential random variables. This is a reasonable assumption in situations where sensor nodes enter a dormant state while harvesting energy from the environment and wake up at random times, or when the communication channel is unreliable and retransmissions are required. Under these conditions, a first-passage path emerges as the path of minimum passage time from a source to a sink. Networks of this type can be described in terms of first-passage percolation [12], [13]. Localization of the source node may be performed by mobile or stationary sinks able to fuse hop count observations to infer the location of the source node, where denotes the a priori pdf of X. By Bayes’ rule, the a posteriori pdf of the source location is , conditioned on observing the hop count z at the sink position. Knowledge of the observation model , that is, the conditional pdf of the hop count, given the source location hypothesis x, is essential for the success of this approach, which may be complicated further by the presence of model parameters whose values are not known a priori and must be learnt on- or off-line. Bayesian localization involves a large number of numerical evaluations of the observation model, due to the typically large space of location hypotheses. This creates a need for observation models with low computational complexity, which may outweigh the need for high accuracy in some applications.
The main contribution of this paper is the formulation of a jump stochastic process whose marginal distribution has a simple analytical form, to model the hop count of the first-passage path from a source to a sink, which is at a distance r. In contrast to earlier works, which use geometric arguments to construct a hop count distribution, our approach utilizes the abstract model of a jump stochastic process and attempts to describe the hop count in its terms. Starting with a process of stationary increments satisfying a strong mixing condition, we make a simplifying independence assumption which allows the hop count to be modeled as a jump Lévy process with drift [14]. We show that, consistent with our assumptions about the hop count, the maximum entropy principle leads to the selection of a translated Poisson distribution as the marginal distribution of the hop count model process.
The paper is structured as follows. In Section 2, we review relevant concepts from stochastic geometry and first-passage percolation and introduce our network model. Our main result, the stochastic process which models the observed hop count distribution at distance r from a source node, is derived in Section 3. We describe how the parameters of this process can be learnt using maximum likelihood estimation. In Section 4, we present simulation results which show that in sensor networks of the type considered in this paper, the marginal distribution of the model process approximates the empirical hop count quite well. Extending work presented in [15], we study the localization error due to the approximation by comparison with a ficticious, idealized network where observations are generated as independent draws from our model. Especially for more densely connected WSNs, we observe a good localization error performance. In the Appendices, we provide proofs of propositions used to derive our model, which were omitted in [15].
Section snippets
System model
The geometry of randomly deployed WSNs is commonly described by Gilbert’s disk model [16]. Given a spatial Poisson point process of density on , two sensor nodes are said to be linked if they are within communication range R of each other. Gilbert’s model induces a random geometric graph with node set and edge set . The node density and the communication range R are related through the mean node degree , so that the graph can be defined equivalently in
Stochastic process model for the hop count
Despite much effort, general results for the limiting distribution of the first-passage time and by extension, the hop count, remain elusive. Moreover, if the objective is to describe the hop count distribution in WSNs, where the distance from the source is relatively small, knowledge of limiting distributions is of little value. Therefore, an attempt is made here to model the hop count by a stochastic process, motivated by several simplifying assumptions. The resulting process is characterized
Hop count distribution
The translated Poisson distribution was applied to model the empirical hop count observed in simulated broadcast WSNs. For this purpose, our simulation generates realizations of random geometric graphs on the unit square , where the number of nodes is a Poisson variable with density nodes per unit area. The source node is located at the center, a setup which minimizes boundary effects. We consider mean node degrees ranging from to , representative of weakly to
Conclusion
We have proposed a new approach to model the hop count distribution between a source and a sink node in broadcast WSNs, in which message propagation is governed by first-passage percolation and node-to-node delays are characterized by i.i.d. exponential random variables. We utilize the abstract model of a jump stochastic process to describe the hop count. By making a simplifying independence assumption and using a maximum entropy argument, the process is shown to have a translated Poisson
Acknowledgement
The authors would like to thank the editor and the anonymous reviewers for their comments and constructive suggestions.
Steffen Beyme received the Diplom (Univ.) degree in electrical engineering from the Humboldt University of Berlin, Germany, in 1991. He is currently a Ph.D. student at the University of British Columbia. His research interests include stochastic control, particularly in the context of wireless sensor networks.
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Steffen Beyme received the Diplom (Univ.) degree in electrical engineering from the Humboldt University of Berlin, Germany, in 1991. He is currently a Ph.D. student at the University of British Columbia. His research interests include stochastic control, particularly in the context of wireless sensor networks.
Cyril Leung received the B.Sc. (First class honours) degree from Imperial College, University of London, England, and the M.S. and Ph.D. degrees in electrical engineering from Stanford University in 1974 and 1976 respectively.
From 1976 to 1979 he was an Assistant Professor in the Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology. During 1979–1980 he was with the Department of Systems Engineering and Computing Science, Carleton University, Ottawa, Canada. Since July 1980, he has been with the Department of Electrical and Computer Engineering at The University of British Columbia, Vancouver, BC, Canada, where he is a Professor and currently holds the PMC-Sierra Professorship in Networking and Communications. His current research interests are in wireless communications systems. He is a member of the Association of Professional Engineers and Geoscientists of British Columbia, Canada.
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This work was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada under Grant OGP0001731 and by the UBC PMC-Sierra Professorship in Networking and Communications. Portions of this paper were presented at the 26th IEEE Canadian Conference on Electrical and Computer Engineering (CCECE), Regina, SK, May 5–8, 2013.