a1 Department of Mathematics and Statistics, University of Melbourne, Victoria, 3010, Melbourne, Australia and INRIA, Paris, France email: firstname.lastname@example.org
For a sensor network, a tractable spatially dependent node deployment model is presented with the property that the density is inversely proportional to the sink distance. A stochastic model is formulated to examine message advancements under greedy routing in such a sensor network. The aim of this work is to demonstrate that an inhomogeneous Poisson process can be used to model a sensor network with spatially dependent node density. Symmetric elliptic integrals and asymptotic approximations are used to describe the random behaviour of hops. Types of dependence that affect hop advancements are examined. We observe that the dependence between successive jumps in a multi-hop path is captured by including only the previous forwarding node location. We include a simple uncoordinated sleep scheme, and observe that the complexity of the model is reduced when sufficiently many nodes are asleep. All expressions involving multi-dimensional integrals are derived and evaluated with quasi-Monte Carlo integration methods based on Halton sequences and recently developed lattice rules. An importance sampling function is derived to speed up the quasi-Monte Carlo methods. The ensuing results agree extremely well with simulations.
(Received March 18 2010)
(Revised January 16 2012)
(Accepted January 16 2012)
(Online publication March 27 2012)