  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. 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 multihop
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 enough nodes are asleep. All expressions involving
multidimensional 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.
