On the k-coverage of line segments by a non homogeneous Poisson-Boolean model

We consider k-coverage of a line by a two-dimensional, non homogeneous Poisson-Boolean model. This has applications in sensor networks. We extend the analysis of [1] to the case for k > 1. The extension requires us to define a vector Markov process that tracks the k segments that have the longest residual coverage at a point. This process is used to determine the probability of a segment of the line being completely covered by k or more sensors. We illustrate the extension by considering the case of k = 2.