On the event distance of Poisson processes with applications to sensors (Q477345)

The author calculates the expected distance between two i.i.d.\ Poisson processes with respective arrival times \(X_1, X_2, \dots \) and \(Y_1, Y_2, \dots\) on a line and derives a closed form formula for the event distances \(E[|X_{k+r}-Y_k|]\), for any \(r \geq 0\), \(k \geq 1\). A typical motivation for studying these quantities could arise in sensor networks with one Poisson process representing resources and the other Poisson process representing events, and is required to match one-by-one resources with events. The distance is represented as a linear combination of the Pochhammer polynomials. An application to optimal matchings and optimal transportation cost for a set of sensors in a line is provided.