Stochastic domination for multirate Poisson processes  

The Poisson process models events arriving randomly in time (e.g. radioactive decays). Poisson processes are used to build many more complex models, including queues or interacting particle systems such as the Ising model or contact process. The standard Poisson process assumes events arrive at a constant rate, with the (random) wait time between events following an exponential distribution. In this talk, our primary goal will be to understand a multirate Poisson process (MPP) where the arrival rate randomly varies between several values according to a background Markov process. This model (the background and foreground processes together) can also be called a continuous-time hidden Markov chain. Our primary goal is to show when the MPP (stochastically) dominates a standard single-rate Poisson process. Stochastic domination between two random variables or stochastic processes means that one is "above" the other and can be understood by its equivalency to the existence of a coupling which preserves the ordering. We study the discrete-time version of the model and domination of product measures first, arriving at the results for the continuous-time model via weak convergence of cadlag step function sample paths. Formulating the problem using matrix theory provides the key to derive our results in a straightforward way. The optimal coupling parameters (the product measure probability parameter and the standard single-rate Poisson process arrival rate) are shown to be related to eigenvalues of matrices constructed from the model parameters in a simple way. This work solves an open problem stated in Broman (2007). In that paper, the case where the MPP alternates between two arrival rates is studied, whereas we provide sufficient conditions for the case where there are an arbitrary number of rates---that the optimal coupling parameter is an eigenvalue is novel and general for the entire class of models (those satisfying our sufficient condition).