A bivariate Poisson queueing process that is not infinitely divisible

Abstract

We are using the term ‘bivariate Poisson process’ to describe a bivariate point process (N1(.), N2(.)) whose components (or, marginal processes) are Poisson processes. In this we are following Milne (2) who amongst his examples cites the case where N1(.) and N2(.) refer to the input and output processes respectively of the M/G/∈ queueing system. Such a bivariate point process is infinitely divisible. We shall now show that in a stationary M/M/1 queueing system (i.e. Poisson arrivals at rate λ, exponential service at rate µ > λ, single-server) a similar identification of (N1(.), N2(.)) yields a bivariate Poisson process that is not infinitely divisible.