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.