A correlated random walk model for a queue with Markovian arrivals and departures

Using a random walk model for single server queues with Bernoulli arrivals and geometric service times is an analysis technique well known in queueing theory. When the arrivals and departures are Markov dependent, the number of customers in the queue can be modelled as a correlated random walk (CRW). Previous investigations of CRW have used mostly transform techniques. In this paper, using a CRW model for the queue, representing it as a bivariate Markov chain, and exploiting the structural properties of its transition probability matrix, explicit results for the equilibrium solution and some first passage characteristics, including the mean busy period, have been obtained in the following cases: (i) with no restrictions on the waiting space and (ii) with limited waiting space.