In the present work, we study a queue with a Markov renewal service process. The objective is to model systems where different customers request different services and there is a setup time required to adjust from one type of service to the next.

The arrival is a Poisson process independent of the service. After arrival, all the customers will be attended in order of arrival. Immediately before a service starts, the type of next customer is chosen using a finite, irreducible and aperiodic Markov chain P. There is only one server and the service time has a distribution function F_ij, where i and j are the types of the previous and current customer in service, respectively. This model will be called M/MR/l.

Embedding at departure epochs, we characterize the queue length and the type of customer as a Markov renewal process. We study a special case where F_ij, is exponential with parameter μ_ij. We prove that the departure is a renewal process if and only if μ_ij = μ , A i j ε E. Furthermore, we show that this renewal is a Poisson process. The type-departure process is extensively studied through the respective counting processes. The crosscovariance and the crosscorrelation are computed and numerical results are shown. Finally, we introduce several expressions to study the interdependence among the type·departure processes in the general case, i.e. the distribution function F_ij, does not have any special form.