The queue GI/M/s with customers of different types or the queue GI/Hm/s

We study the queue GI/M/s with customers of m different types. An arriving customer is of type i with probability p i and the types of different customers are independent. A customer of type i requires a service time which is exponentially distributed with parameter b i . This model is equivalent to the queue GI/H m /s , where H m denotes a mixture of m different exponential distributions. We are primarily interested in the distributions of waiting times and queue lengths. Using a probabilistic argument we reduce the problem to the solution of a system of Wiener-Hopf-type equations. This system is solved by a factorization method. Thus we obtain explicit results for the stationary distributions of waiting times and queue lengths.