Integral equation approach to the M/G/2 queue

We study the stationary distribution of the number of customers in M/G/2 queueing systems. The two servers are allowed to have different service time distributions. We include the elapsed service times of the customers presently served as supplementary variables and obtain the forward equations satisfied by the joint stationary distribution of the number of customers and the elapsed service times. Using a sequence of transformations, we reduce the problem of determining the marginal probabilities of the number of customers present to the solution of a pair of coupled integral equations. When the servers are identical, only a single integral equation must be solved. The solution of the integral equation(s), and with it the stationary distribution of the number of customers, is constructed for several specific service time densities (e.g., Erlang, hyperexponential, and deterministic).