Two Parallel Queues with Dynamic Routing

We consider two parallel
M/M/1
queueing systems where a new arrival (customer, job, message) joins the shorter of the two queues. Such problems arise naturally in computer communications and packet switched data networks. An asymptotic approach is developed to obtain approximations to the steady-state joint distribution of the number of customers in the two systems. We first analyze the case where the two queueing systems are identical and then consider the case when the two servers work at different rates. Our results are shown to agree with the expansions of known exact solutions, when such solutions are available, and to yield new approximations when such solutions are not available.