The shortest queue problem

Abstract

A Poisson stream of customers arrives at a service center which consists of two single-server queues in parallel. The service times of the customers are exponentially distributed, and both servers serve at the same rate. Arriving customers join the shortest of the two queues, with ties broken in any plausible manner. No jockeying between the queues is allowed. Employing linear programming techniques, we calculate bounds for the probability distribution of the number of customers in the system, and its expected value in equilibrium. The bounds are asymptotically tight in heavy traffic.