| The shortest queue (SQ) problem is one of the classic models in queueing theory. We first consider two identical, parallel M/M/infinity queues. A new arrival is routed to the queue with the smaller number of customers. If both systems have equal occupancy, the arrival joins either with probability 1/2. We call this the symmetric case. We analyze this model both numerically and asymptotically. For the latter we consider the limit rho = lambda/mu → infinity, where lambda (resp., mu) is the arrival (resp., service) rate. We give asymptotic formulas the joint steady-state distribution of the number of customers in the two queues. Secondly, we study the non-symmetric version of the problem, where the service rates for the two sets of servers are allowed to be different. We call the two service rates mu1 and mu2. Also, if the two queues are of equal length, the arrival joins the first queue with probability nu1 and the second with probability nu2 = 1 - nu1. The symmetric case is a special case of this model (with mu1 = mu2 and nu1 = nu 2 = 1/2). We now define rho = lambda/mu1 and beta = mu2/mu1, where lambda is the arrival rate. Without loss of generality, we assume that beta > 1. We again study the joint steady state distribution p(m, n), both numerically and asymptotically, for rho → infinity and a fixed beta > 1. |
