On the Finite Capacity Shortest Queue Problem

Abstract

We consider two parallel queues. There is one server tending to each queue and the capacity of each queue is K . The network is fed by a single Poisson arrival stream of rate λ, and the two servers are identical exponential servers working at rate µ. A new arrival is routed to the queue with the smaller number of customers. If both have the same number of customers then the arrival is routed randomly, with the probability of joining either queue being 1/2. If there are more than 2 K customers in the system, further arrivals are turned away and lost. We let ρ = λ/µ and take K →∞, and consider the cases ρ 2 and ρ − 2 = O ( K − 1 ). We shall obtain asymptotic approximations to the joint steady state distribution of finding m customers in the first queue and n in the second. The asymptotic approximations are shown to be quite accurate numerically. We shall identify precisely for what ranges of m and n can the finite capacity model be approximated by the infinite capacity one. We will also show that the marginal distribution of finding n customers in the second queue undergoes a transition when ρ = 4. Key words: Shortest queue problem; Finite capacity; Poisson arrival stream; Analytical approximations