The Longer Queue Model

Abstract

Two queues forming two independent Poisson processes are served by one server with exponential service time. The server always works on the longer queue and, in case that they are of equal length, chooses either one with probability ½. Let πij be the probability that the two queue lengths equal i andj at equilibrium and π(z, w) = ∑πi j Ziwj. We determine π(z, w) and derive from this asymptotic formulas forπij as i, j → ∞. These asymptotic formulas are used to study the interdependence of the queue lengths. In particular, we obtain limit laws for the queue lengths conditioned on each other.