Functional limit theorems for the queue GI/G/1 in light traffic

Abstract

We consider a single GI/G/1 queueing system in which customer number 0 arrives at time t0 = 0, finds a free server, and experiences a service time v0. The nth customer arrives at time tn and experiences a service time vn. Let the interarrival times tn - tn-1 = un, n ≧ 1, and define the random vectors Xn = (vn-1, un), n ≧ 1. We assume the sequence of random vectors {Xn : n ≧ 1} is independent and identically distributed (i.i.d.). Let E{un} = λ-1 and E{vn} = μ-1, where 0 < λ, μ < ∞. In addition, we shall always assume that E{v02} < ∞ and that the deterministic system in which both vn and un are degenerate is excluded. The natural measure of congestion for this system is the traffic intensity ρ = λ/μ. In this paper we shall restrict our attention to systems in which ρ < 1. Under this condition, which we shall refer to as light traffic, our system is of course stable.