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

We consider a single GI / G /1 queueing system in which customer number 0 arrives at time t 0 = 0, finds a free server, and experiences a service time v 0 . The n th customer arrives at time t n and experiences a service time v n . Let the interarrival times t n - t n -1 = u n , n ≧ 1, and define the random vectors X n = ( v n -1 , u n ), n ≧ 1. We assume the sequence of random vectors { X n : n ≧ 1} is independent and identically distributed (i.i.d.). Let E { u n } = λ -1 and E { v n } = μ -1 , where 0 < λ, μ < ∞. In addition, we shall always assume that E { v 0 2 } < ∞ and that the deterministic system in which both v n and u n 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.