Large Deviations for the Empirical Mean of an M/M/1 Queue

J. Blanchet , P. W. Glynn, and S. Meyn

Submitted for publication

Let (Q(k):k≥0) be an M/M/1 queue with traffic intensity ρ∈(0,1). Consider the quantity

for any p>0. The ergodic theorem yields that Sn(p)→μ(p):=E[Q(∞)p], where Q(∞) is geometrically distributed with mean ρ/(1-ρ). It is known that one can explicitly characterize I(ε)>0 such that

In this paper, we show that the approximation of the right tail asymp- totics requires a diĀ¤erent logarithm scaling, giving

where C(p)>0 is obtained as the solution of a variational problem.

We discuss why this phenomenon - Weibullian right tail asymptotics rather than exponential asymptotics - can be expected to occur in more general queueing systems.