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 di¤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.