Large Deviations for the Empirical Mean of an M/M/1 Queue
J. Blanchet , P. W. Glynn, and S. Meyn
Queueing Systems: Theory and Applications, Vol 73, No. 4, 425-446 (2013)
Let be an M/M/1 queue with traffic intensity . Consider the quantity
for any . The ergodic theorem yields that , where is geometrically distributed with mean . It is known that one can explicitly characterize such that
In this paper, we show that the approximation of the right tail asymp-totics requires a different logarithm scaling, giving
where 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.