Large deviations for the empirical mean of an M/M/1 queue

Let $(Q(k):kge 0)$ be an $M/M/1$ queue with traffic intensity $rho in (0,1).$ Consider the quantity $$begin{aligned} S_{n}(p)=frac{1}{n}sum _{j=1}^{n}Qleft( jright) ^{p} end{aligned}$$ for any $p>0.$ The ergodic theorem yields that $S_{n}(p) rightarrow mu (p) :=E[Q(infty )^{p}]$ , where $Q(infty )$ is geometrically distributed with mean $rho /(1-rho ).$ It is known that one can explicitly characterize $I(varepsilon )>0$ such that $$begin{aligned} lim limits _{nrightarrow infty }frac{1}{n}log Pbig (S_{n}(p)0. end{aligned}$$ In this paper, we show that the approximation of the right tail asymptotics requires a different logarithm scaling, giving $$begin{aligned} lim limits _{nrightarrow infty }frac{1}{n^{1/(1+p)}}log Pbig (S_{n} (p)>mu big (pbig )+varepsilon big )=-Cbig (pbig ) varepsilon ^{1/(1+p)}, end{aligned}$$ 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.