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

Let $(Q(k):k\ge 0)$ be an $M/M/1$ queue with traffic intensity $\rho \in (\mathrm{0,1})$ . Consider the quantity $\begin{array}{c}{S}_n(p)=\frac{1}{n}\sum _{j=1}^{n}Q{\left(j\right)}^p\end{array}$ for any $p>0$ . The ergodic theorem yields that $S_n(p)\to \mu (p):=E[Q(\mathrm{\infty }{)}^p]$ , where $Q(\mathrm{\infty })$ is geometrically distributed with mean $\rho /(1‐\rho )$ . It is known that one can explicitly characterize $I(ϵ)>0$ such that $\begin{array}{c}\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}logP(S_n(p)<\mu \left(p\right)-\epsilon )=-I\left(\epsilon \right),\epsilon >0\end{array}$ . In this paper, we show that the approximation of the right tail asymptotics requires a different logarithm scaling, giving $\begin{array}{c}\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n^{1/(1+p)}}logP(S_n(p)>\mu (p)+\epsilon )=-C(p){\epsilon }^{1/(1+p)},\end{array}$ 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.