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

Abstract

Let $(Q(k):k\ge 0)$ be an $M/M/1$ queue with traffic intensity $\rho \in (0,1).$ Consider the quantity

$$\begin{aligned} S_{n}(p)=\frac{1}{n}\sum _{j=1}^{n}Q\left( j\right) ^{p} \end{aligned}$$

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

$$\begin{aligned} \lim \limits _{n\rightarrow \infty }\frac{1}{n}\log P\big (S_{n}(p)<\mu \left( p\right) -\varepsilon \big ) =-I\left( \varepsilon \right) ,\quad \varepsilon >0. \end{aligned}$$

In this paper, we show that the approximation of the right tail asymptotics requires a different logarithm scaling, giving

$$\begin{aligned} \lim \limits _{n\rightarrow \infty }\frac{1}{n^{1/(1+p)}}\log P\big (S_{n} (p)>\mu \big (p\big )+\varepsilon \big )=-C\big (p\big ) \varepsilon ^{1/(1+p)}, \end{aligned}$$

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.