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 $$\rho \in (0,1).$$ 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.