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Published in:
01-04-2013
Large deviations for the empirical mean of an \(M/M/1\) queue
Published in: Queueing Systems | Issue 4/2013
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Abstract
Let \((Q(k):k\ge 0)\) be an \(M/M/1\) queue with traffic intensity \(\rho \in (0,1).\) Consider the quantity 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 In this paper, we show that the approximation of the right tail asymptotics requires a different logarithm scaling, giving 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.
$$\begin{aligned} S_{n}(p)=\frac{1}{n}\sum _{j=1}^{n}Q\left( j\right) ^{p} \end{aligned}$$
$$\begin{aligned} \lim \limits _{n\rightarrow \infty }\frac{1}{n}\log P\big (S_{n}(p)<\mu \left( p\right) -\varepsilon \big ) =-I\left( \varepsilon \right) ,\quad \varepsilon >0. \end{aligned}$$
$$\begin{aligned} \lim \limits _{n\rightarrow \infty }\frac{1}{n^{1/(1+p)}}\log P\big (S_{n} (p)>\mu \big (p\big )+\varepsilon \big )=-C\big (p\big ) \varepsilon ^{1/(1+p)}, \end{aligned}$$