Modelling extremal events: for insurance and finance
Modelling extremal events: for insurance and finance
Control Techniques for Complex Networks
Control Techniques for Complex Networks
Most likely paths to error when estimating the mean of a reflected random walk
Performance Evaluation
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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 $$p0.$$ 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 $$\begin{aligned} \lim \limits _{n\rightarrow \infty }\frac{1}{n}\log P\big (S_{n}(p)0. \end{aligned}$$ 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.