Estimating tail probabilities in queues via extremal statistics (Q2702300)

Let \(X=(X(t) \mid t\geq 0)\) be a stationary and classically regenerative process whose cycle lengths have finite moments of all orders. Under various conditions on the asymptotic behavior of \(\alpha(b) =P(X(0)\geq b)\) the authors derive almost sure and \(L_p\) limit theorems for \(M(t) =\sup\{X(s) \mid 0\leq s\leq t\}\), as \(t\to \infty\); they also study the rate of convergence and first passage times. The results are used to establish the so-called logarithmic consistency of estimators for \(\alpha(b)\) which are based solely on \(M(t)\). In the special case that \(X\) is the workload process of a stationary GF/G/1 queue, these semi-parametric estimators are compared to the natural estimator `fraction of time during \([0,t]\) that \(X\) spends in \((b,\infty)\)', whose asymptotic analysis is based on a compound Poisson limit theorem for the stationary waiting time sequence. Finally, the case of reflecting Brownian motion is treated in detail.NEWLINENEWLINEFor the entire collection see [Zbl 0951.00043].