Exact Asymptotics for a Large Deviations Problem for the GI/G/1 Queue

#### S. Asmussen, J.F. Collamore

1999, v.5, №4, 451-476

ABSTRACT

Let $V$ be the steady-state workload and $Q$ the steady-state queue length in the GI/G/1 queue. We obtain the exact asymptotics for probabilities of the form $\P\{V\ge a(t),\, Q\ge b(t)\}$ as $t\to\infty$. In the light-tailed case, there are three regimes according to the limiting value of $a(t)/b(t)$. Our analysis here extends and simplifies recent work of Aspandiiarov and Pechersky [S. Aspandiiarov and E.A. Pechersky, A large deviations problem for compound Poisson processes in queueing theory, Markov Processes Relat. Fields, 1997, v.3, pp. 333-366]. In the heavy-tailed subexponential case, a lower asymptotic bound is derived and shown to be the exact asymptotics in a regime where $a(t)$, $b(t)$ vary in a certain way determined by the service time distribution.