Exact asymptotics for a large deviations problem for the GI/G/1 queue

Sören Asmussen and Jeffrey F. Collamore

Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology,
Lund University,

ISSN 1403-9338

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 of probabilities of the form P(V > a(t), Q > b(t)) as t tends to infinity. In the light-tailed case, there are three regimes according to the limiting value of a(t)/b(t), and our analysis extends and simplifies recent work of Aspandiiarov and Pechersky (1997). 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.
Key words:
change of measure, conditioned limit theorems, saddlepoint approximations, subexponential distribution, workload