Ng. Duffield et N. Oconnell, LARGE DEVIATIONS AND OVERFLOW PROBABILITIES FOR THE GENERAL SINGLE-SERVER QUEUE, WITH APPLICATIONS, Mathematical proceedings of the Cambridge Philosophical Society, 118, 1995, pp. 363-374
We consider queueing systems where the workload process is assumed to
have an associated large deviation principle with arbitrary scaling: t
here exist increasing scaling functions (a(t), v(t), t epsilon R(+)) a
nd a rate function I such that if (W-t, t epsilon R(+)) denotes the wo
rkload process, then [GRAPHICS] on the continuity set of I. In the cas
e that a(t) = v(t) = t it has been argued heuristically, and recently
proved in a fairly general context (for discrete time models) by Glynn
and Whitt[8], that the queue-length distribution (that is, the distri
bution of supremum of the workload process Q = sup(t greater than or e
qual to 0 W?t) decays exponentially: P(Q > b) similar to e(-delta b) a
nd the decay rate delta is directly related to the rate function I. We
establish conditions for a more general result to hold, where the sca
ling functions are not necessarily linear in t: we find that the queue
-length distribution has an exponential tail only if lim(t-->infinity)
a(t)/v(t) is finite and strictly positive; other:wise, provided our c
onditions are satisfied, the tail probabilities decay like P (Q > b) s
imilar to e(-delta v) (a(-1)(b)). We apply our results to a range of w
orkload processes, including fractional Brownian motion (a model that
has been proposed in the literature (see, for example, Leland et al. [
10] and Norros[15, 16]) to account for self-similarity and long range
dependence) and, more generally, Gaussian processes with stationary in
crements. We show that the martingale upper bound estimates obtained b
y Kulkarni and Rolski [5], when the workload is modelled as Ornstein-U
hlenbeck position process, are asymptotically correct. Finally we cons
ider a non-Gaussian example, where the arrivals are modelled by a squa
red Bessel process.