Tail Behaviour of the Area Under the Queue Length Process of the Single Server Queue with Regularly Varying Service Times

Abstract

This paper considers a stable G I |G I |1 queue with a regularly varying service time distribution. We derive the tail behaviour of the integral of the queue length process Q(t) over one busy period. We show that the occurrence of a large integral is related to the occurrence of a large maximum of the queueing process over the busy period and we exploit asymptotic results for this variable. We also prove a central limit theorem for t 0 Q(s) ds.

Local asymptotics for the area under the random walk excursion

“…Unfortunately, we do not know how to derive a version of (7) for non-lattice random walks. Moreover, we do not know how to derive (8) without local asymptotics. One can derive an upper bound for P(A τ > x) via the exponential Chebyshev inequality.…”

Tail behaviour of the area under a random process, with applications to queueing systems, insurance and percolations

“…The statements ( 2) and (3) were proven in [22] and [9], respectively, under regularly varying assumption of the service time, that is…”

“…The statements (2) and (3) were proven in [22] and [9], respectively, under regularly varying assumption of the service time, that isF (x) = x −α L(x), where α > 1 and L is slowly varying at infinity. Furthermore, in [22] one needs additionally that…”

“…Such integrals appear naturally in analysis of ATM. The reader is referred to references given in [9] and [22]. Recently, in [2], the authors connected mean bit rate in time varying M/M/1 queue with moments of the integral τ 0 Q(u) du in a corresponding standard M/M/1 system.…”

A Note on Skewness in Regenerative Simulation

“…copies of C holds, convergence will be slow. For extensions of these results to GI/GI/1 queues and regularly varying service times, see Kulik and Palmowski (2005). We will also show that as ρ ↑ 1, the distributions of τ and C become totally unbalanced in the sense that asymptotically the probability that they exceed their means tend to zero.…”