A Pollaczek–Khintchine formula for M/G/1 queues with disasters

Abstract

A disaster occurs in a queue when a negative arrival causes all the work (and therefore customers) to leave the system instantaneously. Recent papers have addressed several issues pertaining to queueing networks with negative arrivals under the i.i.d. exponential service times assumption. Here we relax this assumption and derive a Pollaczek–Khintchine-like formula for M/G/1 queues with disasters by making use of the preemptive LIFO discipline. As a byproduct, the stationary distribution of the remaining service time process is obtained for queues operating under this discipline. Finally, as an application, we obtain the Laplace transform of the stationary remaining service time of the customer in service for unstable preemptive LIFO M/G/1 queues.

Time-dependent properties of symmetric queues

“…The idea of the proof is based on the fact that the state of the system at an exponential time, starting from the empty state, has the same distribution as the steady-state distribution of the system where work is removed after exponential times; such a system is referred to as a clearing model. This connection has been noted before, and applied to LCFS by Jain and Sigman [8].…”

“…and given by T i,q , thus extending the explanation of [6] to the time-dependent case, as was done in [8] for LCFS. Based on the expressions given in both (3.8) and (4.9), it seems reasonable to compare the distributions of T q and min(B, E κ(q) ) * .…”

“…In passing, we also obtain a geometric random sum decomposition for the workload of an M/G/1 queue sampled at an exponential time, and derive an interpretation similar to the steady-state M/G/1 queue as given by Cooper and Niu [6], see also [8].…”

Performance estimation of an email contact center by a finite source discrete time Geo/Geo/1 queue with disasters

“…Disaster has no effect on an empty system. Disaster is also called catastrophe, mass exodus (Jain & Sigman, 1996) or queue flushing (Chen & Renshaw, 1997). Queue models with disasters can be used to analyze computer networks with virus infections and breakdowns due to a reset order.…”

“…The presence of disasters in queuing systems was introduced by Jain and Sigman (1996). In the literature related to continuous time queues, numerous papers (Artalejo & GQomez-Corral, 1999;Chen & Renshaw, 1997;Jain & Sigman, 1996;Li & Lin, 2006;Towsley & Tripathi, 1991 and Wang, Liu, & Li, in press) have recently appeared in which a disaster removes all the work present in the system.…”