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.

Decay properties and quasi-stationary distributions for stopped Markovian bulk-arrival and bulk-service queues

“…Within this framework, state-dependent input and output mechanisms have attracted considerable attention. In particular, Gelenbe [17] and Gelenbe, Glynn and Sigman [18] introduced the very useful concept of negative arrival, and this was followed up by other authors, including Bayer and Boxma [5], Harrison and Pitel [20], Henderson [21] and Jain and Sigman [22]. The queueing models with negative arrivals also has close theoretical links with the versatile Markovian arrival processes introduced by Neuts [33], which include several kinds of batcharrival process.…”

A Note on Negative Customers, GI/G/1 Workload, and Risk Processes

“…In Section 2 of the present note, we further explore the possibilities opened by this transformation. We show how it yields an immediate explanation of the remarkable occurrence, first observed by Jain and Sigman [8,9], of a generalized Pollaczek-Khintchine form for the Laplace-Stieltjes transform of the workload distribution in the M/G/1 queue with negative customers.…”

“…Assume that A.+ /3 < 1 + A.-1'; this is the stability condition for this M/G/1 generalization [2,8). One can even allow the case in which negative customers always remove all the work present; this is the so-called disaster model [2,9].…”

Computation of the limiting distribution in queueing systems with repeated attempts and disasters

“…A number of papers [3][4][5]15] have recendy appeared in the queueing literature in which a catastrophe removes all the work present in the system. These disasters can be view as a gênerai breakdown of the system so these models can be used to analyze computer networks with virus infections, and breakdowns due to a reset order.…”