The M/G/1 queue with negative customers

Abstract

We derive expressions for the generating function of the equilibrium queue length probability distribution in a single server queue with general service times and independent Poisson arrival streams of both ordinary, positive customers and negative customers which eliminate a positive customer if present. For the case of first come first served queueing discipline for the positive customers, we compare the killing strategies in which either the last customer in the queue or the one in service is removed by a negative customer. We then consider preemptive-restart with resampling last come first served queueing discipline for the positive customers, combined with the elimination of the customer in service by a negative customer—the case of elimination of the last customer yields an analysis similar to first come first served discipline for positive customers. The results show different generating functions in contrast to the case where service times are exponentially distributed. This is also reflected in the stability conditions. Incidently, this leads to a full study of the preemptive-restart with resampling last come first served case without negative customers. Finally, approaches to solving the Fredholm integral equation of the first kind which arises, for instance, in the first case are considered as well as an alternative iterative solution method.

Wiener-Hopf analysis of an M/G/1 queue with negative customers and of a related class of random walks

“…For the M/G/l queue with negative customers who upon their arrival immediately remove an ordinary customer, the queue-length distribution is analysed by Harrison and Pitel [10]. Their analysis of the generating function for the equilibrium queue-length distribution eventually leads to a Fredholm integral equation of the first kind, that must be solved numerically -which is a notoriously difficult problem.…”

Stationary queue length of a single-server queue with negative arrivals and nonexponential service time distributions

“…By assuming Poisson arrival of negative customers, Harrison and Pitel [8] derived expressions to find the stationary queue length and sojourn time distributions for an M/M/1 queue. The same authors [9] then extended the model to M/G/1 queue and expressed explicitly the stability conditions.…”

“…A set of equations for the stationary probabilities will be derived and the stationary queue length distribution can be obtained by solving the equations. Mean queue length computed by the alternative method will be compared to those obtained by the analytical method in [9] and verified with the simulation results. However, we will only find the stationary queue length distribution using the alternative method introduced in this paper since the derivation of probability generating function (pgf) to find the steady-state distribution using Laplace transform have a complex form of expression for an M/G/1 queue.…”

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

“…In contrast with the ordinary customers, the arrivai of a négative customer implies that one positive customer must be removed, if any présents, from the system. Since their introduction, there has been a growing interest in several directions, including extensions to the case of multiple classes of customers [7,13], networks with triggered customer motion [11], networks with batch and random amount of work removal [4,12] and analysis of single node queues with négative customers [14].…”