Abstract
In this paper, a single-server queue with negative customers is considered. The arrival of a negative customer will remove one positive customer that is being served, if any is present. An alternative approach will be introduced to derive a set of equations which will be solved to obtain the stationary queue length distribution. We assume that the service time distribution tends to a constant asymptotic rate when time t goes to infinity. This assumption will allow for finding the stationary queue length of queueing systems with non-exponential service time distributions. Numerical examples for gamma distributed service time with fractional value of shape parameter will be presented in which the steady-state distribution of queue length with such service time distributions may not be easily computed by most of the existing analytical methods.
Highlights
Queues with negative arrivals, known as G-queues, were first introduced by Gelenbe [1, 2] in modelling of neural networks
The arrival of a negative customer to the queue will remove one positive customer according to some killing disciplines
We will only find the stationary queue length distribution using the alternative method introduced in this paper since the derivation of probability generating function to find the steady-state distribution using Laplace transform have a complex form of expression for an M/G/1 queue
Summary
Known as G-queues, were first introduced by Gelenbe [1, 2] in modelling of neural networks. The arrival of a negative customer to the queue will remove one positive customer according to some killing disciplines. 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. We consider queueing systems with negative arrivals and RCH killing strategy. Numerical method introduced in Koh et al [10] will be applied to find the stationary queue length distribution. 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.
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