An alternative approach in finding the stationary queue length distribution of a queueing system with negative customers

Siew Khew Koh,Tan Ching Ng,Ching Herny Chin,Yi Fei Tan,L.E Teoh,W.S Ng

doi:10.1051/itmconf/20213604001

Siew Khew Koh, Tan Ching Ng + Show 4 more

Open Access

https://doi.org/10.1051/itmconf/20213604001

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Abstract

A single-server queueing system with negative customers is considered in this paper. One positive customer will be removed from the head of the queue if any negative customer is present. The distribution of the interarrival time for the positive customer is assumed to have a rate that tends to a constant as time t tends to infinity. An alternative approach will be proposed to derive a set of equations to find the stationary probabilities. The stationary probabilities will then be used to find the stationary queue length distribution. Numerical examples will be presented and compared to the results found using the analytical method and simulation procedure. The advantage of using the proposed alternative approach will be discussed in this paper.

Highlights

The development of queueing system with negative customers can be dated back to the work of Gelenbe [1, 2]
The simplest form of killing strategy in G-queue is removal of one positive customer at the head (RCH) or at the end (RCE) immediately by a negative customer that arrives to the queue
The numerical method was first proposed in Koh [25] and has been successfully applied in some other queueing systems with different features

Summary

Introduction

The development of queueing system with negative customers can be dated back to the work of Gelenbe [1, 2]. In Harrison and Pitel [11], an M/M/1 G-queue was considered and the sojourn time distribution was found by using Laplace transform. RCH killing discipline is considered and the interarrival time of the positive customers is assumed to have a constant asymptotic rate as time t tends to infinity. We call such distribution a CAR distribution. When exponential assumption of both the interarrival and service time distributions is relaxed, the expressions derived using the analytical method may have a complicated form and will not be easy to solve.

Model description

Stationary probabilities

Stationary queue length distribution

Numerical example

Simulation procedure

Conclusion