A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters

Mahdy S El-Paoumy ,Taha Radwan,Mohammed Alqawba

doi:10.1515/math-2021-0126

Mahdy S El-Paoumy , Taha Radwan + Show 1 more

Open Access

https://doi.org/10.1515/math-2021-0126

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Abstract

Abstract This work considers the infinite multi-server Markovian queueing model with balking and catastrophes where the rates of arrivals, service, balking, and catastrophes are time dependent. The catastrophes arrive as negative customers to the system. The arrival of negative customers to a queueing system removes the positive customers. The catastrophes may come either from another service station or from outside the system. In this paper, we obtained the transient solution of this model using the approach of probability-generating function. Also, we derived an expression of transient probabilities in terms of Volterra equation of the second kind. Furthermore, we obtained a measure for time-dependent expected number of customers in the system.

Highlights

Queueing systems have been used effectively in computer networks, communication networks, hospitals, and manufacturing models
The catastrophes that happen randomly lead to the annihilation of most units in the queueing system
Rakesh [11] studied the transient solution of the M/M/c queueing system with balking and catastrophes, assuming that the arrival, departure, balk, and catastrophes rates are constants

Summary

Introduction

Queueing systems have been used effectively in computer networks, communication networks, hospitals, and manufacturing models. Many researchers studied the models with constant time of arrival rate, service rate, balk, and catastrophe. Rakesh [11] studied the transient solution of the M/M/c queueing system with balking and catastrophes, assuming that the arrival, departure, balk, and catastrophes rates are constants. Zhang and Coyle [13] studied the model without balking and catastrophes, in which he obtained the boundary probability function p0(τ) in the form of Volterra integral equation of the second kind, and presented the numerical solution of the Volterra integral equation using the Runge-Kutta algorithm [14]. Jain and Singh [16] studied the transient model of the Markovian feedback queue subject to disaster and discouragement with other concepts of time-independent parameters. We will derive a measure for time-dependent expected number of customers in the system

Mathematical model

Expressions for the expected queue size

Illustration examples

Special cases

Conclusion