A matrix approach for the transient solution of an M/M/1/N queue with discouraged arrivals and reneging

Abstract

This paper illustrates a computable matrix technique that can be used to derive explicit expressions for the transient state probabilities of a finite waiting space single-server queue, namely (M/M/1/N), having discouraged arrivals and reneging. The discipline is the classical first-come, first-served (FCFS). We obtain the transient solution of the system, with results in terms of the eigenvalues of a symmetric tridiagonal matrix. Finally, numerical calculations are given to illustrate the effectiveness of this technique and system behaviour.

Keywords:

• matrix approach
• transient solution
• finite waiting space
• Laplace transform
• state probabilities
• queue length

2010 AMS Subject Classifications :

• 44A10
• 90B22
• 60K25
• 68M20

Acknowledgement

The authors thank the area editors and the anonymous referees for their valuable suggestions that led to considerable improvement in the presentation of this paper.

Additional information

Notes on contributors

S. A. El-Shehawy

Present address: Department of Mathematics, College of Science, Qassim University, Buraidah 51452, Saudi Arabia