Journal of Probability and Statistics
Volume 2011 (2011), Article ID 812726, 11 pages
http://dx.doi.org/10.1155/2011/812726
Research Article

On a Batch Arrival Queuing System Equipped with a Stand-by Server during Vacation Periods or the Repairs Times of the Main Server

Rehab F. Khalaf,1 Kailash C. Madan,2 and Cormac A. Lukas1

1School of Information Systems Computing and Mathematics, Brunel University, Middlesex UB83PH, UK
2College of Information Technology, Ahlia University, P.O. Box 10878, Bahrain

Received 25 January 2011; Accepted 20 May 2011

Academic Editor: Rongling Wu

Copyright © 2011 Rehab F. Khalaf et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study a queuing system which is equipped with a stand-by server in addition to the main server. The stand-by server provides service to customers only during the period of absence of the main server when either the main server is on a vacation or it is in the state of repairs due to a sudden failure from time to time. The service times, vacation times, and repair times are assumed to follow general arbitrary distributions while the stand-by service times follow exponential distribution. Supplementary variables technique has been used to obtain steady state results in explicit and closed form in terms of the probability generating functions for the number of customers in the queue, the average number of customers, and the average waiting time in the queue while the MathCad software has been used to illustrate the numerical results in this work.