Analysis of the M^x/G/1 Queue with a Random Setup Time

Abstract

We consider an M ^x /G/1 queueing system with a random setup time, where the service of the first unit at the commencement of each busy period is preceded by a random setup time, on completion of which service starts. For this model, the queue size distributions at a random point of time as well as at a departure epoch and some important performance measures are known [see Choudhury, G. An M ^x /G/1 queueing system with setup period and a vacation period. Queueing Sys. 2000, 36, 23–38]. In this paper, we derive the busy period distribution and the distribution of unfinished work at a random point of time. Further, we obtain the queue size distribution at a departure epoch as a simple alternative approach to Choudhury4. Finally, we present a transform free method to obtain the mean waiting time of this model.

Key Words:

• Busy period
• M ^x /G/1 queue
• Queue size
• Setup time
• Unfinished work

2001 AMS Subject Classification:

• Primary 60 k 25
• Secondary 90 B 22
• 68 M 20

Acknowledgments

The first author would like to acknowledge Department of Atomic Energy, Govt. of India for financial support vide NBHM Project No.: 48/2/2001/R&D II/1211, while the work of second author is being supported by U.G.C., Govt. of India, Research Award F-30-97/98 (SA III).