Monotonicity in the Limited Processor-Sharing Queue

Abstract

We study a processor-sharing queue with a limited number of service positions and an infinite buffer. The occupied service positions share an underlying resource. We prove that for service times with a decreasing failure rate, the queue length is stochastically decreasing in the number of service positions, and that for service times with an increasing failure rate, the queue length is stochastically increasing. We show that a similar result holds for the foreground-background queue. The queue length distributions and their decay rates are compared to these in other queueing models with and without restrictions on the number of service positions.

Keywords:

• Failure rate
• Limited processor-sharing
• LPS
• Monotonicity
• Queue length

Mathematics Subject Classification:

• Primary 90B22, 60K25, 68M20

ACKNOWLEDGMENTS

The authors would like to thank Rhonda Righter, Michel Mandjes, and Bert Zwart for many interesting remarks and suggestions on earlier versions of this article, and the referees for many useful comments and questions.