Abstract

In this paper, we consider a queue with compound Poisson arrivals, phase type required service times in which a single processor serves according to the processor-sharing discipline. For this queue, we derive a system of equations for the transform of the queue-length and obtain the moments of the queue-length as a solution of linear equations. We also obtain a system of equations for the joint transforms of the sojourn time and the queue-length and find the moments of the sojourn time as a solution of linear equations. Numerical examples show that the smaller the variation of the required service times becomes, the larger the mean and variance of the sojourn times become.

Keywords: Processor-sharing; Queue-length; Sojourn time; Joint transform; Bulk arrivals

Article Outline

1. Introduction

2. Transform of the queue-length distribution

3. Moments of the queue-length

4. Joint transforms of the sojourn time and the queue-length

5. Moments of the sojourn time

5.1. Conditional moments of the sojourn time

5.2. Unconditional moments of the sojourn time

6. Numerical examples

Acknowledgements

Appendix A. Derivation of (7)

Appendix B. Uniqueness of solutions

B.1. Eq. (9) with , 1≤k,l≤m

B.2. Eq. (10) with , 1≤k,l,r≤m

B.3. Eq. (27)

B.4. Eq. (30) with , 1≤j,k,l≤m

B.5. Eq. (29)

References

Vitae