Abstract

We obtain a decomposition result for the steady state queue length distribution in egalitarian processor-sharing (PS) models. In particular, for multi-class egalitarian PS queues, we show that the marginal queue length distribution for each class equals the queue length distribution of an equivalent single class PS model with a random number of permanent customers. Similarly, the mean sojourn time (conditioned on the initial service requirement) for each class can be obtained by conditioning on the number of permanent customers. The decomposition result implies linear relations between the marginal queue length probabilities, which also hold for other PS models such as the egalitarian PS models with state-dependent system capacity that only depends on the total number of customers in the system. Based on the exact decomposition result for egalitarian PS queues, we propose a similar decomposition for discriminatory processor-sharing (DPS) models, and numerically show that the approximation is accurate for moderate differences in service weights.

Keywords: Processor-sharing queues; Queue length; Decomposition; Permanent customers; Approximation; Generalized discriminatory processor-sharing

Article Outline

1. Introduction

2. Model

3. Decomposition of egalitarian processor-sharing models

3.1. Queue length decomposition

3.2. Sojourn time decomposition

3.3. A feedback network with egalitarian processor-sharing

3.4. Multi-class egalitarian processor-sharing with state-dependent capacity

4. Approximation for discriminatory processor-sharing models

4.1. General approximation method for mean sojourn times

4.1.1. Approximation method for K=2 customer classes

4.1.2. Outline of the approximation method for K>2 customer classes

4.2. Conservation law

4.3. Numerical results

4.3.1. Two-class DPS queue

4.3.2. Three-class DPS queue

4.4. Discussion

5. Conclusion

Acknowledgements

References

Vitae