Abstract

Fluid models have for some time been used to approximate stochastic networks with discrete state. These range from traditional ‘heavy traffic’ approximations to the recent advances in bio-chemical system models. Here we present a simple approximate compositional method for analysing a network of fluid queues with Markov-modulated input processes at equilibrium. The idea is to approximate the on/off process at the output of a queue by an n-state Markov chain that modulates its rate. This chain is parameterised by matching the moments of the resulting process with those of the busy period distribution of the queue. This process is then used, in turn, as a separate Markov-modulated on/off process that feeds downstream queue(s). The moments of the busy period are derived from an exact analytical model. Approximation using two- and three-state intermediate Markov processes are validated with respect to an exact model of a tandem pair of fluid queues — a generalisation of the single queue model. The analytical models used are rather simpler and more accessible, albeit less general, than previously published models, and are also included. The approximation method is applied to various fluid queue networks and the results are validated with respect to simulation. The results show the three-state model to yield excellent approximations for mean fluid levels, even under high load.

Keywords: Fluid queue; Network; on/off process; Analytical model; Approximation

Article Outline

1. Introduction

2. Fluid queueing networks

3. Analytical model for tandem queues

3.1. Laplace transform of the joint fluid level probability density

3.2. Moments

4. Approximation method

4.1. First attempt: n=2

4.2. Second attempt: n=3

4.3. Accumulated error

5. Summary and conclusion

Appendix A. Single fluid queue: Review

Remarks

Appendix B. Tandem pair of fluid queues

B.1. Computation of and

B.2. Joint Laplace transform

References