Abstract

This paper presents an efficient solution technique for the steady-state analysis of the second-order stochastic fluid model underlying a second-order fluid stochastic Petri net (FSPN) with constant flow and transition rates, and a single bounded fluid place. The new solution technique is an extension of existing solution techniques developed for (first-order) Fluid Models; the solution algorithm uses upwind semi-discretization and Matrix Geometric techniques to efficiently compute the steady-state probabilities of the mixed discrete and continuous state space of the model. The effectiveness of our technique is proven first on a simple producer–consumer second-order FSPN model and then on a complex example taken from the literature where the analysis of the completion time distribution and the packet loss probability of short-lived TCP connections is investigated through the decomposition of a model for the whole system into several (simpler) sub-models; the interaction between the different sub-models is handled by iterating their solution until the complete model solution converges according to a fixed point algorithm. The introduction of a second-order FSPN in a fixed point iteration scheme has been made possible thanks to the efficiency of the proposed solution technique.

Keywords: Second order stochastic fluid model; Fluid stochastic Petri nets; Matrix geometric; TCP

Article Outline

1. Introduction

2. Second-order stochastic fluid models

2.1. Boundary conditions

3. Second-order fluid stochastic Petri nets

4. The proposed solution technique

4.1. Boundary conditions

4.2. Solution of the discretized equations

4.3. A simple example

5. An application of second-order FSPN to the analysis of TCP

5.1. The network scenario

5.2. The modelling approach

5.3. The second-order FSPN sub-model of the IP network link

5.4. Model validation

5.5. Model solution complexity

6. Conclusions

References

Vitae