Abstract

Stochastic networks defined by a collection of cooperating agents are solved for their equilibrium state probability distribution by a new compositional method. The agents are processes formalised in a Markovian Process Algebra, which enables the reversed stationary Markov process of a cooperation to be determined symbolically under appropriate conditions. From the reversed process, a separable (compositional) solution follows immediately for the equilibrium state probabilities. The well-known solutions for networks of queues (Jackson’s theorem) and G-networks (with both positive and negative customers) can be obtained simply by this method. Here, the reversed processes, and hence product-form solutions, are derived for more general cooperations, focussing on G-networks with chains of triggers and generalised resets, which have some quite distinct properties from those proposed recently. The methodology’s principal advantage is its potential for mechanisation and symbolic implementation; many equilibrium solutions, both new and derived elsewhere by customised methods, have emerged directly from the compositional approach. As further examples, we consider a known type of fork-join network and a queueing network with batch arrivals.

Author Keywords: Product-forms; Process Algebra; G-networks; Markov processes

Article Outline

1. Introduction

2. Background and previous results

2.1. Process algebraic formalism

2.2. Reversed Markov processes and RCAT

2.3. Application of RCAT in practice

3. Gelenbe networks and extensions

3.1. Two G-queues with triggers

3.2. Multiple cooperations and negative triggers

3.2.1. Two-node G-network with negative triggers

3.3. Split passive actions

3.4. Extensions of G-networks

3.4.1. Networks with reset queues

3.4.2. Gelenbe and Fourneau’s resets

3.5. Generalised G-networks

4. Further product-forms

4.1. A fork-join network

4.1.1. A synchronised join-buffer

4.1.2. Synchronised arrival processes

4.1.3. Chains of positive triggers

4.2. A queueing network with batches

4.3. Batch removals

4.4. Arbitrary batch input and output

5. Conclusion

Appendix A. Reversed Compound Agent Theorem (RCAT)

Appendix B. Balance equations in Section 3.4.1

References

Vitae