Deriving the rate equations characterising product-form models and application to propagating synchronisations

Abstract

Performance engineering often uses stochastic modelling as a powerful approach to the quantitative analysis of real systems. Product-form Markovian models have the property that the steady-state analysis can be carried out efficiently and without the need for solving the system of global balance equations. The Reversed Compound Agent Theorem (RCAT) gives sufficient conditions for the model to have a product-form solution. In this paper we show its application in the case of instantaneous synchronisations of more than two components at the same time. Although examples of this class of product-form models are already known, the results shown here are novel. We introduce the idea of Propagation of Instantaneous Transitions (PITs) to model multi-way synchronisations as successive pairwise ones in the case of product-form. An algorithm that derives the system of equations that must be solved to obtain the steady-state distribution is presented. Two examples of new product-form models are then derived as a consequence of these contributions. The first is a queueing network with finite capacity nodes, a skipping policy, and partial flushing as a congestion handling mechanism. The second is a queueing network with nodes that may have negative queue lengths, where an unbounded customer deletion mechanism is introduced.