In this paper, we deal with the lumpability approach to cope with the state space explosion problem inherent to the computation of the stationary performance indices of large stochastic models. The lumpability method is based on a state aggregation technique and applies to Markov chains exhibiting some structural regularity. Moreover, it allows one to efficiently compute the exact values of the stationary performance indices when the model is actually lumpable. The notion of quasi-lumpability is based on the idea that a Markov chain can be altered by relatively small perturbations of the transition rates in such a way that the new resulting Markov chain is lumpable. In this case, only upper and lower bounds on the performance indices can be derived. Here, we introduce a novel notion of quasi-lumpability, named proportional lumpability, which extends the original definition of lumpability but, differently from the general definition of quasi-lumpability, it allows one to derive exact stationary performance indices for the original process. We then introduce the notion of proportional bisimilarity for the terms of the performance process algebra PEPA. Proportional bisimilarity induces a proportional lumpability on the underlying continuous-time Markov chains. Finally, we prove some compositionality results and show the applicability of our theory through examples.

A new algorithm is given to find product-form solutions for the joint equilibrium probabilities in a class of synchronized Markov processes. This is based on, and proved by, multiple applications of the Reversed Compound Agent Theorem (RCAT) and can describe multi-way synchronizations (seen as chains of pairwise synchronizations) that occur in a prescribed order. The length of the sequence is unbounded but finite with probability 1. Several applications are given to illustrate the methodology, which include various modes of resets in queueing networks with negative customers. In particular, it is shown that there is a type of reset that can propagate further transitions in a chain actively. Furthermore, a number of completely new product-form models, for example, where the transitions in a chain are non-homogeneous, are given.

Quantitative analysis of computer systems is often based\ud on Markovian models. Among the formalisms that are used\ud in practice, Markovian process algebras have found many\ud applications, also thanks to their compositional nature that\ud allows one to specify systems as interacting individual au-\ud tomata that carry out actions. Nevertheless, as with all\ud state-based modelling techniques, Markovian process alge-\ud bras suer from the well-known state space explosion prob-\ud lem. State aggregation, specically lumping, is one of the\ud possible methods for tackling this problem. In this paper we\ud revisit the notion ofMarkovian bisimulation which has previ-\ud ously been shown to induce a lumpable relation in the under-\ud lying Markov process. Here we consider the coarser relation\ud of contextual lumpability, and taking the specic example of\ud strong equivalence in PEPA, we propose a slightly relaxed\ud denition of Markovian bisimulation, named lumpable bisim-\ud ilarity, and prove that this is a characterisation of the notion\ud of contextual lumpability for PEPA components. Moreover,\ud we show that lumpable bisimilarity induces the largest con-\ud textual lumping over the Markov process underlying any\ud PEPA component. We provide an algorithm for lumpable\ud bisimilarity and study both its time and space complexity