Passage-time computation and aggregation strategies for large semi-Markov processes

High-level semi-Markov modelling paradigms such as semi-Markov stochastic Petri nets and process algebras are used to capture realistic performance models of computer and communication systems but often have the drawback of generating huge underlying semi-Markov processes. Extraction of performance measures such as steady-state probabilities and passage-time distributions therefore relies on sparse matrix–vector operations involving very large transition matrices. Previous studies have shown that exact state-by-state aggregation of semi-Markov processes can be applied to reduce the number of states. This can, however, lead to a dramatic increase in matrix density caused by the creation of additional transitions between remaining states. Our paper addresses this issue by presenting the concept of state space partitioning for aggregation. Aggregation of partitions can be done in one of two ways. The first is to use exact state-by-state aggregation to aggregate each individual state within a partition. However, we discover that this approach still causes matrix density problems, albeit on a much smaller scale compared to nonpartition aggregation. A second approach to the aggregation of partitions, and the one presented in this paper, is atomic partition aggregation. Inspired by a technique used in passage-time analysis, this collapses a whole partition into a small number of semi-Markov states and transitions. Most partitionings produced by existing graph partitioners are not suitable for use with our atomic partition aggregation techniques, and we therefore present a new deterministic partitioning method which we term barrier partitioning. We show that barrier partitioning is capable of splitting very large semi-Markov models into a number of partitions such that first passage-time analysis can be performed more quickly and using up to 99% less memory than existing algorithms.