Autoreversibility: Exploiting Symmetries in Markov Chains

Abstract

The computation of the steady-state distribution of Continuous Time Markov Chains (CTMCs) may be a computationally hard problem when the number of states is very large. In order to overcome this problem, in the literature, several solutions have been proposed such as the reduction of the state space cardinality by lumping, the factorization based on product-form analysis and the application of the notion of reversibility. In this paper we address this problem by introducing the notion of auto reversibility which is defined as a symmetric co inductive relation which induces an equivalence relation among the chain's states. We show that all the states belonging to the same equivalence class share the same stationary probabilities and hence the computation of the steady-state distribution can be computationally more efficient. The definition of auto reversibility takes inspiration by the Kolmogorov's criteria for reversible processes and hence requires to test a property on all the minimal cycles of the chain. We show that the notion of auto reversibility is different from that of reversible processes and does not correspond to other state aggregation techniques such as lumping. Finally, we discuss the applicability of our results in the case of models defined in terms of a Markovian process Algebra such as the Performance Evaluation Process Algebra.