Detecting and Exploiting Symmetry in Discrete-state Markov Models

Abstract

Dependable systems are usually designed with multiple instances of components or logical processes, and often possess symmetries that may be exploited in model-based evaluation. The problem of how best to exploit symmetry in models has received much attention from the modeling community, but no solution has garnered widespread support, primarily because each solution is limited in terms of either the types of symmetry that can be exploited or the difficulty of translating from the system description to the model formalism. We propose a new method for detecting and exploiting model symmetry in which 1) models retain the structure of the system, and 2) all symmetry inherent in the structure of the model can be detected and exploited for the purposes of state-space reduction. Composed models are constructed from models through specification of connections between models that correspond to shared state fragments. The composed model is interpreted as an undirected graph, and results from group and graph theory are used to develop procedures for automatically detecting and exploiting all symmetries in the composed model. A statespace generator which implements these algorithms within Möbius [10] is then presented.12th Pacific Rim International Symposium on Dependable Computing (PRDC'06) 0-7695-2724

On the Detection of Symmetries in Compositional Markov Models

“…We begin our discussion by defining the basic aspects of the model and present our findings using the notation of [10].…”

“…What is essential is that B contains a symmetry with respect to its variables b 1 and b 2 that extends into the composed model since those variables are shared with instances of the same model. In [10], Obal et al describe how to recognize lumpability for state sharing composed models with the help of symmetries that are detected for an associated MCG. We closely follow their notation and concept.…”

“…Two vertices are in the same partition element if: 1) they both correspond to private state variable fragments of the same model and contain the same private state variables, OR 2) they both correspond to public state variable fragments of the same model and contain the same shared state variable, OR 3) they are both connection nodes, OR 4) they both correspond to public state variable fragments of the same model and the two shared state variables commute. The first 3 requirements are the same as originally used in [10]. The fourth condition relaxes the second since it allows for state variables that commute by Def.…”

“…Early work dates back to Sanders and Meyer [8] and stochastic activity networks (SANs), which have two composition operations Rep and Join that allow for generation of a reduced, lumped CTMC. Obal, McQuinn and Sanders generalized the concept to a graph composition [9], [10]. A separate thread of results has been obtained for Stochastic Well-formed Nets (SWNs) [11], [12], where the concept of colors in stochastic Petri nets and a particular way of using colors helps to establish theoretical results that guarantee lumpability on the associated CTMC based on structural conditions observed in an SWN.…”

“…In this paper, we extend the approach of Obal, McQuinn and Sanders [10]. The original idea is to map the compositional structure of a composed model to an undirected graph, the model composition graph (MCG), such that symmetries in the graph relate to lumpable states in the associated CTMC.…”