A Stochastic Hybrid Systems framework for analysis of Markov reward models

• Published in: Reliability engineering & system safety 2014-03, Vol.123, p.158-170
• Main Authors: Dhople, S.V., DeVille, L., Domínguez-García, A.D.
• Format: Article
• Language: English
• Subjects: Applied sciences; Differential equations; Exact sciences and technology; Hybrid systems; Mapping; Markov availability models; Markov models; Markov reliability models; Mathematical analysis; Mathematical models; Mathematics; Operational research and scientific management; Operational research. Management science; Performability analysis; Probability and statistics; Probability theory and stochastic processes; Reliability theory. Replacement problems; Reward models; Sciences and techniques of general use; Self-propagating synthesis; Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications); Stochastic hybrid systems; Stochasticity
• ISSN: 0951-8320; 1879-0836
• DOI: 10.1016/j.ress.2013.10.011

In this paper, we propose a framework to analyze Markov reward models, which are commonly used in system performability analysis. The framework builds on a set of analytical tools developed for a class of stochastic processes referred to as Stochastic Hybrid Systems (SHS). The state space of an SHS is comprised of: (i) a discrete state that describes the possible configurations/modes that a system can adopt, which includes the nominal (non-faulty) operational mode, but also those operational modes that arise due to component faults, and (ii) a continuous state that describes the reward. Discrete state transitions are stochastic, and governed by transition rates that are (in general) a function of time and the value of the continuous state. The evolution of the continuous state is described by a stochastic differential equation and reward measures are defined as functions of the continuous state. Additionally, each transition is associated with a reset map that defines the mapping between the pre- and post-transition values of the discrete and continuous states; these mappings enable the definition of impulses and losses in the reward. The proposed SHS-based framework unifies the analysis of a variety of previously studied reward models. We illustrate the application of the framework to performability analysis via analytical and numerical examples.