Effects of the transition rate uncertainty on the steady state probabilities of Markov models using interval arithmetic

Abstract

Markov models are often used to assess the reliability or availability of a system. Such assessment generally involves calculations of the steady state probability for each possible state in the system by solving a linear system of simultaneous equations based on the transition rates between states or using ad hoc algorithms such as the state reduction algorithm. Transition rates are usually subject to uncertainty. This paper proposes the use of interval arithmetic as an alternative approach for dealing with uncertainties in the transition rates that are modelled through lower and upper bounds. Interval arithmetic takes into account the uncertainty of all the parameters and is able to provide strict bounds with only one evaluation. The interval version of the state reduction algorithm, with additional procedures to enhance its precision, is proposed as an alternative method to evaluate uncertainty propagation. Several examples illustrate the proposed approach. Results are compared with other classical approaches, such as Monte Carlo simulation or solving a linear system of simultaneous equations.