Markov Chains-Based Approach for Unavailability Modeling in Multi-State Systems

Safety Instrumented Systems (SIS) are critical for ensuring the safe operation of industrial processes. Evaluating their performance, particularly the average Probability of Failure on Demand ($\mathrm{P F D}_{{avg }}$), is essential for system reliability and safety. This paper models the performance of a multi-state SIS subjected to proof tests using the multiphase Markov chain method. A generic approach based on Markov chains is employed for the assessment of the unavailability of multi-state systems. The unavailability of the $1{\mathrm {oo}}1$ structure is used to determine the performance of different subsystems of the complete SIS. A case study is presented demonstrating the impact of proof test parameters, such as the proof test coverage rate and the probability of failure due to the test, on the overall reliability.