Semi-Markov Approach to Reliability Evaluation in Two-Unit Standby Repairable System with Priority in Use

Abstract

Modern systems incorporate an increasing number of components performing complex functions, necessitating the development of adequate mathematical models for their reliability. Particular attention should be paid to mission-critical units, whose failure may lead to the breakdown of the entire system. Such priority units require enhanced reliability to ensure stable system operation and mitigate the consequences of potential failures. The paper deals with the stochastic analysis of a two-unit repairable system model with cold standby, priority in use and non-identical units. We constructed its semi-Markov model with a discrete-continuous phase space, eliminating certain limitations and yielding generalized results by using probability distribution functions of system unit's failure-free operation and repair times in general form. Applying the method of successive approximations, the stationary distribution of the embedded Markov chain was derived, on the basis of which the stationary reliability characteristics were determined. Under the condition of “fast” restoration of system units, approximate formulas for determining reliability characteristics were obtained. The error of the obtained approximate formulas was assessed in comparison with exact results for exponential distributions of system unit's failure-free operation and repair times. Examples are provided.