Stability analysis of a standby system with an unreliable server and switching failure

Abstract This paper is devoted to analysing stability of a standby system with an unreliable server and switching failure, where both the time-to-repair of units and the time-to-repair of server follow general distributions. By employing $C_0$ semigroup theory, we show that time-dependent solution of the system converges to steady-state solution under a rather mild condition. Further, if component’ repair rate and server’s repair rate satisfy a reasonable assumption, then the system is exponentially stable. Based on the above theoretical foundation, some steady-state reliability indices are obtained and total expected profit function per unit time is developed. Finally, a sensitivity analysis with numerical illustrations is also provided.