Non-Homogeneous 1-Out-of-${N}$ Warm Standby Systems With Random Replacement Times

In standby systems, when an online working element fails, a replacement procedure is initiated to activate a standby unit which will take over the mission task to sustain the system function. Existing works on standby systems have mostly assumed that such replacement procedure takes a negligible or fixed amount of time. This assumption is not practical in many real-world systems, where the replacement procedure can take times that are random and different for different standby elements. This paper makes novel contributions by considering the effects of the random replacement times in analyzing and optimizing 1-out-of- N: G non-repairable warm standby systems. The system elements are not necessarily identical; different elements can have different time-to-failure distributions, different performance levels, and different replacement time distributions. The system is considered failed if the elements cannot complete the specified amount of work (mission task) within the maximum allowed mission time. A numerical algorithm is first proposed to simultaneously evaluate the mission reliability and expected mission completion time of the considered warm standby system. Influences of different mission and element parameters on the mission reliability and expected completion time are investigated. It is revealed that the considered warm standby systems exhibit non-coherent behavior as mission reliability may increase with the decrease of an element's reliability. Due to heterogeneity of the system elements, the order in which the elements are initiated and replaced can affect the mission reliability and actual completion time significantly. Therefore, based on the suggested numerical evaluation algorithm, we further formulate and solve the optimal element replacement sequencing problem for the considered warm standby system subject to random replacement times. Examples are given to demonstrate the considered problems and proposed methodology.