While the majority of existing works on modeling and optimizing standby systems focus on single-phased missions, only a few works consider standby systems with phased-mission requirements, and these works are only applicable to restricted cold-standby configurations with assumptions of negligible replacement times and fixed phase durations. This paper makes new theoretical contributions by suggesting a general model of heterogeneous 1-out-of-  $N$  : G warm standby phased-mission systems (PMS) with dynamic phase durations. The model takes into account diverse, phase-dependent performances and time-to-failure distributions of system elements. It also considers non-zero replacement times specific for activated elements, and for the mission phase when the replacement occurs. Both cold and hot standby PMSs are special cases of the proposed model. An algorithm for evaluating the mission reliability and expected completion time is first presented. The algorithm is applicable to any type of time-to-failure distribution for system elements. The heterogeneous standby PMS can demonstrate non-coherent behavior where the reduction of reliability of some elements can cause the increase of the entire mission reliability. Then it is demonstrated that the sequence of the element activation affects the mission reliability and its expected completion time. Hence the element sequence optimization problem is further formulated and solved for the heterogeneous warm standby PMS. Illustrative examples of mission reliability and expected completion time analysis and optimization are presented.
['Gregory Levitin', 'Liudong Xing', 'Amir Ehsani Zonouz', 'Yuanshun Dai']
Heterogeneous Warm Standby Multi-Phase Systems With Variable Mission Time