Optimal loading of series parallel systems with arbitrary element time-to-failure and time-to-repair distributions

Many real-life systems have series parallel structures, where some components or subsystems work in parallel while some others have to work in series or consecutively. This paper models dynamic performance of multi-state series parallel systems with repairable elements that can function at different load levels. Performance (productivity) and time-to-failure distribution of an operating system element depend on its load level. An element, upon failure, can be repaired with repair time obeying a known distribution. The entire system must satisfy a random demand during a fixed mission time or must complete a fixed amount of work. A discrete numerical algorithm is first proposed for evaluating instantaneous availability of a system element with a particular load level, which further defines stochastic process of the element's performance. A universal generating function technique is then used for assessing system performance metrics including expected system performance, expected probability of meeting system demand, expected amount of unsupplied demand over a particular mission time and expected time needed to perform a given amount of work for the considered system. The proposed methodology is applicable to arbitrary types of failure time and repair time distributions. Another original contribution of this work is formulating and solving elements loading optimization problems, which choose elements load levels to achieve one of the following objectives: maximum system expected performance, maximum expected probability of meeting demand during a time horizon, minimum total unsupplied demand during the mission, or minimum completion time for a given amount of work. As demonstrated through a case study of a power station coal transportation system, optimization results can provide effective guidance on optimal operational load of multi-state series parallel system elements.