Semi-Markov model of a restorable system with elementwise time redundancy

A model that describes the operation of a multi-element restorable system is constructed. After failure, each element of the system remains functionally operable due to an immediately refilled time redundancy. An element is considered to be failed if its restoration time exceeds the time redundancy. In this case, all elements that have a functional link with a failed element are not disconnected. All random variables describing the evolution of the system over time are assumed to have general distributions. This system is studied using the framework of semi-Markov processes with a discrete-continuous state space. The stationary distribution of the embedded Markov chain is found by solving the system of integral equations. Formulas for calculating the stationary availability and mean stationary sojourn times of the system in the operable and failure states are obtained. The stationary characteristics of the system are expressed through the stationary availabilities of its elements and the structural function of the system. An illustrative example of a 3-out-of-4 system is given, and its characteristics are calculated depending on different time redundancies of the elements.