On some generalization of the ROCOF for general multi-state systems with applications

The rate of occurrence of failures (ROCOF) is an important indicator in the reliability analysis of complex random systems. In general, it gives information on whether there are a lot of failures or only a few within a given time frame. The ROCOF was studied for Markov processes (MP) in [3] and [5]. Extension to semi-Markov processes (SMP) was advanced in [4], where the authors derived a formula for the ROCOF and a statistical estimator with an investigation of its asymptotic properties. Recently, a generalization of the ROCOF, named the ROCOF of higher order, was calculated for MP in [1] and successively extended by [2] to a SMP framework with mixed probability distribution for the initial law of the system, taking into account the possible random starting from any state of the system with any duration. In this study, we provide further generalizations of the ROCOF of higher order motivated by practical applications, especially in mathematical finance. The results include previous investigations as particular cases and demonstrate the usefulness of the indicator not only for reliability purposes but for more general scopes beyond the reliability framework. Sensitivity analysis of random systems may convey important information on the dynamical properties of the system. In this paper, we determine the effects of parameters’ perturbation on two dynamic