ROCOF of higher order for semi-Markov processes

In this paper, we study the rate of occurrence of failures (ROCOF) of higher order for con- tinuous time semi-Markov processes (SMP). This indicator gives information on whether there are a lot of failures or only a few within a time interval. It also considers the relative positioning of tuples of failures in time. Here, we consider SMP with a mixed probability distribution for the initial law of the system taking into account the possible random starting from any state with any duration. Furthermore, under suitable assumptions on the transition rates, we determine an explicit formula for the ROCOF of higher order and we recover as particular cases previous results obtained in the literature. A numerical example demonstrates the possibility of using this index in real applications. A possible application in the field of mathematical finance is also discussed.