Assessment of a generalized discrete time mixed [formula omitted]-shock model for the multi-state systems

Abstract

In this paper, the life behavior of a shock model is studied, when the external shocks occur according to a binomial process whose interarrival times between successive shocks follow a geometric distribution. The system transits into a lower partially working state upon the occurrence of each interarrival time between two successive shocks less than a critical threshold, say δ. The system fails when k out of interarrival times between two successive shocks are less than δ, or the magnitude of the shock is larger than the other critical level, say γ. Such a model creates a multi-state system having a number of different states. The probability mass functions of system’s life time, the time spent by the system in a perfectly functioning state, and the total time spent by the system in partially working states are derived for the proposed model. The corresponding probability generating functions are also derived. The case of Markov shock occurrences is also studied. To illustrate the model studied in this paper, some numerical examples are also considered. Finally, an application in insurance is also provided.