Abstract
In this paper, the life distribution behavior of a generalization of the mixed $$\delta$$-shock models in the multi-state systems is studied. In this model, the k out of interarrival times between two successive shocks with a magnitude less than $$\delta$$ have a disaster result on the system which causes a complete failure. In addition to this event, another factor called the magnitude of shock causes the failure of the system, such that if the magnitude of a shock is greater than another critical threshold $$\gamma$$, then the system fails. Such model create a multi-state system with a number of different states. The survival functions of the lifetime, the time spent by the system in a complete working state, and the total time spent by the system in partially working states are derived and the corresponding first two moments are also computed. An application in industry is analyzed to illustrate the proposed methodology. A simulation study is also presented to illustrate the behavior of the survival functions.
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