Reliability Modeling of Three-State (k,ℓ ) -out-of-n Systems Subject to Marshall–Olkin Shocks

In this paper, we consider a (k,ℓ ) -out-of-n system with three states: up, partial performance, and down. The system has n binary components and is in up state if at least (n-k+1) out of its components work. The state of partial performance is defined when the number of working components is at least (n-ℓ +1) and less than (n-k+1) ; k<ℓ . It is assumed that the system is subject to Marshall–Olkin type of shocks where there are n shocks, each of them affects one component and destroys it, and there is one shock that affects all components and destroys simultaneously all of them. Under this scenario, the joint reliability function of state lifetimes and the corresponding singular and absolutely continuous parts are obtained. For the system with the age of t, the mean residual lifetimes of the system states are explored. Some other aging, stochastic, and dependence properties of the system states are investigated, too. We also extend the model for the case where the system is subject to Marshall–Olkin type of shocks in which the arrived shocks may affect one, two,..., or all components and destroy them. Some illustrative examples are also provided to show the applications of the proposed model.