Reliability modeling for a discrete time multi-state system with random and dependent transition probabilities

In a random environment, state transition probabilities of a multi-state system can change as the environment changes. Thus, a dynamic reliability model with random and dependent transition probabilities is developed for non-repairable discrete-time multi-state system in this article. The dependence among the random state transition probabilities of the system is modeled by a copula function. By probability argument and random process theory, we obtain explicit expressions of some reliability characteristics and joint survival function of random time spent by the system in all working states (partially and completely working states). A special case is considered when the state transition probabilities are dependent random variables with power distribution, and the dependence structure is modeled by Farlie-Gumbel-Morgenstern copula. Numerical examples are also presented to demonstrate the developed model and perform a comparison for the models with random and fixed transition probabilities.