Joint interval reliability for Markov systems with an application in transmission line reliability

We consider Markov reliability models whose finite state space is partitioned into the set of up   states UU and the set of down   states DD. Given a collection of k   disjoint time intervals Iℓ=[tℓ,tℓ+xℓ]Iℓ=[tℓ,tℓ+xℓ], ℓ=1,…,kℓ=1,…,k, the joint interval reliability   is defined as the probability of the system being in UU for all time instances in I1∪⋯∪IkI1∪⋯∪Ik. A closed form expression is derived here for the joint interval reliability for this class of models. The result is applied to power transmission lines in a two-state fluctuating environment. We use the Linux versions of the free packages Maxima and Scilab in our implementation for symbolic and numerical work, respectively.