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 U and the set of down states D . Given a collection of k disjoint time intervals I ℓ = [ t ℓ , t ℓ + x ℓ ] , ℓ = 1 , … , k , the joint interval reliability is defined as the probability of the system being in U for all time instances in I 1 ∪ ⋯ ∪ I k . 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.