On a two-server reliability system with one-server idle below a threshold

In this work, we study a two-server k-out-of-n: G repairable system with identical components. Lifetimes of components follow an exponential distribution with parameter λ. For the repair of failed components, we consider two servers called server1 and server2 and the repair times are independent exponential random variables with parameters μ1 and μ2 (μ1 < μ2) respectively for server1 and server2. It is assumed that server1 alone is active if the number of failed components is below a prescribed threshold say N (>1), whereas both servers are active above that threshold. That is, activating server2 only when the number of failed components reaches the threshold N and when it falls below that threshold, switch off the server. It will again reactivate only on the accumulation of N units. We obtained the system state distributions in finite time and the steady-state, by analysing a continuous time Markov chain and derived several other characteristics of the system such as system reliability, mean time to failure, mean busy period of the servers, etc. Also, we investigated a control problem and proved that the total expected cost of the system is convex in N and hence global minimum exists. Some numerical illustrations also provided.