Dynamic group instantaneous replacement policies for unreliable Markovian service systems

This paper considers the group replacement problems for a service system with multiple independent operating servers and a single Markovian queue. The servers are unreliable with identically exponentially distributed failure times and the repair time is assumed negligible. This proposed model is formulated as a continuous time Markov decision process. Two classes of group replacement policies are developed where the decision to trigger the repair processes dynamically depends on the number of customers and the number of operating servers in the system. For the theoretical analysis, it is proved that the optimal group replacement policies have a threshold structure following several significant mathematical properties. Finally two numerical examples are given to illustrate the main theoretical results, and also present two classes of optimal replacement policies for finding the related threshold of number of customers and the number of operating servers to trigger the group replacement.