The NN-policy for an unreliable server with delaying repair and two phases of service

This paper deals with an MX/G/1MX/G/1 with an additional second phase of optional service and unreliable server, which consist of a breakdown period and a delay period under NN-policy. While the server is working with any phase of service, it may break down at any instant and the service channel will fail for a short interval of time. Further concept of the delay time is also introduced. If no customer arrives during the breakdown period, the server becomes idle in the system until the queue size builds up to a threshold value N(≥1). As soon as the queue size becomes at least NN, the server immediately begins to serve the first phase of regular service to all the waiting customers. After the completion of which, only some of them receive the second phase of the optional service. We derive the queue size distribution at a random epoch and departure epoch as well as various system performance measures. Finally we derive a simple procedure to obtain optimal stationary policy under a suitable linear cost structure.