A Two-Phase Service System with Bernoulli Vacation Schedule, Setup Time and N-policy for an Unreliable Server with Delaying Repair

This paper deals with an M/G/1 queueing system with two phases of service and Bernoulli vacation schedule for an unreliable server, which consist of a breakdown period and a delay period, under N-policy and a random setup time. 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 N, the server immediately begins to serve the waiting customers in two successive phases of service. The first phase of service is followed by a second phase of service, after the completion of which, the server may take a vacation or may remain in the system to serve the next unit, if any. We derive the queue size distribution at different points of time, delay busy period distribution and optimal N-policy.