• OPTIMAL STRATEGY ANALYSIS OF AN N-POLICY TWO-PHASE MX/EK/1 GATED QUEUEING SYSTEM WITH SERVER STARTUP AND BREAK DOWNS

Abstract

This paper deals with the optimal operation of a single removable server in an M^X/E_k/1 two-phase queueing system with gating, server startup and unpredictable breakdowns under N-policy.  Arrivals occur in batches of random size X according to a compound Poisson process and waiting customers receive batch service all at a time in the first phase and proceed to the second phase to receive individual service.  After providing the second phase of service, the server returns to the first phase to see if new customers have arrived.  If there are any waiting customers, the server starts the cycle by providing them batch service followed by individual service.  If no customer is waiting, the server leaves for a random period of vacation.  As soon as the queue length reaches a threshold value N (N1) the server is turned on and is temporarily unavailable to serve the waiting batch of customers.  Because, when it comes back from vacation it needs a startup time before the batch service commences in the first phase.  While the server is working with the second phase of individual service, it may breakdown at any instant and is immediately repaired.  In the first phase of service, the batch includes only those customers who are already in the queue.  This criterion is called gating.  Explicit expressions for the steady state distribution of the number of customers in the system and hence the expected system length is derived.  The total expected cost function is developed to determine the optimal threshold of N at a minimum cost.  Numerical experiment is performed to validate the analytical results.  The sensitivity analysis has been carried out to examine the effect of different parameters in the system.