Optimal Control of an N-Policy Two-Phase M/EK/1 Gated Queuing System with Server Startup, Breakdowns and Delayed Repair

This paper deals with the economic behavior of an Npolicy M/Ek/1 queue with server startup, two-phases of compulsory service with gating and unreliable server, which consists of a breakdown period and a delay period. The customers arrive individually according to the Poisson process and all the waiting customers receive batch service at a time in the first phase and proceed to the second phase for individual service. The server is turned off each time the system is emptied. As and when the total number of customers in the system reaches the threshold N (N≥1), the server immediately turned on but is temporarily unavailable to serve the waiting customers. The server requires time for the preparatory work before starting the service. As soon as the startup period is over, the server starts the batch service followed by the individual service to all customers in the batch. The customers who arrive during batch service are not allowed to enter the batch which is in service This criterion is called gating. It is assumed that the server may breakdown while serving in the individual queue according to the Poisson process and there may be a delay in repair due to non availability of the repairing facility. It is assumed that the startup times, the batch service times, the delay times and the repair times follow exponential distribution. Explicit expressions for the steady state distributions of the number of customers in the system are obtained and also the expected system length is derived. This paper also deals with the total expected cost function which is developed to determine the optimal threshold of N at a minimum cost. Sensitivity analysis is carried out with numerical illustrations.