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Non Markovian queue with two types service optional re-service and general vacation distribution. (English) Zbl 1356.60157

Summary: We consider a single server batch arrival queueing system, where the server provides two types of heterogeneous service. A customer has the option of choosing either type 1 service with probability \(p_1\) or type 2 service with probability \(p_2\) with the service times follow general distribution. After the completion of either type 1 or type 2 service a customer has the option to repeat or not to repeat the type 1 or type 2 service. As soon as the customer service is completed, the server will take a vacation with probability \(\theta\) or may continue staying in the system with probability \(1-\theta\). The re-service periods and vacation periods are assumed to be general. Using supplementary variable technique, the Laplace transforms of time dependent probabilities of system state are derived and thus we deduce the steady state results. We obtain the average queue size and average waiting time. Some system performance measures and numerical illustrations are discussed.

MSC:

60K25 Queueing theory (aspects of probability theory)
90B22 Queues and service in operations research
68M20 Performance evaluation, queueing, and scheduling in the context of computer systems