On M[x]G1G21 queue with optional re-service

Department of Statistics, Faculty of Science, Yarmouk University, Irbid, Jordan
Applied Mathematics and Computation (Impact Factor: 1.35). 01/2004; 152(1):71-88. DOI: 10.1016/S0096-3003(03)00545-9
Source: DBLP

ABSTRACT We study a single server queue with batch arrivals and two types of heterogeneous service with different general (arbitrary) service time distributions. The server provides either type of service to customers, one by one, on a first come, first served basis. Just before a service starts, a customer has the option to choose either type of service after completion of which the customer may leave the system or may opt for re-service of the service taken by him. We obtain steady-state results in explicit and closed form in terms of the probability generating functions for the number of customers in the queue and the system, the average number of customers and the average waiting time in the queue as well as the system. Some special cases of interest are discussed and some known results have been derived. A numerical illustration is provided.

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    ABSTRACT: This paper deals with the steady-state behaviour of an M/G/1 queue with an additional second phase of optional service subject to breakdowns occurring randomly at any instant while serving the customers and delayed repair. This model generalizes both the classical M/G/1 queue subject to random breakdown and delayed repair as well as M/G/1 queue with second optional service and server breakdowns. For this model, we first derive the joint distributions of state of the server and queue size, which is one of chief objectives of the paper. Secondly, we derive the probability generating function of the stationary queue size distribution at a departure epoch as a classical generalization of Pollaczek–Khinchin formula. Next, we derive Laplace Stieltjes transform of busy period distribution and waiting time distribution. Finally, we obtain some important performance measures and reliability indices of this model.
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