Retrial Queues With Second Optional Service

Retrial queueing systems are characterized by the feature that the arrivals who find the server unavailable are obliged to join the retrial queue (orbit) and to try again for their requests in random order and at random intervals. It is a branch of queueing system developed quickly in the late of the twenty century, which has attracted the extensive attention of numerous experts and scholars. According to the data, there are about 65% literatures on retrial queue published between 1990-1999, and the international workshops on retrial queues are held extensively in the last ten years.Retrial queueing systems have been widely used to model many problems in telephone switching systems, telecommunication networks and computer systems. For example, in a call center, if the calling customer get a busy signal, he may retry after some random time until get the service needed. We can appropriately control the system, reduce customer waiting time,and improve service quality and efficiency by analyzing the performance of the whole system.According to practical background, this paper establishes and resolves three retrial queue models, that is, an M/G/1 retrial queue with second multi-optional service and server breakdowns, an M/G/1 retrial queue with second multi-optional service, feedback and unreliable server, and an M/G/1 stochastic clearing system with repeated attempts and server breakdowns.In these models, we analyze two queueing disciplines in the retrial orbit. One is the retrial customers are queued in the orbit in accordance with a first-come-first-served (FCFS) discipline; The other is that there is no waiting space in the orbit and therefore it needs not to queue, any retrial customer is independent of others. We also consider three kinds of situations of server breakdowns: Firstly, the normal breakdowns, that is the server fails at an fixed rate; Secondly, the abnormal breakdowns, that is, the server fails immediately after an arrival of catastrophe; Thirdly, the starting failures, that is the server fails when the customer tries to turn on it to get service. Using a supplementary variable method, we get some indexes for both queueing and reliability measures of interest, such as the mean queue length in retrial orbit, availability, failure frequency and the reliability function of the server.