The GI/M/1 Queue With Setup Time And Working Vacations

Vacation queue is the expansion of the classical queueing theory. During the last two decades, the vacation queueing systems have been investigated extensively developed into a study direction of distinguishing features and formed the theoretical framework whose core is stochastic decomposition. In the models with various vacation policies, the server completely stops original service in the vacation period, but he can take the assistant work. The research results of the vacation queues have been applied to various fields, such as the computer systems, communication networks, flexibility manufacture system, asynchrinously transfer mode and electronic commerce.Recently, Servi and Finn first introduced a class of semi-vacation policy: the server can take the original work at a lower speed during a vacation period rather than stopping completely. Such a vacation is called a working vacation (WV). The essence of the working vacation policy is that, when the number of customers is less relatively, a lower speed period is established to economize the operational cost in the system. Servi and Finn use classical methods to obtain the PGF of the number of customers in the system and the LST of the total time in steady state, and applied these results to performance analysis of gateway router in fiber communication networks.In the thesis, firstly, we study a GI/M/1 queue with set-up time and vacations. we explicitly describe the models, using matrix-geometric solution method, we obtain the distributions of the steady-state queue length and its stochastic decomposition and the distributions of the waiting time. Meanwhile, we discuss two special cases to explain the relationship between our model and previous work, and their steady-state queue length and stochastic decomposition.Secondly, we study a GI/M/1 queue with set-up time and working vacations. Simally, we explicitly describe the models, using matrix-geometric solution method, we obtain the distributions of the steady-state queue length and its stochastic decomposition. Meanwhile, we get the distributions of the waiting time in the stationary state. Then, we plot some figures obviously presenting the mutual effects between the system parameters and expected queue length.