The Finite Buffer And Discrete Time Working Vacation Queues

During the last two decades, the vacation queueing systems have been investigated extensively and developed 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 and production management.Recently, Servi and Finn (2002) 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). In the classical vacation queueing models, the server doesn't continue on the original work during the vacation period and such policy may cause the loss or dissatisfaction of the customers. So, the working vacation is more reasonable than the classical vacation policies in some sense. 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(z transform) 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. Therefore, it is necessary to study two continuous-time working vacation queue models in this paper.In the thesis, we will study an finite buffer discrete time Geo/Geo/1/N with single working vacation, an finite buffer discrete time Geo/Geo/1/N with multiple working vacation and finite buffer discrete time Geo/Geo/1/N with multiple working vacation and set-up period respectively. Then, we explicitly describe the models, Using quasi-birth-and-death chain, we give out the transition probability matrix and obtain the stationary distributions of the models. Furthermore, we get distributions of the average number of customers in the system and loss probability and the PDF of the waiting time in the stationary state. At last, we programme with MATLAB and plot some figures obviously presenting the mutual effects between the system parameters and stationary indices. It is found that higher system efficiency achieves by exiguously adjusting the system parameters in a given area. This model has provided academic gist for scientific manufacture management.