The M/M/1 With N-Policy And M/PH/1 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 thesis.In the thesis, we will study an M/M/1 queue with working vacations and N-policy, M/M/1(N-WV) in short and an M/PH/1 queue with multiple working vacations respectively. Then, we explicitly describe the models, and give out the infinitesimal generator of the processes. Using quasi-birth-and-death process and matrix-geometric solution method, we obtain the stationary indices of the models. It is well known that stochastic decomposition results have essential sense in a classical vacation queue where the server completely stops service during a vacation. This thesis indicates that the queue with working vacations where the server serves the customers in a lower rate during a vacation has similar decomposition rule: conditional stochastic decomposition. Thus, we get the conditional stochastic decomposition structures of queue length and waiting time in the stationary state and obtain the distributions of additional queue length and additional delay. Furthermore, we make examples to explain how the model is applied in manufacture management, then, we 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.