The Discrete-Time Queue With Multiple Working Vacations

Vacation queue is the expansion of the classical queuing theory. During the 1980s, vacation queue has developed into a study direction of distinguishing features, and formed a basic theoretical framework, whose core is stochastic decomposition. It has been widely applied to computer and telecommunication networks, flexibility manufacture system, asynchronously transfer mode and electronic commerce.In recent years, Servi and Finn introduced working vacation policy, which plays an important role in the performance analysis of gateway router in optical networks. This thesis studies the Geo/Geo/1 and GI/Geo/1 discrete time queue with working vacations. Using matrix-geometric solution method, the steady-state queue length and waiting time (or the sojourn time) are obtained, and their stochastic decompositions are proved.Firstly, we briefly introduce the structures of the vacation queue theory and the major studying methods. Furthermore, we summarize the classical queue and the vacation queue theory, respectively, and give a concise introduction on development and meaning for choosing the topic.Secondly, we simply introduce the discrete time queue, and give a concise summary on the theory of quasi-birth and death process and the GI/M/1 type matrix, which provide theoretical preparations for the analysis of later models.Thirdly, we deal with the discrete time Geo/Geo/1 queue with multiple working vacations. By means of matrix-geometric solution method, the distributions of the steady-state queue length and the sojourn time are obtained, and their stochastic decomposition properties are also derived. Furthermore, we analyze the busy period and busy cycle.Finally, we investigate the GI/Geo/1 queue with multiple working vacations, in which the inter-arrivals are generally distributed, and this makes the analysis more difficult. Using matrix-geometric solution method, the distribution and stochastic decomposition of the steady-state queue length are obtained. Meanwhile, some properties of the negative binomial distribution are proved, and with them, the distribution of the waiting time for an arbitrary customer is derived. Furthermore, the conditional stochastic decomposition result of the waiting time for a customer arriving during regular busy period is also analyzed. Finally, with symbol computational function and plotting function of Matlab, we give some numerical analysis in effect of vacation parameter and service rate during vacation period on the mean queue length and mean waiting time.