Stationary Analysis On The M/g/1 Queue With Single Working Vacation

In recent years, with the rapid development of modern technology, the operation in management institutions and networks is more and more complex. In order to meet the need of the theoretical analysis and application problems of the systems, new vacation policies are required constantly to introduce and new queuing models are to established. Under this background, the working vacation(WV) policy is introduced, that is, during the vacation period, the system provides service at a lower rate rather than completely stop service.Based on the classical M/G/1 queuing system with vacations, the single working vacation(SWV) and vacation interruption(VI) policies are introduced, and two new vacation queue models are derived. The new results obtained enrich the contents of the working vacation queueing system. And the M/G/1 vacation queuing systems studied before can be seen as the special examples in the thesis.In the whole thesis, Firstly, we show the development history and current state of queueing theory and working vacation queues, and their applications in the up-to-date technical fields.Secondly, we give a concise review about the methods of vacation queue's study, which provides a preparation for the later model analysis in theory and symbols.Thirdly, we describe the model that has been studied. With the Markov chain imbedded in the time that the customers leave the system, its transition probability matrix is expressed in the Block-Jocabi form. We set up the structure model of M/G/1 queue with single working vacation, and discuss the sufficient and necessary condition in which the queueing system reached equilibrium. By exercising matrix analysis and matrix-geometric solution method, We derive the distribution of the stationary queue length and waiting time, and also obtain the corresponding stochastic decomposition results. Besides, some numerical examples of the model we studied above are obtained by Matlab and table making and the feasibility of application of the model is proved.Finally, based on the M/G/1 queue with single working vacation, the vacation interruption(VI) policies are introduced, and the corresponding stationary results are obtained. By the stochastic decomposition theory, the relation between M/G/1 queue with single working vacation and vacation interruption and the classical M/G/1 queueing system without vacation is derived, and the theory results of working vacation are enriched. Various numerical examples are obtained finally, and the system optimization is faciliteted.