Research On Discrete-time Retrial Queueing Models With Vacations And Their Applications

Queueing models with retrials or vacations have been extensively studied and appliedto many service systems such as flexible manufacturing, inventory management, callcenter, and electronic commerce. However, there are few queueing models combiningboth retrial phenomenon and server’s vacations, especially in the discrete-time situation.In recent years, queueing models with both retrial phenomenon and server’s vacations getmore and more attentions due to the fact that there are many practical service systems withretrial phenomenon and server’s vacations. This dissertation is devoted to study severalnew queueing models which combine both retrial phenomenon and server’s vacations.Some performance measures of the models are obtained by using the method of Markovprocess theory, supplementary variable technique and probability generating functionsmethod. In addition, the stochastic decomposition property of the system size is proved.The effects of model’s parameters on some performance measures are also investigatedthrough numerical examples. This research may provide new research ideas andtheoretical basis for some practical service systems with retrial customers and server’svacations. The main work of this dissertation is summarized as follows:Firstly, the concept of second multi-optional service and exhaustive single vacationare introduced into the discrete-time Geo/G/1retrial queue. For this model, someperformance measures of the system are obtained. In addition, the stochasticdecomposition result for the system size is proved. The steady-state probability generatingfunctions of the corresponding continuous-time M/G/1queueing model with exhaustivesingle vacation and second multi-optional service is also derived.Secondly, a discrete-time Geo/G/1retrial queues with general retrial times andexhaustive J vacation policy is investigated where J vacation is a generalization of singleand multiple vacations. For this model, we derived the stability condition of the Markovchain and the joint probability generating functions of the orbit size and state of the server.The stochastic decomposition property for the system size is also proved. The relationshipbetween our model and the corresponding continuous-time model is investigated. Moreover, the effects of model’s parameters on some performance measures of the systemare investigated numerically.Thirdly, two discrete-time Geo/G/1retrial queueing models with different type ofvacations and breakdowns are studied, respectively. In the first model, a discrete-timeretrial queue with Bernoulli vacation and geometric distributed lifetime is considered. Inthe second model, a retrial queue with server’s starting failures and exhaustive J vacationsis studied. By using the supplementary variable technique, some main performancemeasures of the two models are obtained. In addition, the effects of server’s breakdownson performance measures of the two models are given numerically.Fourthly, the retrial queueing model with priority customers and exhaustive singlevacation is studied. For this model, some performance measures such as the probabilitygenerating functions and mean number of the two types of customers are obtained,respectively. In addition, we prove the stochastic decomposition property for the systemsize, and obtain the probability generating function of the system size for thecorresponding continuous-time model.Fifthly, a discrete-time retrial queue with two types of vacations is proposed. It isassumed that the server can also take non-exhaustive random vacation in addition toexhaustive single vacation. The stability condition and some performance measures of themodel are obtained. Moreover, the effects of no-exhaustive random vacation on the mainperformance measures of the model are given through some numerical examples.Finally, the discrete-time retrial queueing models with vacations are used to analyzethe performance measures of electronic commerce system with priority scheduling service.On the basis of research on electronic commerce model with priority scheduling service,the electronic commerce model with mixed priority scheduling service is proposed. Thetheoretical results in the research can be applied to analyze the performance measures ofthis model. In addition, the effects of parameters on the performance measures areinvestigated by some numerical examples.