An M~x/G/1 Queue With Multiple Adaptive Vacation And Set-Up Time

Vacation queue is the expansion and development of the classic queuing theory, which was first studied by Levy and Yechiali in 1975. In 1980s, vacation queue had been developed into a study direction with distinguishing features, and formed a basic theoretical framework, whose core was stochastic decomposition. Now it has been widely used in computer systems, communic- ation systems, manufacturing systems, electronic commerce and manufacture line etc. Queues with"higgling, consultation"have been widely used in manu- facture and practice. In this paper, we deal with the M X /G/1 queue with set-up time and multiple adaptive vacation, which is a complete new vacation queuing model.By means of imbedded Markov chain, this thesis offers transition probab- ility matrix of the system and further deduces the steady-state queue length and other corresponding steady state quota, thus stabling the theory of stochastic decomposition of the steady-state queue length and the basic technique of the system optimization. According to this researching clue, this thesis is divided into four parts:Firstly, we comprehensively introduce the basic knowledge concerning the subject and analyze the historic and current state of vacation queues. And pointed out that the model examined by the theoretical and practical siginficance.Secondly, we explicitly describe the model that has been studied; With the Markov chain imbedded in the time that the first service is finished . we give the transition probability matrix of the system; and compare it with the classic M X /G/1 queue , which paves the way for the analysis of results.Thirdly, we analyze the existent condition of the steady distribution; then deduce the steady-state queue length and the Laplace transform of the steady- waiting time, and deduce the average steady-state queue lengh and the average steady-state waiting time. By the stochastic decomposition of the above results, we obtain the additional steady-state queue length and the Laplace transform of the additional steady-state waiting time, and deduce the average additional steady-state queue lengh and the average additional steady-state waiting time of the M X /G/1 queue with set-up time and multiple adaptive vacation.Fourthly, we give other factors of steady state such as busy period, busy circulation,whole vacation, on-line period etc.