The Single Working Vacation Queue With Batch Arrival And Set-up Time

In recent years, with the high-tech developments in computer communicationnetwork and other fields, a number of complex system designs and control issues areproposed. Vacation queuing problems caused by these practical problems are varying. Thepaper introduces set-up time and batch arrival policy based on the previous study, andresearches on continuous-time and discrete-time queuing model with single workingvacation (SWV). These two models enrich the research contents of working vacationqueuing system. They are also new and promotion working vacation queuing models.This paper uses matrix analysis method to get the PGF expression of steady-state.The paper provides the stochastic decompositions by the classical random decompositionmethod. The upper and lower bounds of the mean waiting time are obtained respectivelyby using the two parameters addition theorem of condition Erlang distribution and the twoparameters addition theorem of condition negative binomial distribution.Firstly, the paper briefly introduces the random decomposition, the two parametersaddition theorem of condition Erlang distribution and condition negative binomialdistribution.Secondly, this paper gives the analysis of a continuous-time M^[x]/M/1(SWV) withset-up time and batch arrival. The PGF and the stochastic decomposition of steady-statequeue length are obtained. The upper and lower bounds of steady-state waiting times aregiven in the under Laplace transform order. This paper also gives the mean queue length,the upper and lower bounds of the mean waiting time, the mean sojourn time.Thirdly, a discrete-time Geo^[x]/Geo/1(SWV) with set-up time and batch arrival isanalyzed. The PGF and the stochastic decomposition of steady-state queue length arepresented. The upper and lower bounds of steady-state waiting times are given in themoment generating function order. The mean queue length, the upper and lower bounds ofthe mean waiting time, the mean sojourn time are also obtained.Finally, some numerical examples of the models are presented by Matlab software.This paper gives the analysis of the effect of system parameters on steady-state index and the analysis of the effect of same parameters on different model, verifies the feasibility ofapplication of the models and rationality of description.