The Queuing Systems With Breakdown In Vacation

The vacation queueing system is a typical model of queuing theory, and it has beenwidely used in machine processing systems, computer systems and communicationsystems. In the classical vacation queueing, the model supposes that service station carryout other auxiliary work without any breakdowns on vacation, but the service station issubject to breakdown in the adjuvant work while in the actual production of machines.This paper studies a vacation queueing model, where the service station is subject tobreakdown while in operation or vacation. Finally, the paper studies a vacation queueingmodel with set-up and breakdown, so that such problems have been further discussed.Firstly, this paper studies a single desk vacation queueing model with breakdown inwork or vacation, where the repair time, uninterrupted service time and length of eachvacation follow exponential distributions. Steady-state probability equations are obtainedby Markov process method, and paper obtains the mean queue length by using thegenerating function. Finally paper gives some numerical analysis.Secondly, this paper studies an M/PH/1vacation queueing model with breakdown inoperation or vacation, where service, vacation, and repair times follow phase-typedistributions. Using the quasi-birth-and-death process method, paper obtains the explicitexpression of the rate matrix and the boundary probability vectors. The expected queuelength of the system is presented. Finally, paper gives the numerical example of severalspecial case of PH distribution.Finally, paper studies a vacation model with set-up time and breakdown while inoperation or vacation. Paper derives the equilibrium condition of the system and thematrix-geometric solution of the steady-state probability vectors. The expected queuelength of the system is presented. By MATLAB calculation, paper gives some numericalanalysis for the results. Paper also studies a vacation model of following PH distributionwith set-up time. Using the quasi-birth-and-death process method, we obtain the explicitexpression of the rate matrix and the boundary probability vectors. The expected queuelength of the system is presented.