A Discrete-Time Vacation Queuing System

Queuing system with working vacations is all emerging research hotspot in queuingsystem in recent years. At the same time, literatures regarding queuing models incorporatingnegative customers are extensively discussed by researchers worldwide. In addition, feedbackis another feature which has been widely studied in queuing system. In this paper, we studyqueuing systems taking together the above mentioned features. The organization of the paperis as follows:The paper deals with Geom/Geom/1feedback queue with working vacations in whichcustomers are either positive or negative. Using Markov chain and matrix geometric solutionmethod, we obtain the transition probability matrix, and then derive the steady-statedistribution for queue length and the mean of the system size of positive customers and theresult of stochastic decomposition of the queue length. And then, simulation numericalexperiment to analysis of several parameters influence on the performance of the system.Firstly, the paper studies the Geom/Geom/1queue with Bernoulli feedback and vacation.Using QBD (quasi birth and death) process and matrix geometric solution, we obtain thesteady-state distributions for the number of customers in the system and the stochasticdecomposition of the steady state queue length.Secondly, we extend the model above, and study the discrete-time Geom/Geom/1queuewith negative customers and server set-up time, multiple working vacation system. Wedescribe the model, by means of the transition probability matrix, we get the steady-statedistribution for queue length and the stochastic decomposition of the queue length and thebusy cycle and the mean values of busy period.Finally, the paper analyzes of the special cases corresponding to each model, theaccuracy of the theory is validated by several numerical examples.