M/G/1-type Queue With Working Vacation And Threshold

Working vacation refers to that the staffs do not really stop working during the vacation,they provide service with a lower speed.The strategy can reduce the waiting time of the customer effectively,what's more,it can avoid the loss of resources caused by the long queue.However,frequent conversion between working vacation and regular service period will also increase the cost of services.The control of threshold N makes the cost of the system decrease and improves the customer satisfaction.Based on the analysis of classical M/G/1 queuing system with vacations,this paper analyzes M/G/1-type queue with multiple working vacations and threshold.The main contents of this paper are as follows:Firstly,this paper introduces the method of M/G/1-type structure matrix,then a brief introduction about the elements of the discrete time queue and the discussion of the access are presented,which have made a good preparation for the analysis of the latter sections.Secondly,this paper analyzes the M/G/1 working vacation queue with threshold N.Considering the customer's departure time as the reconstruction point of queue length,the transition probability matrix of the embedded Markov chain is obtained.By using M/G/1-type structure matrix method we obtain the recurrence condition of the embedded Markov chain and the steady state distribution of the system.Using the PGF of the queue length,the average queue length and the stochastic decomposition form are derived.What's more,the average waiting time at the steady state is obtained by establishing the relationship between the queue length and the waiting time.In the end,the model is applied to the commercial operation of the bank,the effect of the parameters on the performance of the system is given.Lastly,this paper studies the discrete time Geo/G/1 working vacation queue with threshold N.The queue length of the embedded Markov chain can be analyzed by the M/G/1-type matrix,by supposing that the service time follows geometric distribution,then the effects of different working vacation rate and vacation rate on the system length and customer waiting time are also derived,so the application of accessing the web site is briefly introduced.