Promote T Policy M/g/1 System With Multiple Vacations Geom ~ X/g/1 Repairable System Analysis

When queueing theory is applied widely in the area of random optimal control, there are quite many different policies in the various application backgrounds.Firstly,I generalized the traditional T policy to two kinds of policies:One is the modified (t,T) policy,which means after the system turns empty,the queueing system would turn down for a period of stochastic time.It is supposed that both the service time and the delay time follow arbitrary discrete distribution.By using a concise decomposition method,the transient and steady-state distribution of the queue length are studied,and the stochastic decomposition property of steady-state queue length has been proved.In addition,the service costs(including the running cost and adjusting cost) of the system under the policy in unit time and the sojourn cost of the customs in the system are considered.On the basis of those,the model of cost structure is obtained.Finally,we give a specific example of cost structure of the system with optimal solution and the optimal policy.The other generation is the modified(p,T) policy,which means on the end of the server's busy period the server will close for a fixed period of time with probability p,or stay idle and wait for the next customer with probability (1-p).Firstly,we obtain the steady mean queue length in the system and the expectation of server busy cycle which is the sum of consecutive idle and busy period in the view of systematic function indexes.On the basis of that,the operating costs, including the system running and switching costs(starting up and shutting down),and the holding cost for each customer in the system per unit time under the policy are discussed.Finally,we give a numerical example to illustrate the application of the cost structure model,in which the relationship among total cost,vacation probability and vacation time is discussed.Extending the approach to practice can provide much information for managerial decision-making in industrial management.This paper also discussed the reliability and queueing indices in discrete time Geom~x/G/1 repairable queue system with delayed multiple vacations in detail.It's assumed that both the inter-arrival times and the life of the service station are independent random variables with geometric distribution,while the service time,the delayed vacation time,the vacation time and the repair time follow general discrete distribution.By introducing the server busy period and using the total probability decomposition technique we discuss the transient properties of the queue length from the beginning of the any initial state i(i=0,1,…),and obtain the z-transformation of the transient distribution of the queue length.Furthermore,I obtained the recursion expressions of the steady distribution and the stochastic decomposition of the queue length at a random point in equilibrium.Especially we can obtain some corresponding results of the discrete queue system under some especial cases.As for the reliability indices,the following reliability indices results of the service station are obtained:1) The probability that the service station is during the "generalized busy period";2) The point unavailability at time and the steady unavailability;3) The expected failure number during;4) The expected failure number during the "generalized busy period".