M/G/1(Repairable)Queueing System With Adaptive Multistage Vacation And Min(N,V)-Policy

Firstly,we consider the M/G/1 queueing system with adaptive Multistage va-cation and Min(N,V)-policy.By using the total probability decomposition technique and the Laplace transformation tool,the transient distribution of the queue length from the beginning of the any initial is discussed.Both the expressions of the Laplace transformation of the tran-sient queue length distribution and the recursive expressions of the steady-state queue length distribution are obtained.Meanwhile,we obtain the probability function of the steady-state queue length distribution and the expression of average queue length.Moreover,some more concise explicit expressions of the corresponding steady-state queue length distribution are ob-tained under some special cases such as the vacation number obeys the geometric distribution or the vacation number is a fixed positive integer M.By the numerical example we illustrate the important value of the explicit expressions of the steady-state queue length distribution in the optimal design of the system capacity.And the explicit expression of the long-run expected cost rate is derived under a given cost structure.Furthermore,through numerical calculation,we determine the optimal control policy N*for minimizing the long-run expected cost per unit time as well as the combined control strategy(N*,M*)when the vacation number is a fixed positive integer M.Secondly,"the server can be invalidated and repairable" is introduced into the system in this thesis,by considering the server's "generalized busy period" as the server's busy period,we reconsider the queuing indexes of the system.Furthermore,we focus on dis-cussing some reliability indexes of the service station,such as the distribution of the first failure time,the unavailability at the time t and the average failure number during the time(0,t].