The N-Policy M/G/1 Queueing System With Start-Up Time And Delay Vacation

Whether it is in the continuous time or discrete-time queueing system research, the majority of literature studies at home and abroad based on the system already in a steady state has gained the probability generating function of the steady-state queue length distribution and the average queue length of the system. But by the probability generation function for the steady-state queue length distribution, it is difficult to obtain the specific expressions for the steady-state queue length distribution.This paper, to promote the N-policy M/G/1 queueing system, studies N-policy M/G/1 queueing system with start-up time and delay vacation. By the introduction the "server busy period", using the total probability decomposition technique, the Laplace transform and Laplace-Stieltjes transform tool, this paper firstly discusses the transient distribution of the queue length that starting from any initial state of the system and obtains Laplace transform expressions on the time t. And then, through the transient queue length distribution, it is very simple to gain the steady-state distribution expressions of the queue length of the system, as well as the probability generating function for the steady-state queue length distribution and the average additional queue length. Secondly, by discussing the corresponding steady-state waiting time and sojourn time, the average steady-state waiting time and the average sojourn time have been gotten. In addition, in some special cases, the steady-state queue length distribution expressions are obtained, which are more specific and easier to calculate.