The M/M/1 Queue With Setup Time And Working Vacations

During the last two decades, the vacation queuing theory has been investigated extensively and developed the theoretical framework whose core is stochastic decomposition. It is the expansion of the classical queuing theory. Recently, Servi and Finn firstly introduced a class of semi-vacation policy: the server can take the original work at a lower speed during a vacation period rather than stopping completely. Such a vacation is called a working vacation (WV). The essence of the working vacation policy is that, when the number of customers is less relatively, a lower speed period is established to economize the operational cost in the system.Firstly, we briefly introduced the foundational theory of the continuous-time Markov chain and the quasi-birth and death process and gave a concise summary on the theory of the GI/M/1 type matrix and the matrix-geometric solution method, which provided the theoretical preparations for the analysis of later models.Secondly, we studyed an M/M/1 queuing model with set-up time and multiple working vacations. By means of the theory of the quasi-birth and death process and the matrix-geometric solution method, we obtained the distribution for the stationary queue length, the average member of the customers and the average sojourn time in the stationary state.Thirdly, we sdudyed an M/M/1 queuing model with single working vacation and set up-closed down times, using quasi and death prosess and matrix-geometric solution method to obtain the distribution for the stationary queue length and the waiting time of a customer in the system. Furthermore, we got the stochastic decomposition structures of stationary indices. Meanwhile, we obtained the distribution of the additional queue length and additional delay. At the same time, we investigated the influence of parameters on the additional queue length by means of numerical examples.Finally, we investigated an M/M/1 queue with single working vacation serving at a slower rate in the set-up time, using quasi and death prosess and matrix-geometric solution method to obtain the distribution for the stationary queue length and waiting time of a customer in the system. Furthermore, we got the stochastic decomposition structures of stationary indices and obtained the distribution of the additional queue length and additional delay.