Research On Queueing System With Working Breakdowns

On the basis of the important function in telecommunication, flexible manufacturing, e-commerce and supply-chain systems, the working breakdowns policy become a burgeoning research since introduced into the queueing system. Combining with vacations, variable arrival rate, impatient customers, server set-up period, this thesis is based on the classic M/M/1 queueing system. We establish and analyze two kinds of mathematical models, and derive several meaningful performance measures at the same time. The main contents of this thesis are organized as follows:Firstly, preface and preliminary knowledge would be given. We would show you the historical background of the queue model and their applications in the actual life. What’s more, the queueing theory has established a relatively complete theoretical system and formed a series of research methods after the development of more than one century. The research status of several kinds of queue model and mainly research methods that this topic mentioned would be given accordingly.Secondly, this thesis studies an M/M/1 queueing system with server working breakdowns, variable arrival rate and multiple vacations at the base of the classical queue model. It assumed that the server works at a lower rate instead of doing nothing during the working breakdowns period and the arrival rate varies according to the state of the server. By giving the state transition rate matrix of the system and using the quasi birth and death process and matrix-geometric solution method, we may gain the distribution of the stationary queue length. Furthermore, we derive the formulae of expected queue length and expected regular busy period. Lastly, some numerical examples are given for analyzing the effect of parameters on the performance measures of the system.Finally, we put impatient customers and server set-up period into the M/M/1/N vacation queueing system with working breakdowns. Within the working breakdowns period, the customers may become impatient for waiting time increase or even leave the system. The addition of server set-up period can reduce the service costing. If no customers in the system, we can just close down the service facilities. Once the customers came, they could be served after the server set-up period. We can build a quasi birth and death process model at the base of the state transition rate matrix of the system. Then we obtain the distribution of the stationary queue length and the formulae of expected working breakdowns period probability by using matrix-geometric solution methods. Some numerical results are provided to illustrate the effect of the parameters on several performance measures at last.