The Research Of The Queueing System With(N, N)-preemptive Priority

The preemptive and nonpreemptive disciplines are both extreme cases with respect to preemption condition: they “always” or “never” allow preemption when a high-priority customer arrives during the provision of service to a low-priority customer. For this reason, the hybrid type of the preemptive and nonpreemptive priority disciplines has been vitally concentrated by more and more scholars and has gradually developed into a new research direction. At present, the studies about this kind of queue system mostly takes directly into account the system state pertaining to low-priority customers in the system for the preemption condition. However, to effectively control the quality of service for high-priority customer has more realistic significance in practical application. This paper focuses on the(N, n)-preemptive priority queueing system, under this system, the preemption of the service of a low-priority customer is determined by two thresholds of N and n of the queue length of high-priority customers.Firstly, the M/M/1 queueing system with(N, n)-preemptive priority policy is studied. By analyzing the schematic diagram of the service for the two types of customers and the state transition diagram, the transition matrix is deduced in detail. By using the quasi birth and death process and matrix-geometric solution theory, the distribution of the stationary queue length is obtained. Furthermore, some performance measures of the system are obtained. Finally, some numerical examples are given for analyzing the effect of parameters on the performance of the system.Secondly, we consider a Geom/Geom/1 discrete time queue with(N, n)-preemptive priority discipline. By analyzing the state transition diagram of the model in detail, the state transition probability matrix is deduced. By using the quasi birth and death chain and matrix-geometric solution method, the distribution of the stationary queue length is obtained. Furthermore, the average queue length of the two types and the busy period of the system and the probability of a customer I being preempted are researched. Finally, the changing curves of the system performance measures against parameters are obtained by using the MATLAB, and the main reason which leads to these changes is analyzed.Finally, the queueing system with(N, n)-preemptive priority discipline and negative customers is studied: negative customers are introduce into the M/M/1 queueing system with(N, n)-preemptive priority discipline. By analyzing the schematic diagram of the service for the two types of customers and the influence of negative customers to low-priority customers, the transition matrix is deduced in detail. Then the distribution of the steady-state queue length is obtained by using the method of matrix-geometric solution. And then some performance measures of the system are obtained. Finally, numerical examples are given by MATLAB to illustrate the influence of the parameters on several performance indicators.