The Research Of Geom/Geom/1+1Queueing System With Changing Service Rate

The discrete time queueing models have been applied widely in manufacturingsystem and computer communication system. Various kinds of queueing models havebeen studied that formed a relatively complete theory of the discrete time queueing system.Based on the Geom/Geom/1queueing system, we add a queue in which the capability islimited. This system is built and considered in this paper.Firstly, we investigate the Geom/Geom/1+1queueing system with changing servicerate. With the memoryless property of geometric distribution, a Quasi-Birth-Death modelis established and the state transition probability matrix is derived. Using the matrixgeometric solution method, the expressions for some performance measures are obtained,such as the average queue lengths of two queues, the average waiting time of a new arrivaland so on. Numerical illustrations are provided to analyze the influence of parameters onthe system performance measures.Secondly, negative customers are introduced into the Geom/Geom/1+1queueingsystem. With the RCH (Removal of Customers at the Head) and the RCE (Removal ofCustomers at the End) killing strategies, the three diagonal form and the four diagonalform of the transition probability matrices are obtained. By the matrix geometric solutionmethod, the stationary queue length of the system is obtained. Then some performancemeasures are derived, such as the average waiting time of positive customers, theprobabilities of sever states and so on. Through the changing curves of the performancemeasures against system parameters, we make an analysis of the main reason which leadsto these changes.Finally, another strategy which called working vacation is introduced into theGeom/Geom/1+1queueing system. When the system is empty, these two severs enter intovacation together and they can take service at the lower rates when a new customer arrivesin this system. The matrix-block form of the transition probability matrix is obtained byestablishing a three-dimensional Markov chain. Using the matrix analytic method, thematrix geometric solution form of the stationary distribution is obtained. Furthermore, theprobabilities of a new arrival without waiting and the system in busy period are derived. The numerical examples are presented to show how sensitive performance measures areversus changes in parameters of the system.