M/M/R/N Queueing Systems With Balking And Reneging Under Working Vacations

In daily life, we often encounter a lot of queuing phenomenons. A commonphenomenon is that the arriving customer may not enter the system if he sees many peoplein front of the line. Another one is that the customer may leave the system if he has waitedfor a long time in the queue. In the practical queueing problems, we also see thephenomenon of the server works with a lower service rate rather than completely stopsservice during the vacation period. This vacation strategy is called working vacation. Thus,the M/M/R/N queueing systems with balking, reneging and working vacations haveimportant theoretical significance and application value.In this paper, the N-policy working vacations, asynchronous single working vacationsand synchronous working vacations are considered respectively.Firstly, The queueing system was considered with balking, reneging and synchronousN-policy multiple working vacations. By using the matrix solution method, the matrixsolution of steady-state probability presented by the inverse of two matrices is obtained.We also derive some system characteristics and give the numerical analysis.Secondly, we analyze an queueing system with balking, reneging and asynchronoussingle working vacations. By using the Markfov process method and the matrix solutionmethod, the system steady-state probability is obtained. Especialiy, it derives the averageservers duing idle period. Then it creats a cost model of the system.Finally, An queueing system is considered with balking, reneging and synchronousworking vacations. It introduces two different vacation policy: synchronous multipleworking vacation and synchronous single working vacation. It establishes a unified model.By using the Markfov process method and the matrix solution method, the matrix solutionof steady-state probability presented by the inverse of two matrices is derived. Thecomputing of the inverse of the two matrices is discussed. In addition, some performancemeasures of the system are presented such as the expected number of customers in thesystem, the probability of the servers in the working vacation period and the average rateof the customer who does not enter the queue. Finally, the effect of the vacation service rate and the vacation rate on the expected queue length are investigated by numericalexamples.