Multi-Server Queuing Systems With Balking, Reneging And Vacation

The multi-server queuing systems are typical models of the queuing theory. The servers' taking vacation or other assistant work has important impact on the performance measures and the economic profit of the system. The queuing models with vacation have been widely applied to many fields such as manufacturing system, computer and communication systems. In many practical queuing problems, we often see the phenomenon of the customers' balking and reneging. Thus, the multi-server queuing systems with balking, reneging and server vacations have important theoretical significance and application value.In this paper, the N-policy vacation, multiple synchronous vacations and synchronous vacation of partial servers for the multi-server queuing systems with balking and reneging are considered, respectively.First, the N-policy M/M/R/K queuing system with balking, reneging and multiple synchronous vacations is studied. By the Markov process method, the steady-state probability equations are developed. By rewriting the transition rate matrix as a blocked one, the matrix form solution of the steady-state probability is obtained by using the inverse of the block matrixes. Then some performance measures of the system such as the expected number of customers in the system, the expected number of customers in the queue and the average rate of the customer loss are obtained.Second, the M/M/R/N queuing system with balking, reneging and multiple synchronous vacations is studied. The distribution of conditional waiting time for customers who enter the system and eventually acquire service is derived. On the base of this, the distribution of conditional waiting time for customers who enter the system and eventually acquire service is derived.Finally, the M/M/R/N queuing system with balking, reneging and single synchronous vacation of partial servers is studied. By Markov process method, the steady-state probability equations are developed. By rewriting the transition rate matrix as a blocked one properly, a matrix form solution of the steady-state probability is obtained by using the inverse of three block matrixes. Some performance measures of the system such as the expected number of customers in the system, the expected number of customers in the queue and the average rate of the customer loss are also presented. In addition, the distribution of conditional waiting time for customers who enter the system and eventually acquire service is derived. Then a cost model is developed by taking the number of servers on vacation as the control variable and the affect of the changes in specific values of the system parameters on the optimal number of servers on vacation and the optimal cost of the system is investigated.