Two-Phase Service Queuing System With Balking, Reneging And Vacation

Balking, Reneging as well as two-phase service is the phenomenon which meets frequently in the actual queuing system. The queuing models with vacation have been widely applied to many fields such as communication, computer system and manufacturing system. Joining the consideration in vacation queuing models with balking, reneging as well as two-phase service can portray the actual problem accurately. Therefore, the two-phase service queuing model with balking, reneging and vacation has the important theory significance and the application value.In this paper, limited waiting space and infinite waiting space queuing system with balking, reneging, two-phase service and multiple vacations are studied respectively.Firstly, a limited waiting space M/M/1/N queuing system with balking, reneging, two-phase service and multiple vacations is studied. Using the Markov process theory, the system steady-state probability equations are developed. By suitably rewriting the transition rate matrix as a blocked one, the matrix form solution of the steady-state probability is derived. 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,the server busy probability and taking vacation probability as well as the average rate of the customer loss are obtained.Secondly, the explicit expression of the steady-state probabilities for the special case of N=2 was obtained by Matlab software. Next, we develop a cost model where the service rate is the control variable, thus we investigate the effect of the parameters of the system such as customer arrival rate and server vacation rate on the optimal service rate and the optimal cost by three numerical examples. Finally, an infinite waiting space M/M/1 queuing system with balking, reneging, two-phase service and multiple vacations is studied. Using the Markov process theory, the system steady-state probability equations are developed. By probability generating function method, the analytic expression of steady-state probability generating function is derived,also the sufficient condition of system steady-state probability solution existing is obtained . Then some performance measures of the system such as the expected number of customers in the system and the average rate of the customer loss are obtained.