Queuing Systems With Breakdowns Or Vacations Of Two Heterogeneous Servers

The queuing systems of multi-server especially two heterogeneous servers are typical models of the queuing theory. Meantime, the breakdowns or vacations of the servers, and the phenomena of the customers'balking or reneging have a important impact on the performance measures and the economic profit of the system. Thus, the study on the queuing systems of two heterogeneous servers with integrated above mechanism has important theoretical significance and application value.In this paper, we consider the queuing model of two heterogeneous servers with breakdowns and the phenomena of balking or reneging, as well as the model with vacation.Firstly, we investigate a finite waiting room M/M/2/N repairable queuing system with balking, reneging and heterogeneous servers, in which one server is unreliable and with different breakdown rates in busy and idle time. By Markov process method, we develop the steady-state probability equations, and obtain the simple and obvious iterative formulas of the steady-state probability vectors by using a method of blocking matrix. Then we obtain some performance measures of the system and reliability indices of the unreliable server. The numerical analysis for the results is also given.Secondly, we study an infinite waiting room M/M/2 repairable queuing system that balking depends on breakdown process and heterogeneous servers, in which one server is unreliable and with different breakdown rates in busy and idle time. Using the quasi-birth-and-death process method, we derive the existing condition of steady-state equilibrium, and the matrix-geometric solutions of the steady-state probability vectors. We then obtain some performance measures of the system and reliability indices of the unreliable server. The numerical analysis for the results is also given. Finally, we study the M/M/2 queuing system with asynchronous multiple vacation and heterogeneous servers. By using the quasi-birth-and-death process method, we obtain the explicit expression of the rate matrix and the boundary probability vectors, and derive the joint distribution of queue length and the state of servers. Furthermore, we give the analytical expression of the waiting queue length and waiting time of the arrival customers when all of the servers are busy, and also prove the conditional stochastic decomposition properties of them.