Matrix-Geometric Solution Of A Repariable Queueing System With Two Heterogeneous Servers

The repairable queuing system of two heterogeneous servers is a typical model of queuing theory, and it has been widely used in machine processing systems, computer systems and communication systems. In our actural applications, the two servers may be different in service rate and failure rate because of the abrasions and degradations. Therefore it has an important impact on the performance measures and economic profit. Thus, the study on the queuing systems of two heterogeneous servers with integrated above mechanism has important theoretical significance and application value.Firstly, we investigate an M/(Ek,M)/2 repairable queuing system of two heterogentous servers with service time following Erlang distribution and exponential distribution, in which one server is subject to breakdown at busy time When the server breakdowns, the customers are still waiting in the queue. It continues to provide service for the customers after repairing and the time used is available. 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. 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 M/(PH,M)/2 repairable queuing system of two heterogeneous servers with service time following PH distribution and exponential distribution, in which one server is subject to breakdown in busy time. Once the server breakdowns, the customers return to the head of the queue and accept service. The time used is invalid. 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. Then we obtain some performance measures of the system and reliability indices of the unreliable server. Combing with serval special circumstances following PH distribution, we give some results of numercial analysis.Finally, we study an M/M/2 repairable queuing system of two servers with the service time following exponential distribution, in which the servers have different service rate. Once the server breakdowns, the customer return to the head of the queue to wait for being served again.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 some performance measures and reliability indices of the system. By MATLAB calculation, we give some numerical analysis for the results.