Performance Analysis Of The Two-Phases-Service M/M/1 /N Queuing System With The Server Breakdown And Vacations

Repairable queuing is the expansion of the classical queuing theory, which service stations may be breakdown and can be repaired during the service period. In recent years, with the quick development of comunication systems, computer techniques, services and finance, there are many kinds of complicated queuing systems. The queuing systems are widely applied in many fields, especially that with two phases of service. The breakdowns or vacations of the servers have an important impact on the performance measures and the economic profit of the system. Thus, the study on the queuing systems with integrated above mechanism has important theoretical significance and application value.In this paper, we consider the M/M/1/N queuing system with two phases of service and multiple vacations. They are expansions of other models in the documents , the main contents are as follows:Firstly, we respectively investigate two two-phases-service M/M/1/N queuing systems with the server breakdown in the first service and the second service. We derive the steady-state equations by the Markov process method. By writing the transition rate matrix as block matrix, we get the matrix form solution of the steady-state probabilities and present a algorithm for calulating the steady-state probabilitie. In this paper, we derive the probability expressions of the system service station during the busy period and the vacation period respectively. We also derive the matrix expressions of the stationary queue length of the system at an arbitrary time. In addition, when N=5, using the Matlab software, we analyze the influence of the parameters of the system to the system performance measuires.Secondly, we investigate a two-phases-service M/M/1/N queuing system with the server breakdown in both service phases and multiple vacations. First, by writing the transition rate matrix as block matrix, we get the matrix form solution of the steady-state probabilities. In this paper, we derive the probability expressions of the system service station during the busy period and the vacation period respectively. We also derive the matrix expressions of the stationary queue length of the system at an arbitrary time . However, the espression is too complex to get the explicit expression of the steady-state probabilities. So, when N=5, using the Matlab software, we analyze the influence of the parameters of the system to the system performance measuires by several numerical examples.