The M/G/1Repairable Queueing System With Delay Close-time And Min(N,V)-Policy

We consider the M/G/1system in which the server takes a vacation if the system becomes empty..If the number of waiting customers in the queue does not reaches N, the server begin to work after the vacation. Whenever the number of waiting customers in the queue reaches N, the server interrupts its vacation and returns to the system immediately. This initiates a new busy period and the server works until the system becomes empty. We call this Min(N,V)-policy. In practice, the server station is not closed at the end of the service, but a closed preparation time, namely "delayed closed time". Due to the preparation work is not fixed before the closure of the service. Therefore the delayed closure time is general distributionIn this paper we study M/G/1queueing system with delay close-time and Min(N,V)-policy, supposing that the service station may fail and be repaired. By total probability decomposition and Laplace transform technique, we discuss the following questions: 1)The transient queue length distribution, and obtain the recursion expressions of Laplace transform of the transient queue length distribution.2)The steady queue length distribution, and obtain the recursion expression of the steady queue length distribution and the probability generating function expression.3)The distribution of the first failure time;4)The probability that the time t is in the "server generalized busy time", and obtain its Laplace transform and the steady result.5)The unavailability of the service station at any time t, and obtain its Laplace transform and the steady result.6)The average failure number of the service station during the interval (0,t], and obtain its Laplace-Stieltjes transform and the steady result.