Analysis Of The M/G/1 Repairable Queueing System With N-Policy And Multiple Vacations

Under the condition that both the service time and vacation time are continuous distributions that is they both have probability density functions, paper[1] using supplementary variable method studies the M/G/1 queue with N-policy and multiple vacations, and gets the generating function of steady-state queue length distribution. However, it is hard to retransform the generating function of steady-state queue length distribution to it's explicit expression. Moreover, paper[1] doesn't study the transient-state queue length distribution and consider the situation that the server may break down during it's work time.So this paper studies a M/G/1 repairable queueing system with N-policy and multiple vacations, in which the server may break down and can be repaired, and the service time and vacation time are all generally distributed but not must have probability density functions. By introducing the "general service time" and "general service busy period", using a method of total probability decomposition and renewal process theory and transforming techniques such as L-transformation, LS-transformation and generating function transfor-mation, we first study the transient-state characters of this system from which obtain the recursion formula of the steady-state queue length distribution and the steady-state equal queue length. On the other hand, considering the server may break down, this paper also studies the reliability indexes of this system as follow:(1) the distribution of the first broken-down time;(2) the probability that the service station is in failure at time t that is the transient-unavailability of the system, and the steady-unavailability;(3) the number of failure time during the server's "general service busy period"; the expectation of failure times during (0, t];(4) the mean downtime during the server's "general service busy period".