With Exhaustive Service Delayed Multiple Adaptive Vacation Geo ~ X/g/1 Repairable Queuing System

This paper studies discrete-time repairable Geo~X/G/1 queue with exhausted service principle and multiple adaptive delayed vacations with a method of total probability decomposition and renewal process theory.In queueing study,the explicit expressions of queue length distribution are usually hard to be obtained,while their transforming forms are easier to be obtained with various transforming techniques.Moreover,it is also hard to retransform these transformations to the explicit expressions.Therefore,though L-transformation, LS-transformation and z-transformation are important in queueing research,results only relied on using them are not satisfied.This paper starts from studying the transient state characters,by introducing "general service busy period",this paper obtains main results as follows:1) We obtain the z-transformation of queue length distributions in transient state of this system,and the recursion formulae and the generating function of the steady-state queue length distribution.The stochastic decomposition property of steady-state queue length has been proved.We demonstrate that several common queue models are special cases of the queueing model presented in this study.2) We obtain reliability indexes including:(a) The probability that the service station is in failure at timen,that is,the unavailability of system;(b) The number of failure times during the server's "generalized busy period";(c) The expectation of failure times during(0,n];(d) The mean downtime during the server's "generalized busy period" and the approximate expression of the average downtime during(0,n].3) Based on the results of queue length,we propose practical numerical calculation method of queueing indexes.By the method,we obtain the numerical results of an example with application background,and propose optimal control and design strategies by the practical meanings of the results.