Batch Arrival M ~ X/g/1 Reach The Discrete-time Vacation Queuing System

Queueing systems with different vacation policies and exhaustive service have been studied by many references and obtained some valuable steady state and transient queueing indices. But at the same time, we note that considerable domestic and foreign references about M~X/G/1 vacation queue have seldom discussed the steady state waiting time of an arbitrary customer in queue. In the Chapter 3 of this paper, we consider M~X/G/1 queueing system with multiple adaptive vacation policy and setup/entire close-down times as our mathematical model, by applying the full probability decomposition technique and the ball and jar modeling analytic method, we study this problem in detail and obtain some new results as follows:1) The expression of L transform of the transient queue length distribution at arbitrary epoch t and the recursion expression of the steady state distribution of the queue length;2) The probability generating function of the steady state queue length and its stochastic decomposition;3) The LS transform, the mean value and the stochastic decomposition structure of the steady state waiting time of an arbitrary customer in queue;4) The PH structure of the additional delay time in M~X/G/1 queueing system with multiple adaptive vacation policy and setup/entire close-down times and some common M~X/G/1 vacation queueing systems.Usually, to the Geom/G/1 discrete time queueing system with vacations and its variants, we often employ the embedded Markov analytic technique to study, while, to the GI/Geom/1 discrete time queueing system with vacations, we often use the matrix geometric solutions method to research. In the Chapter 4 of this paper, due to the singularity of the research method for studying discrete time queueing system and the complexity of theoretic basis of the matrix geometric solutions method, we introduce a new method for analysis the queuing model with the matrix of GI/M/1 type and canonical form, that is, differencing of the invariant probability measure equation technique. We apply this method to investigate the GI/Geom/1 discrete time vacation queueing system with setup and closedown times and obtain some new results as follows:1) The steady state queue length distribution just before customer arrival instant and its mean value;2) The steady state sojourn time of a customer in queue and its mean value;3) The stochastic decomposition structure of the steady state queue length and the steady state sojourn time;4) The PH structure of the additional queue length and the additional delay time...