Stationary Theory Of The M/G/1 Type Queueing System With Multiple Adaptive Vacation

Along with the development of information technology, computer technology and management system etc., queue theory has been applied to the performance analysis of above fields. But, because the M/G/1 type queue with exhaustive service is limited to model the complicated problems, more and more scholars from various fields have beginning to study nonexaustive service policy and established some M/G/1 type vacation queue models with nonexhaustive service. Queue theory has been enriched by some theories about these models, and has acted as a key roles on the performance analysis about some systems. A class of M/G/1 type queues with multiple adaptive vacation policy is studied, and some stochastic decomposition results of the stationary indices are obtained. Furthermore, the trend of the stationary indices are described for some queue systems by numerical examples, some differences of these indices among systems are found. Thereby, stationary theory of the M/G/1 type queueing system with multiple adaptive vacation is established.Firstly, three vacation queues are summarized, including the M/G/1 queue with multiple adaptive vacation, the M/G/1 boundary state variation model with exhaustive service and the Geom/G/1 queue with multiple adaptive vacation. And the stochastic decomposition results for their performance indices are shown, the busy period of systems is analyzed in detail and the probabilities in every states are obtained. The universality of the stochastic decomposition results for their stationary performance indices is exhibited by some special cases. The Geom/G/1 boundary state variation model with exhaustive service is presented, and the stochastic decomposition results for its stationary queue length and stationary waiting time are proved. Parallel to the analytic results of the continue time M/G/1 queue, the above queue models have more perfect theoretical framework and enrich the theory of the M/G/1 type vacation queue with exhaustive service.Secondly, combining gate service policy with multiple adaptive vacation policy, a new M/G/1 type model is presented firstly. By the regeneration cycle approach, the stochastic decomposition results of the stationary performance indices are proved, and the mean service period is analyzed. The effect of system parameters on system performance indices is presented by numerical examples. Many kinds of M/G/1 vacation queues with gated ser- vice were studied before are the special examples of such model, such as M/G/1 queues with multiple vacations or single vacation with gated service were studied by Takagi, and M/G/1 queues with gated service by Leung and Alfa et al. The results indicate that M/G/1 type vacation queues with gate service policy have more prefect theoretical system and have more extensive application background.Finally, in order to make the models more significant and practical, limited service policy is combined with multiple adaptive vacation policy. A new M/G/1 type queue is established. By the embedded Markov chain method and the regeneration cycle approach to study M/G/1 type queues with multiple adaptive vacation and general limited service, their stochastic decomposition results are gained, and mean service period is analyzed.In a word, this paper establishes the entire theoretical framework for the M/G/1 type queue with multiple adaptive vacation, and some research results of vacation queue with nonexaustive service are their special cases.