【作者】 李涛；

【导师】 王志忠；

【作者基本信息】 中南大学 ， 概率论与数理统计， 2012， 博士

【摘要】 本篇博士论文主要研究了几类带有Bernoulli休假中断的GI/M/1排队系统。本文主要结构安排如下：第二章分析了一个带有Bernoulli休假中断的GI/M/1排队模型。在工作休假期间,当一个顾客完成服务时,如果系统中还有顾客,服务员则以概率p(0≤p≤1)停止休假(而不是以概率1),回到正常的工作状态,或者以概率1-p继续工作休假。显然,工作休假和休假中断分别对应着p=0和p=1两种情形。此时,我们可以认为休假中断是由Bernoulli控制的。所以,工作休假和休假中断我们完全可以放在一起讨论,而没必要像以前的作者一样分开来单独研究。与带有工作休假但没有休假中断的GI/M/1模型相比,即使休假时间没有结束,我们的服务员仍然有可能停止休假,回到正常的工作状态。而与带有休假中断的GI/M/1模型相比,即使在工作休假期间一个顾客完成服务时系统不空,我们的服务员仍然以概率1-p继续休假。利用矩阵分析的方法,得到了系统在顾客到达时的平稳分布,并对系统队长进行了随机条件分解。利用半马尔可夫过程的性质,得到了系统在任意时刻的平稳分布。利用不同的方法,分别得到了等待时间和逗留时间的Laplace Stieltjes变换。最后,以平均队长作为研究对象,我们给出了几个数值例子来说明一些参数对系统的影响。第三章是在第二章模型的基础上,考虑了一个启动期。当休假中断发生时,服务员不能立刻回到正常的工作状态,而是要经过一段随机长度的时间,我们把这段时间称为启动期。事实上,现实生活中的一些情况是带有启动期的。例如,对一个工厂里正在工作的机器而言,猛然提高它的生产率有时是办不到的,有可能会需要一段时间。在第四章,我们考虑了Bernoulli规则控制下的休假和休假中断,把普通休假、工作休假和休假中断综合在一起考虑。如果令参数取合适的值,许多带有休假的GI/M/1模型就是我们研究模型的特殊情形。第五章则是讨论了一个带有开始期和单重工作休假的GI/M/1模型,休假期间的中断也是由Bernoulli控制的。同时,在每一章的最后,我们都简单的叙述了一下上面四个模型对应的离散情形,利用相同的方法,这些离散模型也可以进行研究。更多还原

【Abstract】 In this Ph.D. thesis, we investigate some GI/M/1queues with Bernoulli vacation interruption. This thesis is organized as follows.In Chapter2, a GI/M/1queue with Bernoulli vacation interrup-tion is analyzed. During the working vacation period, if there are customers at a service completion instant, the vacation can be inter-rupted and the server is resumed to a regular busy period with prob-ability p(0≤<p≤1)(not with probability1), or continues the vacation with probability1-p. Obviously, if p=0or p=1, we will get the same results for GI/M/1queue with working vacations and without or with vacation interruption, respectively. And, we regard the vacation inter-ruption is controlled by Bernoulli. Thus, we can investigate working vacation and vacation interruption at the same time, which is differ-ent from the situation many authors considered before. Compared with a GI/M/1queue with working vacations but without vacation interruption, even if the vacation time is not end, it’s still possible for our server to stop the vacation and come back to the normal busy period. On the other hand, compared with a GI/M/1queue with vacation interruption, even if there are customers at a service comple-tion instant during the working vacation period, our server can still continue the vacation with probability1-p. Using the matrix-analytic method, we get the steady-state distribution for the queue length at arrival epochs. The stochastic decomposition structure for the queue length is also derived. Using the theory of semi-Markov process, the steady-state distribution for the queue length at arbitrary epochs is obtained. Using different methods, the LST of waiting time and the LST of sojourn time are derived. Finally, taking the expected queue length as an example, we perform some numerical examples to study the effect of various parameters on the system’s characteristics. In Chapter3, we consider a set-up period on the basis of the model in Chapter2. When the vacation interruption happens, the server cannot come back to the regular busy period immediately, but begins a set-up period of random length. In fact, the set-up period exists in some practical situations. For example, in a production sys-tem, improving the productivity of a machine may experience some time before processing the job. Chapter4investigates a GI/M/1queue with Bernoulli-schedule-controlled vacation and vacation inter-ruption, we consider ordinary vacation, working vacation and vacation interruption at the same time. Let parameters take proper values, many GI/M/1vacation queues will be the special cases of the model we consider. Chapter5studies a GI/M/1queue with start-up pe-riod and single working vacation, and the vacation interruption is also controlled by Bernoulli. Furthermore, we describe the correspond-ing discrete-time queue at the end of each Chapter. Using the same method, these discrete-time queues can be analyzed in a similar way.更多还原