工作休假与马尔可夫到达过程的排队系统分析
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摘要
本篇学位论文分别研究了马尔可夫到达过程和几种工作休假策略的排队模型,包括：可中止工作休假、单重工作休假策略以及可变多重工作休假策略。全文除绪论外由5部分组成,分别如下：
第二章研究了可中止工作休假的M/G/1排队模型,首先利用Foster准则和Kaplan条件得到系统稳态分布存在的充分必要条件。然后利用补充变量法,建立稳态下系统模型的微分方程,结合矩阵分析方法和概率母函数方法,获得稳态下系统状态和顾客人数的联合概率母函数,进而求得稳态下系统所处不同状态的概率及其他性能指标。另外,本章还对系统顾客的等待时间进行了分析,给出稳态下任意顾客等待时间概率分布函数的拉普拉斯-斯蒂阶变换(LST)。最后,分析了工作休假期内的服务率对系统队长的影响,并给出具体模型的数值例子。
第三章研究了具有单重工作休假的M/G/1排队模型,采用补充变量法建立稳态下系统的微分方程组。利用矩阵分析方法,得到稳态下系统所处服务状态和顾客人数的联合概率母函数、整个系统顾客人数的概率母函数以及系统的平均队长和其他性能指标。最后,通过特例分析说明对经典休假模型的一般性。
第四章考虑了具有有限等待场所和可变多重工作休假的GI/M/1/N排队模型,利用补充变量法,写出嵌入马氏链的转移概率,然后利用概率母函数方法,得到顾客到达前夕和任意时刻的队长分布、稳态等待时间分布和消失概率等结果。进一步研究了单重(H=1)工作休假GI/M/1/N排队系统,通过数值例子分析了工作休假服务率对系统队长和消失概率的影响。
第五章研究了到达过程不是Possion到达,而是马尔可夫到达过程(MAP)的可中止工作休假排队模型。利用RG-分解和Cencoring技术得到了稳态下系统状态和顾客人数联合概率密度、任意时刻和顾客到达前夕时刻系统稳态队长分布及顾客等待时间的拉普拉斯-斯蒂阶变换。
第六章考虑了可修的并具有反馈机制的BMAP/G/1重试排队系统,其中服务台遭受启动失效。若顾客到达系统时服务台在忙或处于修理状态,则立刻进入重试轨道,按照FCFS规则进行重试。顾客服务完以概率p(p<1)立即回到重试轨道,等待重新服务,或者以概率q=1-p永远离开系统。同样利用RG-分解和Cencoring方法研究了任意时刻系统队长分布；利用更新过程的理论,得到了系统平均忙期。
In this thesis, we study the queue with Markovian arrival process and some working vacation policies, which include the policy of working vacations and vacation interruption, the single working vacation and the variant of multiple working vacations. The thesis is organized as follows.
In chapter 2, an M/G/1 queue with working vacations and vacation interruption is analyzed. The necessary and sufficient condition for the stability of the system is derived. We obtain the queue length distribution and service status at an arbitrary epoch under steady state conditions by using the method of a supplementary variable and the matrix-analytic method. Further, we provide the Laplace-Stieltjes transform (LST) of the stationary waiting time. Finally, numerical examples are presented.
In chapter 3, we discuss an M/G/1 queue with single working vacation. Using the method of supplementary variable and the matrix-analytic method, we obtain the queue length distribution and service status at the arbitrary epoch under the steady state conditions. Further, we derive expected busy period and expected busy cycle. Finally, several special cases are presented.
In chapter 4, we analyze the GI/M/1/N queue with a variant of multiple working vacations. We analyze the Markov chain underlying the considered queueing system and obtain the transition probabilities. We obtain the queue length distribution at pre-arrival and arbitrary epochs with the method of supplementary variable and the embedded Markov chain technique. Finally, several performance measures and numerical results are obtained when the parameterH=1.
In chapter 5, we consider the MAP/G/1 queue with working vacations and vacation interruption. We obtain the queue length distribution with the method of supplementary variable, combined with the RG-factorization and censoring technique. We also obtain the system size distribution at pre-arrival epoch and the LST of waiting time.
In chapter 6, we discuss the BMAP/G/1 retrial queue with feedback and starting failures. If an arriving customer finds the server busy or down, the customer leaves the service area and enters the orbit and makes a retrial at a later time.When a customer is served completely, he will decide either to join the orbit again for another service with probability p or to leave the system forever with probabilityq=1-p. We obtain the queue length distribution with the method of supplementary variable, combined with the RG-factorization and censoring technique. The mean length of the system busy period of our model is obtained by the theory of regenerative process.
第二章研究了可中止工作休假的M/G/1排队模型,首先利用Foster准则和Kaplan条件得到系统稳态分布存在的充分必要条件。然后利用补充变量法,建立稳态下系统模型的微分方程,结合矩阵分析方法和概率母函数方法,获得稳态下系统状态和顾客人数的联合概率母函数,进而求得稳态下系统所处不同状态的概率及其他性能指标。另外,本章还对系统顾客的等待时间进行了分析,给出稳态下任意顾客等待时间概率分布函数的拉普拉斯-斯蒂阶变换(LST)。最后,分析了工作休假期内的服务率对系统队长的影响,并给出具体模型的数值例子。
第三章研究了具有单重工作休假的M/G/1排队模型,采用补充变量法建立稳态下系统的微分方程组。利用矩阵分析方法,得到稳态下系统所处服务状态和顾客人数的联合概率母函数、整个系统顾客人数的概率母函数以及系统的平均队长和其他性能指标。最后,通过特例分析说明对经典休假模型的一般性。
第四章考虑了具有有限等待场所和可变多重工作休假的GI/M/1/N排队模型,利用补充变量法,写出嵌入马氏链的转移概率,然后利用概率母函数方法,得到顾客到达前夕和任意时刻的队长分布、稳态等待时间分布和消失概率等结果。进一步研究了单重(H=1)工作休假GI/M/1/N排队系统,通过数值例子分析了工作休假服务率对系统队长和消失概率的影响。
第五章研究了到达过程不是Possion到达,而是马尔可夫到达过程(MAP)的可中止工作休假排队模型。利用RG-分解和Cencoring技术得到了稳态下系统状态和顾客人数联合概率密度、任意时刻和顾客到达前夕时刻系统稳态队长分布及顾客等待时间的拉普拉斯-斯蒂阶变换。
第六章考虑了可修的并具有反馈机制的BMAP/G/1重试排队系统,其中服务台遭受启动失效。若顾客到达系统时服务台在忙或处于修理状态,则立刻进入重试轨道,按照FCFS规则进行重试。顾客服务完以概率p(p<1)立即回到重试轨道,等待重新服务,或者以概率q=1-p永远离开系统。同样利用RG-分解和Cencoring方法研究了任意时刻系统队长分布；利用更新过程的理论,得到了系统平均忙期。
In this thesis, we study the queue with Markovian arrival process and some working vacation policies, which include the policy of working vacations and vacation interruption, the single working vacation and the variant of multiple working vacations. The thesis is organized as follows.
In chapter 2, an M/G/1 queue with working vacations and vacation interruption is analyzed. The necessary and sufficient condition for the stability of the system is derived. We obtain the queue length distribution and service status at an arbitrary epoch under steady state conditions by using the method of a supplementary variable and the matrix-analytic method. Further, we provide the Laplace-Stieltjes transform (LST) of the stationary waiting time. Finally, numerical examples are presented.
In chapter 3, we discuss an M/G/1 queue with single working vacation. Using the method of supplementary variable and the matrix-analytic method, we obtain the queue length distribution and service status at the arbitrary epoch under the steady state conditions. Further, we derive expected busy period and expected busy cycle. Finally, several special cases are presented.
In chapter 4, we analyze the GI/M/1/N queue with a variant of multiple working vacations. We analyze the Markov chain underlying the considered queueing system and obtain the transition probabilities. We obtain the queue length distribution at pre-arrival and arbitrary epochs with the method of supplementary variable and the embedded Markov chain technique. Finally, several performance measures and numerical results are obtained when the parameterH=1.
In chapter 5, we consider the MAP/G/1 queue with working vacations and vacation interruption. We obtain the queue length distribution with the method of supplementary variable, combined with the RG-factorization and censoring technique. We also obtain the system size distribution at pre-arrival epoch and the LST of waiting time.
In chapter 6, we discuss the BMAP/G/1 retrial queue with feedback and starting failures. If an arriving customer finds the server busy or down, the customer leaves the service area and enters the orbit and makes a retrial at a later time.When a customer is served completely, he will decide either to join the orbit again for another service with probability p or to leave the system forever with probabilityq=1-p. We obtain the queue length distribution with the method of supplementary variable, combined with the RG-factorization and censoring technique. The mean length of the system busy period of our model is obtained by the theory of regenerative process.
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