几类复杂排队系统的研究
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摘要
本篇博士学位论文研究了几类复杂的排队系统.这些排队系统主要包括G-排队系统、重试排队系统、休假排队系统、可修排队系统、离散时间排队系统、马尔可夫到达排队系统及其混合.全文由如下七部分组成.
第一章是绪论,简要地介绍了排队论的历史背景、研究内容、发展现状以及本文所做的主要工作和主要的创新点.
第二章分析了一个具有非空竭服务随机休假策略的M／G／1重试可修G-排队系统.正、负顾客的到达服从两个独立的泊松过程.其中负顾客的到达不仅移除一个正在服务的顾客,而且致使服务台损坏.服务台采取随机休假策略,即服务台的工作时间服从指数分布.休假时间和修理时间均服从任意分布.我们发展了一个新方法,即通过构造一个具有吸收态的马尔可夫链且利用经典M／G／1排队系统的稳态条件求得了该系统稳态存在的充分必要条件.利用补充变量方法建立了系统的稳态方程组,然后利用取母函数得到了系统状态和重试组中顾客数联合分布的概率母函数,进而求得了系统在稳态下的一系列重要的排队性能指标.而后,我们对系统进行可靠性分析.通过将服务台发生故障视为马尔可夫链的吸收状态,构造了一个新的排队系统,由此得到了一些重要的可靠性指标.此外,我们还分析了系统的忙期和一些有趣的特例.最后,我们还给出了几个数值实例来说明某些参数对系统性能指标的影响.
第三章研究了一个具有不耐烦顾客、灾难、N策略休假和反馈机制的MX／G／1重试可修G-排队系统.利用Foster准则和Kaplan条件得到了系统稳态存在的充分必要条件.利用补充变量法,建立了系统稳态时的平衡方程组.通过求解这些方程,得到了系统状态和重试组中顾客数联合分布的概率母函数.进而求得了系统在稳态下的一系列重要的排队性能指标.而后,我们对系统进行可靠性分析,得到了一些重要的可靠性指标.最后还对一些特例进行了分析.
第四章分析了一个具有抢占优先权和冲突机制且服务台不可靠、修理滞后的M／G／1重试可修G-排队系统.假设重试策略为线性重试策略,负顾客移除一个正在服务的正顾客同时引起服务台损坏,但损坏的服务台不会立即得到修理.推导出了系统稳态存在的充分必要条件.利用取母函数方法,得到了服务台状态和重试组中顾客数联合分布的概率母函数.进而求得了系统在稳态下的一系列重要的排队性能指标.而后,我们对系统进行可靠性分析,得到了一些重要的可靠性指标.此外,我们还讨论了随机分解性质,以及分析了几个特例.最后,给出了几个数值实例.
第五章考虑了一个一般重试时间的离散时间Geo/G/1重试G-排队系统.其中负顾客带走一个正在服务的顾客,而对重试组中的顾客无影响.通过补充剩余服务时间,写出了嵌入马氏链的转移概率.而后,利用取母函数法求解Kolmogorov方程,得到了重试组队长和系统队长的概率母函数.进而得到了一系列重要的排队指标.此外,我们还推导出了系统的稳态存在条件.以及对无负顾客时的特例进行了分析.最后我们还通过几个具体的数值实例来演示一些参数对系统关键性能指标的影响.
第六章讨论了一个在马尔可夫随机环境中运行的有限源MAP/PH/N重试G-排队系统.其中正、负顾客的到达都服从马尔可夫到达过程.负顾客到达系统时,随机移除一个正在服务的顾客.基于状态空间的不同排序,我们研究了描述系统演化行为的两个不同的多维马尔可夫链.分别写出了对应这两个不同的多维马尔可夫链的转移率矩阵.给出了计算这两个马尔可夫链平稳分布的算法.得到了一些重要的排队性能指标.最后,我们还提供了几个数值实例来说明某些参数对系统性能指标的影响.
第七章分析了一个具有二次可选服务和多重休假的BMAP/G/1 G-排队系统.其中正顾客的到达服从批马尔可夫到达过程,而负顾客的到达服从马尔可夫到达过程.正顾客经过第一阶段的服务(初次必选服务)后有可能进入第二阶段的服务(二次可选服务).负顾客到达系统时移除一个正在服务的正顾客(若有).一旦系统空,服务台立即进行多重休假.利用补充变量法和删失技术以及RG-分解方法,得到了队长的分布.利用更新过程的理论,得到了平均忙期长度.
In this Ph.D. thesis, we investigate several classes of complex queueing systems which include G-queueing systems, retrial queueing systems, vacation queueing systems, discrete-time queueing systems, Markov arrival queueing systems and their mixture. This Ph.D. thesis is organized as follows.
Chapter 1 is preface. The historical background, the subject, the recent develop-ment of queueing theory and the main results and innovative contributions of this thesis are introduced.
In Chapter 2, we analyze an M/G/1 retrial queue with negative customers and non-exhaustive vacations subject to the server breakdowns and repairs. Arrivals of both positive customers and negative customers are two independent Poisson processes. A breakdown at the busy server is represented by the arrival of a negative customer which causes the the customer being in service to be lost. The server takes a vacation of random length after an exponential time when the server is up. It is assumed that the server has arbitrary repair time and vacation time distributions. We develop a new method to discuss the stable condition by finding absorb distribution and using the stable condition of classical M/G/1 queue. By applying the supplementary variables method, we obtain the system of equilibrium equations. By taking generating functions of these equations, we derived the solutions for queueing measures. Furthermore, we analyze the system reliability by regarding a new queueing system where the failure states of the server are assumed to be absorbent states. We also analyze the busy period of the system. Some special cases of interest are discussed and some known results have been derived. The effects of various parameters on the system performance are analyzed numerically.
In Chapter 3, we discuss an MX/G/1 retrial queue with impatient customers and feedback under N-policy vacation schedule subject to disastrous failures at which times all customers in the system are lost. The necessary and sufficient condition for the stability of the system is derived. Applying Foster's criterion and Kaplan's condition, we study the ergodicity of the embedded Markov chain. By applying the supplementary variables method, we obtain the system of equilibrium equations. By taking generating functions of these equations, we obtain the steady-state distributions of the server state and the number of customers in the orbit along with various performance measures. In addition, we analyze some reliability problems. Further, some special cases of interest are discussed.
In Chapter 4, we consider an M/G/1 retrial G-queue with preemptive resume prior-ity and collisions under linear retrial policy subject to the server breakdowns and delayed repairs. A breakdown at the busy server is represented by the arrival of a negative cus-tomer which causes the the customer being in service to be lost. The stability condition of the system is derived. Using generating function technique, the steady-state distri-butions of the server state and the number of customers in the orbit are obtained along with some interesting and important performance measures. The stochastic decompo-sition property is investigated. Further, some special cases of interest are discussed. Finally, numerical illustrations are provided.
In Chapter 5, we deal with a discrete-time Geo/G/1 retrial queue with negative customers and general retrial times. Negative customers will make the customer being in service lost but has no effect to the orbit. We analyze the Markov chain underlying the considered queueing system. By applying the supplementary variables method, we derive Kolmogorov equations. By taking generating functions for these equations, we obtain the system state distribution as well as the orbit size and the system size distributions in terms of their generating functions. These generating functions yield exact expressions for different performance measures. Besides, the stability condition of the system is derived. Further, the special case of no negative customers is discussed. Finally, some numerical examples are provided to illustrate the impact of several parameters on some crucial performance characteristics of the system.
In Chapter 6, we study the MAP/PH/N retrial queue with finite number of sources and MAP arrivals of negative customers operating in a finite state Markovian random environment. The arrival of a negative customer with equal probability goes to any busy server to remove the customer being in service. Two different types of multi-dimensional Markov chains describing the behavior of the system based on state space arrangements are investigated. The special features of the two formulations are discussed. The algorithms for calculating the stationary state probabilities are elaborated. Main performance measures are obtained. Illustrative numerical examples are presented.
In Chapter 7, we investigate an BMAP/G/1 queues with negative customers, sec-ond optional service and multiple vacations. Arrivals of positive customers and negative customers follow a batch Markovian arrival process (BMAP) and Markovian arrival pro- cess (MAP) respectively. After completion of the essential service of a customer, it may go for a second phase of service. The arrival of a negative customer removes the customer being in service. The server leaves for a vacation as soon as the system empties and is allowed to take repeated (multiple) vacations. By using the supplementary variables method, the censoring technique and the RG-factorization, we obtain the queue length distributions. We obtain the mean of the busy period based on the renewal theory.
第一章是绪论,简要地介绍了排队论的历史背景、研究内容、发展现状以及本文所做的主要工作和主要的创新点.
第二章分析了一个具有非空竭服务随机休假策略的M／G／1重试可修G-排队系统.正、负顾客的到达服从两个独立的泊松过程.其中负顾客的到达不仅移除一个正在服务的顾客,而且致使服务台损坏.服务台采取随机休假策略,即服务台的工作时间服从指数分布.休假时间和修理时间均服从任意分布.我们发展了一个新方法,即通过构造一个具有吸收态的马尔可夫链且利用经典M／G／1排队系统的稳态条件求得了该系统稳态存在的充分必要条件.利用补充变量方法建立了系统的稳态方程组,然后利用取母函数得到了系统状态和重试组中顾客数联合分布的概率母函数,进而求得了系统在稳态下的一系列重要的排队性能指标.而后,我们对系统进行可靠性分析.通过将服务台发生故障视为马尔可夫链的吸收状态,构造了一个新的排队系统,由此得到了一些重要的可靠性指标.此外,我们还分析了系统的忙期和一些有趣的特例.最后,我们还给出了几个数值实例来说明某些参数对系统性能指标的影响.
第三章研究了一个具有不耐烦顾客、灾难、N策略休假和反馈机制的MX／G／1重试可修G-排队系统.利用Foster准则和Kaplan条件得到了系统稳态存在的充分必要条件.利用补充变量法,建立了系统稳态时的平衡方程组.通过求解这些方程,得到了系统状态和重试组中顾客数联合分布的概率母函数.进而求得了系统在稳态下的一系列重要的排队性能指标.而后,我们对系统进行可靠性分析,得到了一些重要的可靠性指标.最后还对一些特例进行了分析.
第四章分析了一个具有抢占优先权和冲突机制且服务台不可靠、修理滞后的M／G／1重试可修G-排队系统.假设重试策略为线性重试策略,负顾客移除一个正在服务的正顾客同时引起服务台损坏,但损坏的服务台不会立即得到修理.推导出了系统稳态存在的充分必要条件.利用取母函数方法,得到了服务台状态和重试组中顾客数联合分布的概率母函数.进而求得了系统在稳态下的一系列重要的排队性能指标.而后,我们对系统进行可靠性分析,得到了一些重要的可靠性指标.此外,我们还讨论了随机分解性质,以及分析了几个特例.最后,给出了几个数值实例.
第五章考虑了一个一般重试时间的离散时间Geo/G/1重试G-排队系统.其中负顾客带走一个正在服务的顾客,而对重试组中的顾客无影响.通过补充剩余服务时间,写出了嵌入马氏链的转移概率.而后,利用取母函数法求解Kolmogorov方程,得到了重试组队长和系统队长的概率母函数.进而得到了一系列重要的排队指标.此外,我们还推导出了系统的稳态存在条件.以及对无负顾客时的特例进行了分析.最后我们还通过几个具体的数值实例来演示一些参数对系统关键性能指标的影响.
第六章讨论了一个在马尔可夫随机环境中运行的有限源MAP/PH/N重试G-排队系统.其中正、负顾客的到达都服从马尔可夫到达过程.负顾客到达系统时,随机移除一个正在服务的顾客.基于状态空间的不同排序,我们研究了描述系统演化行为的两个不同的多维马尔可夫链.分别写出了对应这两个不同的多维马尔可夫链的转移率矩阵.给出了计算这两个马尔可夫链平稳分布的算法.得到了一些重要的排队性能指标.最后,我们还提供了几个数值实例来说明某些参数对系统性能指标的影响.
第七章分析了一个具有二次可选服务和多重休假的BMAP/G/1 G-排队系统.其中正顾客的到达服从批马尔可夫到达过程,而负顾客的到达服从马尔可夫到达过程.正顾客经过第一阶段的服务(初次必选服务)后有可能进入第二阶段的服务(二次可选服务).负顾客到达系统时移除一个正在服务的正顾客(若有).一旦系统空,服务台立即进行多重休假.利用补充变量法和删失技术以及RG-分解方法,得到了队长的分布.利用更新过程的理论,得到了平均忙期长度.
In this Ph.D. thesis, we investigate several classes of complex queueing systems which include G-queueing systems, retrial queueing systems, vacation queueing systems, discrete-time queueing systems, Markov arrival queueing systems and their mixture. This Ph.D. thesis is organized as follows.
Chapter 1 is preface. The historical background, the subject, the recent develop-ment of queueing theory and the main results and innovative contributions of this thesis are introduced.
In Chapter 2, we analyze an M/G/1 retrial queue with negative customers and non-exhaustive vacations subject to the server breakdowns and repairs. Arrivals of both positive customers and negative customers are two independent Poisson processes. A breakdown at the busy server is represented by the arrival of a negative customer which causes the the customer being in service to be lost. The server takes a vacation of random length after an exponential time when the server is up. It is assumed that the server has arbitrary repair time and vacation time distributions. We develop a new method to discuss the stable condition by finding absorb distribution and using the stable condition of classical M/G/1 queue. By applying the supplementary variables method, we obtain the system of equilibrium equations. By taking generating functions of these equations, we derived the solutions for queueing measures. Furthermore, we analyze the system reliability by regarding a new queueing system where the failure states of the server are assumed to be absorbent states. We also analyze the busy period of the system. Some special cases of interest are discussed and some known results have been derived. The effects of various parameters on the system performance are analyzed numerically.
In Chapter 3, we discuss an MX/G/1 retrial queue with impatient customers and feedback under N-policy vacation schedule subject to disastrous failures at which times all customers in the system are lost. The necessary and sufficient condition for the stability of the system is derived. Applying Foster's criterion and Kaplan's condition, we study the ergodicity of the embedded Markov chain. By applying the supplementary variables method, we obtain the system of equilibrium equations. By taking generating functions of these equations, we obtain the steady-state distributions of the server state and the number of customers in the orbit along with various performance measures. In addition, we analyze some reliability problems. Further, some special cases of interest are discussed.
In Chapter 4, we consider an M/G/1 retrial G-queue with preemptive resume prior-ity and collisions under linear retrial policy subject to the server breakdowns and delayed repairs. A breakdown at the busy server is represented by the arrival of a negative cus-tomer which causes the the customer being in service to be lost. The stability condition of the system is derived. Using generating function technique, the steady-state distri-butions of the server state and the number of customers in the orbit are obtained along with some interesting and important performance measures. The stochastic decompo-sition property is investigated. Further, some special cases of interest are discussed. Finally, numerical illustrations are provided.
In Chapter 5, we deal with a discrete-time Geo/G/1 retrial queue with negative customers and general retrial times. Negative customers will make the customer being in service lost but has no effect to the orbit. We analyze the Markov chain underlying the considered queueing system. By applying the supplementary variables method, we derive Kolmogorov equations. By taking generating functions for these equations, we obtain the system state distribution as well as the orbit size and the system size distributions in terms of their generating functions. These generating functions yield exact expressions for different performance measures. Besides, the stability condition of the system is derived. Further, the special case of no negative customers is discussed. Finally, some numerical examples are provided to illustrate the impact of several parameters on some crucial performance characteristics of the system.
In Chapter 6, we study the MAP/PH/N retrial queue with finite number of sources and MAP arrivals of negative customers operating in a finite state Markovian random environment. The arrival of a negative customer with equal probability goes to any busy server to remove the customer being in service. Two different types of multi-dimensional Markov chains describing the behavior of the system based on state space arrangements are investigated. The special features of the two formulations are discussed. The algorithms for calculating the stationary state probabilities are elaborated. Main performance measures are obtained. Illustrative numerical examples are presented.
In Chapter 7, we investigate an BMAP/G/1 queues with negative customers, sec-ond optional service and multiple vacations. Arrivals of positive customers and negative customers follow a batch Markovian arrival process (BMAP) and Markovian arrival pro- cess (MAP) respectively. After completion of the essential service of a customer, it may go for a second phase of service. The arrival of a negative customer removes the customer being in service. The server leaves for a vacation as soon as the system empties and is allowed to take repeated (multiple) vacations. By using the supplementary variables method, the censoring technique and the RG-factorization, we obtain the queue length distributions. We obtain the mean of the busy period based on the renewal theory.
引文
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