基于马尔可夫骨架过程的排队模型及其在Web信息系统中的应用
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摘要
随着Internet技术的飞速发展,各种Web信息系统大量出现,对其进行性能分析成为迫切的现实需要。本文从Web信息系统的运行机理出发,建立了系统的性能分析模型,然后借助于马尔可夫骨架过程理论,研究了Web服务器的休假排队模型。
首先,研究了Web信息系统的信息传输和处理的一般过程和系统规律特点,将一个Web信息系统抽象为一个排队网络系统,构建了系统的性能分析模型。其次,总结分析了排队系统中的马尔可夫骨架过程方法。最后,研究了Web服务器的休假排队模型。现有分析都假定“顾客”输入的时间间隔为独立同分布(负指数分布)的随机变量,而采用经典排队模型M/M/N来刻画。在实际网络信息系统中,“顾客”的输入常常出现一些与经典模型大不一样的情况,因此有必要研究更一般的排队模型。本文重点研究了4类排队模型:同步单重休假的GI/G/N排队系统、同步多重休假的GI/G/N排队系统、带d-策略休假的GI/G/N排队系统、异步多重休假的GI/G/N排队系统。利用马尔可夫骨架过程方法,求得了这些排队模型队长的瞬时分布。本文模型的到达时间间隔和服务时间均相互独立但服从一般分布,且引入了多种休假规则,使得该模型能更好地刻画实际问题。
本文的主要结果有:
(1)建立了Web信息系统多服务器休假排队模型。本文模型放宽了现行建模的假设,即不要求Web请求、Web服务时间服从负指数分布,并引入GI/G/N模型来刻画系统,从而克服了以往Web信息系统逻辑建模的一些缺陷。
(2)借助于马尔可夫骨架过程理论,给出了同步单重休假的GI/G/N排队系统队长的瞬时分布所满足的方程组,并得到其概率分布是这些方程的最小非负解。
(3)借助于马尔可夫骨架过程理论,给出了同步多重休假的GI/G/N排队系统队长的瞬时分布所满足的方程组,并得到其概率分布是这些方程的最小非负解。
(4)借助于马尔可夫骨架过程理论,给出了带d-策略休假的GI/G/N排队系统队长的瞬时分布所满足的方程组,并得到其概率分布是这些方程的最小非负解。
(5)借助于马尔可夫骨架过程理论,给出了异步多重休假的GI/G/N排队系统队长的瞬时分布所满足的方程组,并得到其概率分布是这些方程的最小非负解。
With the rapid development of Internet technology, a variety of Web information systems have emerged and the analyses of their performance have become urgent and practical needs. Based on the operational mechanism of Web information system, this thesis establishes a systematic model of performance analysis and researches the vacation-queuing model of Web server using the Markov skeleton-processing theory. Firstly, this thesis probes the Web information system's rules and characteristics together with the general process of transmitting and transacting information, and builds a systematic model of performance analysis, regarding a Web information system as an abstract queuing network system. Secondly, the Markov skeleton-processing theory in queuing system is analyzed and summarized. Finally, the vacation queuing model of Web server is also under research. All the previous researches assume that the intervals of customers' inputs are random variables distributed independently and identically (the distribution of negative exponential), and the system is portrayed by the classical queuing models, such as M/M/N. In the actual network information systems, however, some customers' inputs are usually of great difference from the classical models. Consequently, it is necessary that a more general queuing model be researched. In this thesis 4 queuing models are focused on, namely, GI/G/N queuing system with synchronous single vacation, GI/G/N queuing system with synchronous multiple vacation, GI/G/N queuing system with d-policy vacation and GI/G/N queuing system with asynchronous multiple vacation. By means of the Markov skeleton process theory, the transient distribution of the queue length of these queuing models is achieved. In this thesis, the arrival intervals and the service time in the model are independent of each other though kept to the general distribution, and various vacation regulations are adopted so that the practical problems can be portrayed better in the model.
The major findings of this thesis include:
(1) The multiple server vacation queuing model of Web information system is established. The system model in this thesis broadens the existing modeling assumptions, that is, Web requests and the Web service time are not required to keep to the distribution of negative exponential. Besides, the GI/G/N model is introduced to portray the system so that some previous defects of logical modeling in Web information system are corrected.
(2) With the Markov skeleton process theory, the equations are put forward that accord with the transient distribution of the queue length of GI/G/N queuing system with synchronous single vacation, and it proves that the probability distribution is the smallest non-negative solution of these equations.
(3) With the Markov skeleton process theory, the equations are presented that accord with the transient distribution of the queue length of GI/G/N queuing system with synchronous multiple vacation, and it proves that the probability distribution is the smallest non-negative solution of these equations.
(4) With the Markov skeleton process theory, the equations are brought forward that accord with the transient distribution of the queue length of GI/G/N queuing system with d-policy vacation, and it proves that the probability distribution is the smallest non-negative solution of these equations.
(5) With the Markov skeleton process theory, the equations are proposed that accord with the transient distribution of the queue length of GI/G/N queuing system with asynchronous multiple vacation, and it proves that the probability distribution is the smallest non-negative solution of these equations.
首先,研究了Web信息系统的信息传输和处理的一般过程和系统规律特点,将一个Web信息系统抽象为一个排队网络系统,构建了系统的性能分析模型。其次,总结分析了排队系统中的马尔可夫骨架过程方法。最后,研究了Web服务器的休假排队模型。现有分析都假定“顾客”输入的时间间隔为独立同分布(负指数分布)的随机变量,而采用经典排队模型M/M/N来刻画。在实际网络信息系统中,“顾客”的输入常常出现一些与经典模型大不一样的情况,因此有必要研究更一般的排队模型。本文重点研究了4类排队模型:同步单重休假的GI/G/N排队系统、同步多重休假的GI/G/N排队系统、带d-策略休假的GI/G/N排队系统、异步多重休假的GI/G/N排队系统。利用马尔可夫骨架过程方法,求得了这些排队模型队长的瞬时分布。本文模型的到达时间间隔和服务时间均相互独立但服从一般分布,且引入了多种休假规则,使得该模型能更好地刻画实际问题。
本文的主要结果有:
(1)建立了Web信息系统多服务器休假排队模型。本文模型放宽了现行建模的假设,即不要求Web请求、Web服务时间服从负指数分布,并引入GI/G/N模型来刻画系统,从而克服了以往Web信息系统逻辑建模的一些缺陷。
(2)借助于马尔可夫骨架过程理论,给出了同步单重休假的GI/G/N排队系统队长的瞬时分布所满足的方程组,并得到其概率分布是这些方程的最小非负解。
(3)借助于马尔可夫骨架过程理论,给出了同步多重休假的GI/G/N排队系统队长的瞬时分布所满足的方程组,并得到其概率分布是这些方程的最小非负解。
(4)借助于马尔可夫骨架过程理论,给出了带d-策略休假的GI/G/N排队系统队长的瞬时分布所满足的方程组,并得到其概率分布是这些方程的最小非负解。
(5)借助于马尔可夫骨架过程理论,给出了异步多重休假的GI/G/N排队系统队长的瞬时分布所满足的方程组,并得到其概率分布是这些方程的最小非负解。
With the rapid development of Internet technology, a variety of Web information systems have emerged and the analyses of their performance have become urgent and practical needs. Based on the operational mechanism of Web information system, this thesis establishes a systematic model of performance analysis and researches the vacation-queuing model of Web server using the Markov skeleton-processing theory. Firstly, this thesis probes the Web information system's rules and characteristics together with the general process of transmitting and transacting information, and builds a systematic model of performance analysis, regarding a Web information system as an abstract queuing network system. Secondly, the Markov skeleton-processing theory in queuing system is analyzed and summarized. Finally, the vacation queuing model of Web server is also under research. All the previous researches assume that the intervals of customers' inputs are random variables distributed independently and identically (the distribution of negative exponential), and the system is portrayed by the classical queuing models, such as M/M/N. In the actual network information systems, however, some customers' inputs are usually of great difference from the classical models. Consequently, it is necessary that a more general queuing model be researched. In this thesis 4 queuing models are focused on, namely, GI/G/N queuing system with synchronous single vacation, GI/G/N queuing system with synchronous multiple vacation, GI/G/N queuing system with d-policy vacation and GI/G/N queuing system with asynchronous multiple vacation. By means of the Markov skeleton process theory, the transient distribution of the queue length of these queuing models is achieved. In this thesis, the arrival intervals and the service time in the model are independent of each other though kept to the general distribution, and various vacation regulations are adopted so that the practical problems can be portrayed better in the model.
The major findings of this thesis include:
(1) The multiple server vacation queuing model of Web information system is established. The system model in this thesis broadens the existing modeling assumptions, that is, Web requests and the Web service time are not required to keep to the distribution of negative exponential. Besides, the GI/G/N model is introduced to portray the system so that some previous defects of logical modeling in Web information system are corrected.
(2) With the Markov skeleton process theory, the equations are put forward that accord with the transient distribution of the queue length of GI/G/N queuing system with synchronous single vacation, and it proves that the probability distribution is the smallest non-negative solution of these equations.
(3) With the Markov skeleton process theory, the equations are presented that accord with the transient distribution of the queue length of GI/G/N queuing system with synchronous multiple vacation, and it proves that the probability distribution is the smallest non-negative solution of these equations.
(4) With the Markov skeleton process theory, the equations are brought forward that accord with the transient distribution of the queue length of GI/G/N queuing system with d-policy vacation, and it proves that the probability distribution is the smallest non-negative solution of these equations.
(5) With the Markov skeleton process theory, the equations are proposed that accord with the transient distribution of the queue length of GI/G/N queuing system with asynchronous multiple vacation, and it proves that the probability distribution is the smallest non-negative solution of these equations.
引文
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[3]Crovella M E,Bestavros A.Self-Similarity in World Wide Web traffic:Evidence and possible causes.IEEE/ACM Transactions on Networking,1997,5(6):835-846
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