基于极大代数的TCP协议分析

The Analysis of TCP Protocol Based on Max-Plus

TCP协议是Internet中的核心协议之一.TCP协议动态行为对于研究网络的服务质量有重要的意义.本文用离散事件动态系统的方法研究TCP协议的动态行为,将Baccelli等提出的极大代数模型推广到包含有重发机制的一般情况,详细阐述了如何用极大代数系统矩阵的形式描述TCP协议的重发行为;并以此为基础,进一步讨论了TCP协议中发送窗口与TCP链路有效分组传输率之间的关系.理论分析和实验仿真表明:极大代数方法具有描述TCP协议的各种重要动态行为（其中包括具有普遍意义的重发机制）的能力.其中作为例子,基于该方法导出的有效分组传输率与发送窗口之间的显示表达式,证实了即使只考虑单一连接情形下,TCP链路中仍存在最大发送窗口的合理性.极大代数模型的优点在于可对多路由器形成的链路（而不是仅限于单个瓶颈路由器）上的TCP传输进行定量分析,为分析网络环境下的TCP动态行为提供了一条新的途径.

This paper gives the particular representation in different conditions when data are normally transmitted, retransmitted one times and retransmitted several times. Furthermore, the explicit relationship between the goodput and the sender's window size are drived based on this model as follows. Under the assumption that the system model has deterministic parameters, the sequence of reference windows will become ultimately periodic, with values in an integer interval of form [min(W), max(W)], where min(W) and max(W) are the minimal and maximal sizes of sender's window in a cycle, respectively. By calculating the eigenvalue and the eigenvector of Max-Plus algebra matrix of the sequence in a cycle, authors establish the expression of the good-put of a TCP link as a function of the maximal window size. This function reveals the tendency of change in capability of transmission of TCP link with respect to the maximal window size. Theoretical analysis and simulations show that May-Plus method has the potential to describe all the important aspects of dynamic behaviors of TCP protocol (including retransmission) and that the explicit relationship derived based on this model justifies the existence of an optimal maximum window size, even for a single TCP link. The advantage of this Max-Plus model is that TCP transportation over links containing several routers, not only a single router, can be quantified, therefore provides a promising way to analyze the dynamic behaviors of TCP in a network environment.