第二章 计算机网络建模理论 上章回顾 网络设计与优化非常重要 网络设计、资源分配与流量控制需要计算机网络理论分析 第二章 计算机网络建模理论 上章回顾 网络设计与优化非常重要 网络设计、资源分配与流量控制需要计算机网络理论分析 业务的随机性是造成网络拥塞及性能恶化的主要原因 计算机网络性能分析需要建立概率模型 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking.

一个典型的计算机网络

一个典型的排队模型

本章主要内容(1/2) 2.1 计算机网络建模的对象与原则 2.2 通信业务源的概率模型化 2.2.1 随机事件的概率特征量及物理意义 2.2.2 典型概率分布及随机过程 2.2.3 纯随机事件的概率模型 2.2.4 平滑事件的概率模型 2.2.5 突发事件的概率模型 2.2.6 业务源概率模型参数的匹配

本章主要内容 2.3 实际业务源的建模思考 2.4 计算机网络的排队模型化 2.5 典型计算机网络的建模 2.6 小结 2.4.2 计算机网络的排队网络模型 2.4.1 通信处理单接点的排队模型 2.4.3 排队过程的马尔可夫过程描述 2.5 典型计算机网络的建模 2.5.1 电路交换网 2.5.2 移动通信网 2.5.3 分组交换网 2.5.4 ATM网 2.6 小结

网络系统的建模对象 （1）网络结构(Structure) （2）网络控制机制(Strategy) （3）业务量特性、尤其是随机性(Traffic) MAN Stochastic Traffic User demands MACHINE Deterministic Structure Hardware Strategy Software

2.1 计算机网络建模的基本准则 真实性(real and precise) 可操作性(implementable) 尽量精确地描述实际业务的概率特征 可操作性(implementable) 要易于进行数学分析或计算机仿真 通用性(unified) 同时能描述多种不同业务的概率特征 可匹配性(matchable) 模型参数应能容易地从实际业务中拟合出来 保守性(conservative) 近似计算或仿真得到的网络性能应不劣于实际网络性能（安全近似，做最坏的打算）

2.2 业务源的概率模型化 计算机网络性能分析需要概率模型 业务的随机性是网络性能恶化的主要原因 业务需求的产生是随机的、而且往往是突发的 业务所需要的服务时间也是随机的 有时甚至可得到的网络资源也是随机的 业务的随机性是网络性能恶化的主要原因 如果业务需求是确定性的，网络不会发生拥塞（网络的设计容量永远大于需求） 业务需求的随机（不确定）性越大、网络性能的恶化越严重

随机过程与随机服务过程概论 2.1 概率空间 1、随机试验： 2、基本事件ω：表示试验的一个最基本的不可再分解的结果 2.1 概率空间 1、随机试验： 在相同试验条件下可重复进行 每次试验结果不止一个 每次试验之前不能预先精确确定哪一种结果发生 2、基本事件ω：表示试验的一个最基本的不可再分解的结果 （由若干基本结果组成的事件称为复合事件） 3、样本空间Ω：表示一切基本事件所组成的总体，即Ω={ω} 4、事件：它是样本空间的子集 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking.

5 几个概率 1）补 当且仅当A不发生的事件 2）并 事件A或事件B至少有一个发生 3）交 当且仅当事件A与B同时发生 4）必然事件：集合Ω称为必然事件 5）空集Φ：不包含任何元素的集合。 6）不相容： 7）互不相容：若多个事件中任意两个事件都不相容，则称这多个事件是互不相容的 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking.

6 集合论与概率论术语比较 记 号 集合论 概率论  空间 全集 样本空间 必然事件  空集 不可能事件  元素 基本事件 A 记 号 集合论 概率论  空间 全集 样本空间 必然事件  空集 不可能事件  元素 基本事件 A 的子集 事件 Ac A的余集 A的对立事件（补） A B A是B的子集 A发生B必发生 A=B A与B相等 事件A与B相等 A B A与B的和集（并集） 事件A与B至少有一个发生 AB A与B的交集（积集） 事件A与B同时发生 A-B A与B的差集 事件A发生而事件B不发生 AB= A与B没有公共元素 事件A与B互不相容（互斥） The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking.

7 概率的性质 性质1 性质2（有限可加性） 性质3（加法公式） 一般地，若 称为多除少补原理 性质4 性质5 若 性质6（连续性） 性质5 若 性质6（连续性） The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking.

“事件”与“概率” 事件：粗略地可视为实验的结果,而严谨定义基于集合论,使得事件间的关系和运算可借用集合的关系和运算。 概率：严谨的定义基于测度论，简洁定义足以使其担当“量度”事件发生可能性大小的角色。

随机事件的两种描述法 (1) 随机事件发生间隔的概率分布描述法 (2) 随机事件的点过程描述(记数过程)法

定义2-1： 设有样本空间Ω={ω}，F是由Ω的一些子集A（一般是不可列的）组成的集合。 若F满足以下条件： 则称F是Ω中的一个σ代数 显然由条件1，2知 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking. 实质上，F即为一个随机事件族，将Ω与定义在Ω上的σ代数F一起称为可测空间，记（ Ω ，F ），并简称F中的元素为事件

定义2-2： 设对于任一事件 ，P(A)是定义在σ代数F上的实值集函数，若P(A)满足以下条件： 则称P是F上的概率测度，简称概率 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking.

2.2 条件概率 定义2-3：设概率空间 若满足： 则称P(A/B)为在事件B已发生的条件下，事件A发生的条件概率 定义2-3：设概率空间 若满足： 则称P(A/B)为在事件B已发生的条件下，事件A发生的条件概率 1、如果P(B)>0，则P(A/B)可由（2.2.2）唯一确定；若P(B)=0,则P(A/B)可在[0，1]中任意取值。 当P(B)>0时，给定B后，P(A/B)即为A的函数，则由定义2知： The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking.

2、若P(A/B)=P(A),则有 满足（2.2.4）的A、B为统计独立事件 推广到n个事件：： The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking. 则称A1, A2,······, An是相互独立的

3、设B1, B2,······, Bn是互不相容的事件，即 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking.

4、设B1,B2,······, Bn是样本空间Ω的一个划分， 式（2.2.7）称为贝叶斯公式，P(Bi)为先验概率，P(Bi/A)为后验概率 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking.

2.3 随机变量和随机过程 定义2-4 设某随机试验的概率空间为（ Ω ， F ，P）,若对于每一次试验结果 ，均有某实值函数 与之对应，即X(ω)是试验结果ω的一个函数，且对于任意实数x，集合 ，则称X(ω)为随机变量，E称之为状态集或称为状态空间。 最常见的状态集合E有： 非负整数集合N+={0,1,2,······} 整数集合N={···,-2,-1,0,1,2,···} 实数集合 非负实数集合 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking. 1）、2）两种情况称为可列无限集合 当E为可列有限集（如，E={1,2,···,N}）或可列无限集时，称X(ω)为离散随机变量

例2-2 “测试12灯泡的平均寿命” 样本空间Ω={ω： ω=（ ω1，ω2 ，···，ω12 ）} 例2-2 “测试12灯泡的平均寿命” 样本空间Ω={ω： ω=（ ω1，ω2 ，···，ω12 ）} 定义2-5 一族无穷多个随机变量组成的集合 称为一个随机过程，其中集合T称为参数集，各个Xt是定义在相同的概率空间（ Ω ， F ，P）上，各随机变量Xt均在同一状态空间E中取值。 另一个定义：随机过程是以参数集T为定义域，以随机变量为值的“算子” The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking. 1）离散随机过程 2）连续随机过程

2.4 随机变量的分布函数和随机过程的概率分布 2.4.1 设X为一离散随机变量，其可能的一切取值为 显然有： 概率分布函数定义为： The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking.

2.4 随机变量的分布函数和随机过程的概率分布 2.4.2 当X为一连续随机变量时，则其分布往往由它的分布密度f(x)给出 则f(x) 称为X的概率密度函数；F(x)为X的分布函数 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking. 2.4.3 分布函数F(x)具有下述性质 1）F(X)为单调非降函数 2）F(x)为右连续的 3）

2.4.4 二维随机变量（X，Y） 则F(x,y) 称为二维随机变量（X，Y）的联合概率分布函数 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking. 则F1(x)和 F2(x)分别称为二维随机变量（X，Y）关于X和关于Y的边际概率分布函数

则f1(x)和f2(x)分别称为二维随机变量（X，Y）关于X和关于Y的边际概率密度函数 6、任意有限维随机变量 则称F(x1, x2,···, xn)为n维随机变量(X1, X2,···, Xn)的联合概率分布函数 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking. 则称f(x1, x2,···, xn)为n维随机变量(X1, X2,···, Xn)的联合概率密度函数

7、在随机过程中， 为一个随机过程 对于固定时刻 为一个随机变量，则称 为随机过程 的一维分布函数 7、在随机过程中， 为一个随机过程 对于固定时刻 为一个随机变量，则称 为随机过程 的一维分布函数 则f1(t1;x1) 称为随机过程X的一维概率密度函数 8、为刻画随机过程在不同时刻状态之间的联系，引入随机过程多维分布函数 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking. 则F2(t1, t2;x1, x2) 称为随机过程X的二维概率分布函数 则f2(t1, t2;x1, x2) 称为随机过程X的二维概率密度函数

9、对n维情况，其n维联合分布为： 随机过程有限维分布族为： The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking.

2.5 数学期望值和母函数 定义2-6 设X为离散随机变量，其所有可能的取值为 ，相应的概率为 。若 存在，称 E[X]为X的数学期望值，简称均值。 定义2-7 设X为连续型随机变量，且具有分布密度函数f(x), 若积分 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking. 则称 为连续型随机变量X的数学期望值

例2-3：如果有： 则 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking.

例2-4 某元件的寿命X具有如下分布： 则 例2-5 某随机变量X具有分布： 则 2.5.4 若Y=g(x), 则 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking. 2.5.4 若Y=g(x), 则 若[Y，X]为连续型随机变量, 则

定义2.8 随机变量X的方差 定义2.9 设X为整数型随机变量，其概率分布为： 则 称为随机变量X的母函数 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking. 称为随机变量X的母函数

定义2.10 设 为一随机过程，若对于每一个 ，随 机变量 的均值与方差均存在，令 定义2.10 设 为一随机过程，若对于每一个 ，随 机变量 的均值与方差均存在，令 则称 为随机变量X的均值函数与方差函数 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking.

矩、协方差 定义2.11 设r.v.X有 称 为X的k阶原点矩 称 为X的k阶绝点矩 称 为X的k阶中心矩 称 为X的k阶绝对中心矩 定义2.12 若E{[X-E(X)][Y-E(Y)]}存在，称 cov(X,Y)=E {[X-E(X)][Y-E(Y)]} 为随机变量X和Y的协方差，称 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking. 为随机变量X和Y的相关系数；若 为随机变量X和Y不相关， 表示X与Y的线性相关程度

随机事件特征值的物理意义 自相关系数是衡量随机事件之间相互关联性的重要参数

矩、协方差 定义2.13 设r.p.X(t),我们定义 称为r.p.X(t)的协方差函数 定义2.14 设r.p.X(t),有 R(s,t)=E[X(s)X(t)] 称为r.p.X(t)的相关函数 定义2.15 设r.p.X(t),有 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking. 称为r.p.X(t)的相关系数

矩、协方差 定义2.16 设r.p.X(t)和Y(t),我们定义 称为r.p.X(t)的互协方差函数 RXY(s,t)=E[X(s)Y(t)] 称为r.p.X(t)和Y(t)的互相关函数 定义2.18 设r.p.X(t)和Y(t),有 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking. 称为r.p.X(t)和Y(t)的互相关系数 如果r.p.X(t)和Y(t)相互独立，则它们一定互不相关；反之，两个随机过程 互不相关，一般不能推出它们相互独立

随机事件的两种描述法 (1) 随机事件发生间隔的概率分布描述法 (2) 随机事件的点过程描述(记数过程)法 结论：记数过程描述法包含更多的概率信息，它可以描述随机过程在不同时间尺度(time scale)内的概率特征，而时间间隔描述法只能描述随机事件的长时间特征 (2) 随机事件的点过程描述(记数过程)法

随机事件的概率特征描述 事件发生间隔{Xn}的特征量 随机事件点过程{N(t)}的特征量(m(t), v(t), u3(t)) 均值m=1/λ; 方差v2； 三阶中心矩μ3 方差系数 自相关系数 歪度系数 随机事件点过程{N(t)}的特征量(m(t), v(t), u3(t)) Index of Dispersion for Interval (IDI) Index of Dispersion for Count (IDC)

随机事件特征量的物理意 业务强度是衡量随机事件发生强度的基本参数 方差系数是衡量随机事件抖动的重要参数

随机事件特征量的物理意义 歪度系数是衡量随机事件对称性的重要参数

2.6 随机服务过程的基本概念 1、服务系统：顾客和服务员 2、排队现象或拥挤现象 2.6 随机服务过程的基本概念 1、服务系统：顾客和服务员 2、排队现象或拥挤现象 3、顾客的到达过程是随机过程，服务完一个顾客的服务时间也是一个随机过程，因此，服务系统的整个过程也是随机的。正是由于这种随机性，才不可避免地导致了排队现象，这类服务系统称为随机服务系统或简称排队系统，相应的理论方法称之为随机服务理论或排队论 4、存在合理平衡问题：顾客的等待和服务机构的数量（或服务水平）之间存在一个合理平衡的问题 5、设计最优的随机服务系统和最佳的网络系统设计 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking.

2.7 随机服务系统的组成部分 随机服务系统 服务机构 1、组成： 排队 顾客到达 排队规则 服务时间分布 2.7 随机服务系统的组成部分 排队 顾客到达 服务机构 排队规则 服务时间分布 随机服务系统 1、组成： 顾客到达过程：主要描述各类顾客按什么样的规律抵达服务系统。 Mt表示[0，t）时间内到达系统中的顾客数，连续时间参数，离散状态空间； （1）顾客总体数是有限还是无限；（2）顾客到达方式是单个还是成批；（3）顾客到达的概率特性。 排队规则：主要描述服务机构是否允许顾客排队，顾客对排队长度、时间的容忍程度以及在排队队列中等待服务的顺序。 （1）损失制系统 （2）等待制系统：先到先服务（FCFS）；后到先服务（LCFS）；随机服务；优先级服务 （3）混合制系统：缓存有限；时间有限 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking.

3） 服务过程 2、排队系统的描述 （1）服务员数目：在多个服务员情况下，是串联还是并联 （2）是逐个进行服务还是成批服务 （3）服务时间的概率分布 例1-6 M/G/1/ M:顾客到达过程为Possion过程 G:服务时间为一般分布 1:服务员数为1 ：系统容量为无穷大的等待制排队系统 例1-7 GI/Ek/c/K/ /FcFs 一般混合制排队系统 GI:独立到达，即相继到达顾客的时间间隔相互独立，服从相同的一般分布 :服务时间服从k阶爱尔朗分布 c：服务员数 K：系统容量为有限值 ：顾客源为无穷大 FcFs:先来先服务 2、排队系统的描述 A/B/C/D/E/F A:到达过程的概率特性 B:服务时间分布 C:服务员数 D:系统最大顾客数 E:描述顾客总体数 F:排队规则 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking.

2.8 随机服务过程的几个主要数量指标 1、N(t)：表明t时刻在系统中的顾客数或称t时刻系统中的队长 2.8 随机服务过程的几个主要数量指标 1、N(t)：表明t时刻在系统中的顾客数或称t时刻系统中的队长 Nq(t)：表明t时刻在系统中的排队等待顾客数或称t时刻系统中的排队队长或等待队长 Nv(t)：表明t时刻在系统中的正在接受服务的顾客数 一般均为连续参数，离散 离散状态的随机过程。 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking. 同样，若Nq(t)的分布为： 则平均等待队长为：

4 在统计平衡状态下，如极限 等待时间 则，稳态平均队长L和稳态平均队长Lq为 从顾客抵达服务系统起直到开始接受服务为止这段时间，记Tq 4 在统计平衡状态下，如极限 则，稳态平均队长L和稳态平均队长Lq为 等待时间 从顾客抵达服务系统起直到开始接受服务为止这段时间，记Tq The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking. 统计平衡条件下的等待时间分布为： 平均等待时间为：

6 逗留时间：从顾客抵达系统起直到他接受服务完成为止这段时间，记T 7 忙期 从顾客抵达空闲服务系统开始直到服务系统再一次变为空闲状态为止，这段服务 系统连续忙的时间段。 8 闲期 服务系统处于空闲状态的时间段 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking.

本章主要内容 2.1 通信网建模的基本准则 2.2 通信业务源的概率模型化 2.2.1 随机事件的概率特征量及物理意义 2.2.2 典型概率分布及随机过程 （1）连续性概率分布 （2）离散型概率分布 （3）典型随机过程 2.2.3 纯随机事件的概率模型 2.2.4 平滑事件的概率模型 2.2.5 突发事件的概率模型 2.2.6 业务源概率模型参数的匹配

连续型概率分布(1)－定长分布 定长分布(deterministic distribution) 概率特征： 主要应用： 方差为0 周期性到达事件 定长服务系统（e.g., ATM网络）

连续型概率分布(2)－均匀分布 均匀分布(uniform distribution)

连续型概率分布(2)－均匀分布 Uniform (虚线：Mean=0; Std. Dev.=2.3094

连续型概率分布(3)－高斯分布 高斯分布(Gaussian distribution) 概率特征： 主要应用： 三阶矩为0 大量微小随机事件和的分布（中心极限定理、大数定理） 主要应用： 广泛应用于各种噪声、干扰及统计计算中

连续型概率分布(3)－高斯分布 Gaussian (虚线：Mean=0; Std. Dev.=8;

连续型概率分布(3)－二维高斯分布 Joint (Bivariate) Gaussian (theta=0.4)

连续型概率分布(3)－二维高斯分布 Joint (Bivariate) Gaussian (theta=0.9)

连续型概率分布(4) －瑞利分布 瑞利分布(Rayleigh distribution) 概率特征： 主要应用： 主要用于无线蜂窝通信系统的描述(多径信道)

连续型概率分布(4) －瑞利分布 Rayleigh (虚线：Mean=4.4311; Std. Dev.=2.3163;

连续型概率分布(5)－负指数分布 定义 若X为连续型随机变量，其概率密度函数为 则称X是一个服从参数为λ的负指数分布的随机变量。其分布函数为 1、性质：1）由式（1）知，负指数分布的密度函数是单调减少的，因此有对任意t>0,随机变量X落在区间[0,s)内的可能性要大于落在区间[t,t+s)中的可能性。最大可能性是落在t=0附近。 2）无记忆性或无后效性、马尔可夫性：“将来”只与“现在”有关，而与“过去”无关 解释：1）若将x看成为某顾客的服务时间，则无记忆性的意思为，不论该顾客已经接受服务的时间为多少，剩余的服务时间的条件分布仍与原分布相同，且与已服务多少时间无关；2）若将x看成为相继顾客的到达间隔，则无记忆性即指，在下一顾客尚未到达之前，无论到达间隔已持续多长时间，剩余到达间隔的条件分布与到达间隔的原分布相同，且不受已经历的到达间隔的影响。 在连续型随机变量中仅有负指数分布的随机变量具有无记忆性 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking.

定理 ：负指数分布具有无记忆性，即若X服从参数λ>0的负指数分布，则： 由条件概率公式得： The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking. 2、均值： 3、方差：

4、均方差或标准差 5、方差系数： The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking.

连续型概率分布(5)－负指数分布 负指数分布(Exponential distribution) 概率特征： 主要应用： 方差系数＝1 只有一个参数 无记忆性 主要应用： 广泛用于军事、社会、交通、通信等领域

连续型概率分布(5)－负指数分布 Exponential (Mean=5; Std. Dev.=5)

连续型概率分布(5)－负指数分布 Exponential (虚线：Mean=5; Std. Dev.=5;

连续型概率分布(6) －爱尔朗（Erlang）分布 定义 若随机变量X的概率密度函数为 则称X服从k阶爱尔朗分布。其分布函数为 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking. 1、均值 2、方差 3、均方差 4、方差系数

连续型概率分布(6) －爱尔朗（Erlang）分布 T3 T1 T4 T5 T2 T6 t T’3 T’1 T’2 简单流：相邻时间发生的时间间隔服从负指数分布,具有无记忆特性 在简单流中每隔(k-1)个“点”的“点”组成的流叫k阶爱尔朗分布，记为Ek，有无记忆特性 即 f(t) 当k=1, Ek为负指数分布，完全随机；当k足够大时，Ek分布近似于正态分布；当 X以概率1取值 ，即为定长分布，因此Ek分布可以看作随机模型和非随机模型的中间状态 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking. 20 6 理解：对于k个串联的服务台，每个服务台的服务时间相互独立，均服从负指数分布，则每个顾客总的服务时间服从Ek分布 5 4 3 2 k=1 t

连续型概率分布(6) －爱尔朗（Erlang）分布 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking.

连续型概率分布(6) －爱尔兰分布 爱尔兰分布(Erlangian distribution) 概率特征： 主要应用： 方差系数小于1 由r个负指数随机变量的和构成，r 趋于无穷大时逼近于定长分布 主要应用： 多级服务系统；描述平滑（规则）随机事件流

连续型概率分布(6) －爱尔兰分布 Sum of 6 Exponential Variables

Erlang Distributions Erlang-k distributions with mean value equal to one. The case k=1 correspond to an exponential distribution (density functions)

连续型概率分布(7) －超指数分布 超指数分布(Hyper-exponential distribution) 概率特征： 方差系数大于1

连续型概率分布(7) －超指数分布 α1 μ1 α2 μ2 α1 μ1 α2 μ2 αi μi αR μR 1 2 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking.

Coxian (Phase) Distribution α1 αR αi riμi rRμR r1 rR 1 2 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking.

Coxian (Phase) Distribution A Cox-distribution is a generalised Elrlang-distribution having exponential distributions in both parallel and series.

离散型概率分布(1)－几何分布 设随机变量X的概率分布律为： 其中0<p<1，则称X服从参数为p的几何分布 1、均值 2、方差 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking. 定理2-1：几何分布具有无记忆性，即对于任意非负整数n、m，有

离散型概率分布(1)－几何分布 几何分布(Geometric distribution) 两次相邻成功试验（事件的发生或退去）之间的间隔服从负指数分布； 几何分布相当于负指数分布离散化的结果，是离散型概率分布函数中惟一具有无记忆性的概率分布

离散型概率分布(1) －几何分布 Geometric (Mean=0.5385; Std. Dev.=0.9102)

离散型概率分布(1) －几何分布 Geometric (Mean=1.8571; Std. Dev.=2.3035)

几何分布的无记忆性及其应用

几何分布的无记忆性及其应用 几何分布可以用来描述某一顾客的到达间隔或服务持续时间 每单位时间执行一次贝努里试验，“失败”则令服务继续，“成功”则令服务完成 首次“成功”之前需要持续（等待）的时间就可以看成是相应的到达间隔或服务持续时间。

离散型概率分布(2) －负二项分布 负二项分布(Pascal distribution) 概率特征： 相邻n次成功试验（事件的发生或退去）之间的间隔服从特殊的Gamma (n-Erlang)分布；

离散型概率分布(2) －Pascal分布 Pascal (Mean=2.8571; Std. Dev.=2.3035)

离散型概率分布(2) －Pascal分布 Pascal (Mean=17.1429; Std. Dev.=5.6424)

离散型概率分布(3)－二项式分布 设进行n次独立试验，每次试验结果有：p=P(事件 A发生)，q= P(事件 B发生)，且p+q=1 若在n次试验中，事件A发生m次，则事件B发生(n-m)次，则称随机试验为n次贝努利试验，这样二项式分布为： The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking. 1、均值 2、方差

离散型概率分布(3) －二项分布 二项分布(binomial distribution) 概率特征: n次贝努里试验中k次试验成功（事件的发生或退去）的概率服从二项分布； n次贝努里试验的时间段(0,t)内有k次成功试验的概率服从泊松分布

离散型概率分布(3) －二项分布 Binomial (Mean=0.5; Std. Dev.=0.6892)

离散型概率分布(3) －二项分布 Binomial (Mean=14; Std. Dev.=2.0494)

离散型概率分布(4) －泊松分布(Possion) 若随机变量X服从参数为λ的泊松分布,则有 1、均值 2、方差 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking. 例1：若顾客到达系统数服从参数为λ的泊松分布，在t时间内到达k个顾客的概率为多少？

离散型概率分布(4) －泊松分布 泊松分布(Poisson distribution) 概率特征： n􀃆∞, np 􀃆λ时的2项（贝努利）分布逼近于泊松分布 事件发生的间隔服从负指数分布的连续型随机过程，其计数过程N(t)服从泊松分布

举例：泊松分布与二项分布

离散型概率分布(4)－泊松分布 Poisson(Mean=5; Std. Dev.=2.2361)

离散型概率分布(4)－泊松分布 Poisson(Mean=10; Std. Dev.=3.1623)

其它分布 均匀分布 正态分布（高斯分布） 分布 t 分布 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking. 分布 t 分布

2.6 独立过程 定义2.6-1 如果对任意的正整数n及任意的 ，随机变量 相互独立，则称随机过程 为独立过程。 相互独立，则称随机过程 为独立过程。 特别，如果X(n),n=1,2,···是相互独立的随机变量，则称{X(n),n=1,2,···}为独立随机序列。 2.7贝努利过程 随机过程{Xn,n=1,2,···}称为具有“成功”概率为p的贝努利过程，若 （a） X1, X2, ···相互独立 （b）对所有n均有P{Xn=1}=p,P{Xn=0}=q=1-p，其中p,q>0 均值 E[X]=p 方差 D[X]=pq 相关函数 协方差函数 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking.

2.8 独立增量过程 定义2.8-1 设有随机过程 ，如果对任意正整数 ，随机过程的增量： 是相互独立的的随机变量，则称 为独立增量过程 定义2.8-1 设有随机过程 ，如果对任意正整数 ，随机过程的增量： 是相互独立的的随机变量，则称 为独立增量过程 不失一般性，可设P{X(0)=0}=1或0 定义2.8-2 如果独立增量过程 ，对所有的 有相同的概率分布，则称 为平稳独立增量过程 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking. 平稳独立增量过程 的增量 的概率分布仅依赖于 而与t无关，即仅与时间区间的长度 有关，而 与起点无关，具有平稳性，即增量具有平稳性

2.9 泊松过程(Possion Process) 计数过程：如果N(t)是取非负整数值的随机变量，满足若s<t，则有 ，称 为计数过程 如果对t1<t2，s>0,N(t2+s)-N(t1+s)与N(t2)-N(t1)有相同的概率分布，则该计数过程是有平稳增量的 定义2.9-1 计数过程 称为齐次Poisson过程，如果满足条件： （1） （2）具有独立增量 （3） The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking.

2.9 泊松过程(Possion Process) 定义2.9-2 计数过程 称为齐次Poisson过程，如果满足条件： （1） （2）具有平稳独立增量 （3） （4） The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking.

定义2.9-3 若到达过程 为Poission过程，则满足： （1）平衡性（齐次性）：在时间区间[t0, t0+t]内有k个顾客到达的概率与时间区间起点t0无关，而仅与到达数k和时间长度t有关，即： （2）独立增量性（无后效性）：在任意不相交的区间内到达的顾客数相互独立，即任取 和状态k1, k2有： （3）有限性：对任意的有限区间[t0, t0+t]内，到达有限多个顾客的概率为1，即： （4）普通性（单个性）：在[t0, t0+t]内至少到达两个顾客的概率 是关于t的高阶无穷小，即 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking.

泊松过程概率分布与数字特征 1、一维概率分布及均值和方差 均值 方差 2、二维概率分布及均方差函数和相关函数 均方差函数 相关函数 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking. 均方差函数 相关函数

泊松过程的性质 性质1 泊松过程是平稳独立增量过程 性质2 泊松过程是马尔可夫过程 性质3 泊松过程是生灭过程 性质1 泊松过程是平稳独立增量过程 性质2 泊松过程是马尔可夫过程 性质3 泊松过程是生灭过程 性质4 泊松过程是非平稳过程，但为平稳增量过程 设N(t)表示区间[0，t)内事件出现的次数， 为参数为 的泊松过程，设 分别表示事件第一、二、···、n次出现的时间，我们称 为事件第k次出现的时间； 表示事件第k-1次出现到第k次出现之间的时间间隔 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking. 性质5 设 为参数为 的泊松过程，{Tn,n=1,2,···}为时间间 隔序列，则Tn,n=1,2,···是相互独立同分布的随机变量，且都服从参数为 的指数分布

证明性质5 又 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking.

2.10 生灭过程 定义2.10-1 设有一个系统，具有状态集合E={0,1,2,···}或E={0,1,2,···，K}。令Nt表示系统在时刻t所处的状态。若对任意t>=0,有： （1） 为常数，有限状态时i=0,1,···,K-1。可数状态时，i=0,1,2,···。 （2） 为常数，有限状态时i=0,1,···,K-1或i=0,1,2,···。 （3） 则称系统状态随时间变化的过程 为生灭过程。 The seminar is organized in five Chapters. Chapter 1 presents an overview of ATM Traffic Management (TM) by showing the need for ATM TM. Chapter 2 is an introduction to the ATM TM basics and strategy. In Chapter 3, we focus on the nodal level control mechanisms such as cell/frame switching, Intelligent buffer management, traffic shaping, and usage parameter control (UPC). In Chapter 4, we focus on the network level controls such as routing, flow control, and connection admission control. Chapter 5 is a collection of new and merging ATM TM features such as ABR, IMA, and Frame Relay to ATM interworking.

泊松过程 泊松过程(Poisson process) 随机特性： 主要应用： 定义：计数过程N(t)服从泊松分布的随机过程 方差系数等于1的纯随机过程 具有无后效性 只需要确定一个参数 事件发生间隔之间服从负指数分布 主要应用： 广泛应用于各种随机事件的描述或近似

泊松过程是一个非常实用的随机过程 Number of Internet dial-up calls per second. The theoretical values are based on the assumption of a Poisson distribution. A statistical test accepts the hypothesis of Poisson distribution.

泊松过程的叠加和分解 泊松过程的独立叠加/独立分解仍然是泊松过程 对通信业务源的建模至关重要

泊松过程可用来近似大量随机事件的叠加 By superposition of n point processed we obtain under certain assumptions a process which locally is a Poisson process

一般更新过程理论 定义 更新过程F(x)的前(后)向递归时间非常重要 时间间隔{Xi; i=0,1,2,…}相互独立且服从同一概率分布(i.i.d.:independent, identically distributed)的随机过程 Xi之间存在记忆性，但它对历史的记忆是有限的，只要追溯到前一个更新点即可 一般简写为GI (General Independent) 更新过程F(x)的前(后)向递归时间非常重要 forward recurrent time or residual life time

有关Hippy的一个悖论 假设 􀀹 (1)乘车者(Hippy)随机地到达车站 􀀹 (2)乘车者平均等待5分钟等到下一辆车 问题 􀀹 (1)乘车者距离上一趟车离去有多长时间？ 􀀹 (2)汽车行驶间隔的平均为几分钟

更新过程的前(后)向递归时间

定理2.3的证明 因为：

定理2.3的证明（续） 所以：

为什么残余时间的均值会大于总寿命的均值呢？ 某个区间被随机到达的顾客所“选取”的概率应与该区间的长度和出现的频度两者成正比 举例：如果区间长度的概率分布为P(X=5)=1/2 and P(X=15)=1/2，即两者的时间差为3倍，但出现的频度均等，则顾客到达时选取X=15的概率将是选取X=5的3倍。而选择了X=5或X=15后的残余时间将服从均匀分布，故有hb=hf=5/2 x 1/4 + 15/2 x3/4=6.25 但是，若P(X=5)=3/4 and P(X=25)=1/4，即两者的时间差为5倍，但出现的频度差为1:3, 则有hb=hf=5/2 x3/8 + 25/2 x5/8=8.75

名言中的数学哲理 “天灾在你忘记的时候降临” “福无双至、祸不单行” 如果假设“天灾”、“福至”、或是“祸来”的发生是纯随机的，依据负指数分布的无记忆性可知：长时间不发生的概率不可忽视的同时，短时间内连续出现的概率更不可低估！ 实际上，天灾在你没有忘记的时候也会发生！

悖论的解释：观测区间概率分布 观测区间(covering intervals) * 注意到某个区间被随机地选择的概率应与该区间的长度成正比，因此有 Y和Z相互独立、且均服从负指数分布时： I 服从2阶爱尔兰分布