第2章 随机过程与排队轮基础

1、信息与通信工程学院信息与通信工程学院通信网技术教研中心通信网技术教研中心Communication Networks Theory Education & Research Center 2021-11-3第二章 随机过程与排队轮基础无线通信与网络研究室通信网基础BUPT Information Theory & Technology Education & Research Center 2021-11-32准备知识: 随机过程BUPT Information Theory & Technology Education & Research Cente

2、r 2021-11-33准备知识: 随机过程BUPT Information Theory & Technology Education & Research Center 2021-11-34准备知识: 计数过程随机过程N(t),t0称为一个计数过程。N(t)表示到时刻t为止已发生的“事件”的总数。 N(t)0； N(t)是整数； N(t) N(s)，当t s； N(t) - N(s)代表时间区间t, s)中发生的“事件” 数BUPT Information Theory & Technology Education & Research Center 202

3、1-11-35随机事件的两种描述法BUPT Information Theory & Technology Education & Research Center 2021-11-36随机事件的概率特征描述BUPT Information Theory & Technology Education & Research Center 2021-11-37随机事件特征量的物理意义BUPT Information Theory & Technology Education & Research Center 2021-11-38随机事件特征量的物理意

4、义BUPT Information Theory & Technology Education & Research Center 2021-11-39Poisson过程 定义：计数过程N(t)服从泊松分布的随机过程，即长度为t的时间内到达k个事件的概率为 其中0是泊松流的强度，表示平均到达率； 且N(0) = 0；不相交区间上增量相互独立，即对一切 0t1t2tn，N(t1), N(t2)-N(t1), N(t3)-N(t2), , N(tn)-N(tn-1)相互独立。 应用：广泛用于各种随机事件的描述或近似，可用来描述完全不可预测的随机事件和大量随机事件的叠加。etkkkt

5、tp!)()( , 2 , 1 , 0kBUPT Information Theory & Technology Education & Research Center 2021-11-310 （1）平稳性:在区间 内有k个事件到来的概率与起点a无关，只与时间区间的长度有关，这个概率记为 （2）无记忆性：不相交区间内到达的事件数是相互独立的； （3）稀疏性:令 表示长度为t的区间内至少到达两个事件的概率， 则 （4）有限性：在任意有限区间内到达有限个事件的概率为1，即taa,)(),(tPtaaPkk)(t1)(0tPkk)()(tot 0tPoisson过程BUPT Info

6、rmation Theory & Technology Education & Research Center 2021-11-311例题分析设电话呼叫按设电话呼叫按30次次/小时的泊松过程进行，求小时的泊松过程进行，求5分钟分钟间隔内，间隔内，(1)没有呼叫的概率；没有呼叫的概率；(2)呼叫呼叫3次的概率。次的概率。解：按题意= 30次/h = 0.5次/min t = 5min，分别计算k = 0或k=3 214. 0! 35 . 2)5(082. 0)5(55 . 03355 . 00ePePBUPT Information Theory & Technology

7、 Education & Research Center 2021-11-312Simon Denis Poisson Born: 6/21/1781-Pithiviers, France Died: 4/25/1840-Sceaux, France “Life is good for only two things: discovering mathematics and teaching mathematics.”BUPT Information Theory & Technology Education & Research Center 2021-11-313S

8、imon Denis Poisson Poissons father originally wanted him to become a doctor. After a brief apprenticeship with an uncle, Poisson realized he did not want to be a doctor. After the French Revolution, more opportunities became available for Poisson, whose family was not part of the nobility. Poisson w

9、ent to the cole Centrale and later the cole Polytechnique in Paris, where he excelled in mathematics, despite having much less formal education than his peers.BUPT Information Theory & Technology Education & Research Center 2021-11-314Poissons education and work Poisson impressed his teacher

10、s Laplace and Lagrange with his abilities. Unfortunately, the cole Polytechnique specialized in geometry, and Poisson could not draw diagrams well. However, his final paper on the theory of equations was so good he was allowed to graduate without taking the final examination. After graduating, Poiss

11、on received his first teaching position at the cole Polytechnique in Paris, which rarely happened. Poisson did most of his work on ordinary and partial differential equations. He also worked on problems involving physical topics, such as pendulums and sound.BUPT Information Theory & Technology E

12、ducation & Research Center 2021-11-315Poissons accomplishments He has many mathematical and scientific tools named for him, including Poissons integral, Poissons equation in potential theory, Poisson brackets in differential equations, Poissons ratio in elasticity, and Poissons constant in elect

13、ricity. He first published his Poisson distribution in 1837 in Recherches sur la probabilit des jugements en matire criminelle et matire civile. Although this was important to probability and random processes, other French mathematicians did not see his work as significant. His accomplishments were

14、more accepted outside France, such as in Russia, where Chebychev used Poissons results to develop his own.BUPT Information Theory & Technology Education & Research Center 2021-11-316泊松泊松(Poisson)过程的期望与方差过程的期望与方差()(),0,1,2,!kttP Xkekk 0kkkpxXE1()1 !ktktek11()1 !ktkttekt0()!ktktkekBUPT Informa

15、tion Theory & Technology Education & Research Center 2021-11-317 022kkkpxXE1( )1 !ktktkektt2)(1( )1 11 !ktktkek 1!1)(kktkktke 22EXXEXD t20()!ktktk ek泊松泊松(Poisson)过程的期望与方差过程的期望与方差1()11 !ktktkekBUPT Information Theory & Technology Education & Research Center 2021-11-318Poisson过程的叠加和分解B

16、UPT Information Theory & Technology Education & Research Center 2021-11-319Poisson过程的叠加 性质2-1：m个Poisson流的参数分别为 ， ， ，并且它们是相互独立的，合并流仍然为Poisson流，且参数为 。 这个性质也就是说独立的Poisson过程是可加的。12mm21 = 1 +212BUPT Information Theory & Technology Education & Research Center 2021-11-320 性质2-2：参数为 的Poisson

17、流到达交换局A后，每个呼叫将独立去两个不同方向，且去两个方向的概率分别为 则Poisson流被分解为两个独立的Poisson流，参数分别为 2121和iiP 2 , 1iPoisson过程的分解BUPT Information Theory & Technology Education & Research Center 2021-11-321 设N(t) 表示一个Poisson 过程， 假设t ,i = 0,1,2, i 为相应的呼叫到达时刻，考虑呼叫的间隔： 根据Poisson 的特性 令随机变量X 满足PX t = et ， 或分布函数为：PX t= Pmin(T1, T2)t= PT1t, T2t=PT1t PT2t 又因为 ， 所以11tP Tte22tP T