第2章 随机过程与排队轮基础

k =0
k =0
(λ t ) − λ t (λ t ) = ∑e ( k − 1) ! k! k =1
k
∞
k
∞
= λ te
− λt
∑
k =1
(λt ) = λt ( k − 1) !
16
k −1
2012年3月
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其中λ>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 0≤t1<t2<…<tn，N(t1), N(t2)-N(t1), N(t 3)N(t2), …, N(tn)-N(t n-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。
2012年3月
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Siméon Denis Poisson
• Born: 6/21/1781Pithiviers, France • Died: 4/25/1840Sceaux, France • “Life is good for only two things: discovering mathematics and teaching mathematics.”
∞
∞
= ∑ ( k − 1) e
k =1
− λt
(λ t ) −λt (λt ) = ( λt ) 2 + λt + ke ( k − 1) ! k=1 k ⋅ (k −1) !
k
∞
k
∑
D( X ) = E X − ( EX ) = λt
2 2
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( )
Poisson过程的叠加和分解
2012年3月
8
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Poisson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
(λt ) k pk (t ) = k!
e
−λt
k = 0,1,2,⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
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2012年3月
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Poisson过程的叠加
2012年3月 13
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Poisson’s education and work
• Poisson impressed his teachers Laplace and Lagrange with his abilities. • Unfortunately, the École 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, Poisson received his first teaching position at the École 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.
准备知识: 计数过程
随机过程{N(t),t≥0}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
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随机事件的两种描述法
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Poisson过程的分解
• 性质2-2：参数为 λ = λ1 + λ2 的Poisson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 λi P = • i = 1,2 i λ • 则Poisson流被分解为两个独立的 Poisson 流，参数分别为 λ1和λ2
2012年3月
14
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Poisson’s accomplishments
• He has many mathematical and scientific tools named for him, including Poisson's integral, Poisson's equation in potential theory, Poisson brackets in differential equations, Poisson's ratio in elasticity, and Poisson's constant in electricity. He first published his Poisson distribution in 1837 in Recherches sur la probabilit é des jugements en matière criminelle et matière civile. Although this was important to probability and random processes, other French mathematicians did not see his work as significant. His accomplishments were more accepted outside France, such as in Russia, where Chebychev used Poisson ’s results to develop his own.
第二章 随机过程与排队轮基础
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准备知识: 随机过程
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2012年3月 12
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Siméon Denis Poisson
• Poisson’s 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 went to the École Centrale and later the École Polytechnique in Paris, where he excelled in mathematics, despite having much less formal education than his peers.
ψ (t ) = o (t )
t→0
∞
• （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 ∑ P ( t ) = 1
k k = 0
2012年3月
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例题分析
设电话呼叫按30次/小时的泊松过程进行，求5分钟 间隔内，(1)没有呼叫的概率；(2)呼叫3次的概率。 解：按题意λ= 30次/h = 0.5次/min t = 5min ，分别计算k = 0或k=3
P0 (5) = e −0.5×5 = 0.082
2.53 −0.5×5 P3 (5) = e = 0.214 3!