包网络性能指标+无线网络概率基础

以包为单位进行数据传输的网络模型中，一般会有这三个表示网络性能的指标：包时延，丢包率和网络吞吐量。

• 包时延（packet delay）

1. 处理时延：误差检测，决定输出链路
2. 排队时延：等待传输的输出链路，以来与路由器处理拥堵的水平（当路由缓冲区有包在排队（等待转发）时，当包到达率超过链路容量）
3. 传输时延：将比特推送到链路的时间<-包长度/链路带宽（bps）
4. 传播时延：信号到达终点的时间<-物理链路的长度/传输速度（传输媒介）

• 丢包率（packet loss rate）

包到达时缓冲队列已满，即丢包/由传输层协议决定丢包率

• 吞吐量（throughput）

成功传输的比特率；端到端路径会限制端到端吞吐量，从而产生瓶颈链路。

无线网络中为了方便研究，会经常使用无记忆性的指数分布和均匀概率的泊松分布来对随机事件进行建模。

• 指数分布https://en.wikipedia.org/wiki/Exponential_distribution

   

从概率论上说，以上三个式子（概率密度函数，期望和方差）就可以完全表示一个服从指数分布的连续随机变量。但无线网络中偏爱用指数分布的重要原因是它具有无记忆性，即其历史状态不会影响将来状态的性质（Past history has no influence on the future）。这可以极大简化实际的研究。

Physical meaning:

• If we think of X as being the lifetime of some instrument, then equation states that the probability (that the instrument lives for at least x+t hours given that it has survived t hours) is the same as the initial probability (that it lives for at least x hours). 
• In other words, if the instrument is alive at time t , then the distribution of  the remaining amount of time that it survives is the same as the original lifetime distribution; that is, the instrument does not remember that it has already been in use for a time t .

注：只有连续的指数分布才有无记忆的性质。

• 泊松分布https://en.wikipedia.org/wiki/Poisson_distribution

从概率论上说，以上三个式子（概率密度函数，期望和方差）就可以完全表示一个服从泊松分布的离散随机变量。经常用来对一定事件间隔类发生随机事件的数量进行建模。泊松分布还有一个性质是，多个服从泊松分布的变量之和还是服从泊松分布，其参数的变化为：

从上面的性质还可以推导出，对泊松分布进行采样之后的性质

Wide applicability in modeling the number of random events that occur during a given time interval 

• Customers that arrive at a post office during a day
• Wrong phone calls received during a week
• Students that go to the instructor’s office during office hours
• … and packets that arrive at a network switch 

泊松过程

Poisson process widely used to model packet arrivals in numerous networking problems. Justification: provides a good model for aggregate traffic of a large number of “independent” users:

• n traffic streams, with independent identically distributed (iid) interarrival times with PDF F(s) – not necessarily exponential
• Arrival rate of each stream l/n
• As n→∞, combined stream can be approximated by Poisson under mild conditions on F(s) – e.g., F(0)=0, F’(0)>0

Most important reason for Poisson assumption:Analytic tractability of queueing models.

小理论

• 系统的平均数量=每个顾客在系统花费的平均时间*顾客到达率
• 排队的顾客数量=每个顾客排队的平均时间*顾客到达率

概率论

1/conditional probabilities: P(E|F)=P(EF)|P(F)
2/Independent events.
   Definition: Two events E and F are said to be independent if P(EF) = P(E)P(F)
                       The above equation implies that E and F are independent if P(E|F) = P(E)
                       [which also implies that P(F|E) = P(F)]. 
   Remarks: E and F are independent if knowledge that F has occurred does not affect the probability  that E occurs.
3/Law of total probability
   Let E and F be events. We may express E as E = EF ∪ EFc
   Because in order for a point to be in E, it must either be in both E and F, or it must be in E and not in F. Since EF and EFc are mutually exclusive, we have that
    P(E) = P(EF)+P(EFc)
            = P(E|F)P(F) +P(E|Fc)P (Fc)
            = P(E|F)P(F) +P(E|Fc)(1 −P(F))