NS by Example 笔记（13）Pareto On/Off

Mini-HOWTO: Pareto On/Off Traffic Generator

 

描述 Description

 

Pareto(柏拉图) On/Off Traffic Generator (POO_Traffic) 是一种流量生成器traffic generator (一种应用程序) 包含在OTcl类Application/Traffic/Pareto。 POO_Traffic产生网络流量依照Pareto On/Off distribution。 数据包在闭合周期（on periods）内以固定的速率发送， 并且在间歇期间（off period）停止发包。 on和off periods由固定的数据包大小的Pareto distribution得到。 这些数据源可以用来产生聚集的呈现long range dependency的流量（aggregate traffic that exhibits long range dependency）。 详见NS manual.

 

 

 

实例 Usage Example

下面是建立一个新的Pareto On/Off traffic generator的代码:

        set p [new Application/Traffic/Pareto]
        $p set burst_time_ 500ms
        $p set idle_time_ 500ms
        $p set rate_ 200k
        $p set packetSize_ 210
        $p set shape_ 1.5

 

详细解释 Detailed Explanation

1. 输入参数 Input parameters:

     burst_time_ : Mean On (burst) time
     idle_time_  : Mean Off (idle) time
     rate_       : Send rate during burst
     packetSize_ : Packet size (fixed application frame size)
     shape_      : Pareto shape parameter
     

2. 起始计算 Initial computation:

     interval = packetSize_ * 8 / rate_
     burstlen = burst_time_ / interval (in packets of given packetSize_)
     

3. 在每个On/Off round循环， 计算2个独立的(independent) Pareto random variables:

     next_burstlen  : # of packets to be transmitted in the next burst period
                      (ex: next Web object size in packets of given packetSize_)
     next_idle_time : next idle period length in seconds
                      (ex: next think time)
     

4. Pareto Traffic Generator 算法:
    
   1. Compute next_burstlen given the mean burstlen and the Pareto shape_ parameter.
   2. Send all next_burstlen packets (each packet transmission interval is interval seconds).
   3. Compute the next_idle_time using the mean idle_time_ and the Pareto shape_ parameter.
   4. Go to sleep for next_idle_time and go back to step 1.
       
5. Pareto Distribution 分布:

   In general, if X is a random variable from a Pareto distribution
   (let f be the pdf, and E be the expected value)

   f(x) = a *b^a / x^{a + 1} for x ≥ b

   E(X) = b *a / (a − 1) if a > 1

   where, a is called the Pareto "shape parameter" (shape_) and b is called the Pareto "scale parameter".

6. How Pareto is used in the Pareto Traffic Generator:
   (The Pareto shape_ parameter is denoted as a in the following equations.)

   burstlen    = E(X) = b1 *a / (a − 1)
   idle_time_ = E(Y) = b2 *a / (a − 1)

   Therefore:

   b1 = burstlen * (a − 1) / a
   b2 = idle_time_ * (a − 1) / a

   NS has a Pareto random number generator that takes the scale and shape parameter:

    

         double pareto(double scale, double shape)
     

   When the Pareto traffic generator needs to compute next_burstlen, it does:

         int next_burstlen = int(pareto(b1, a) + 0.5);
         /* next_burstlen should be at least 1 packet */
         if(next_burstlen == 0) next_burstlen = 1;
     

   When the Pareto traffic generator needs to compute next_idle_time, it does:

         double next_idle_time = pareto(b2, a);
     

Discussion

The Pareto traffic generator can be used as a Web traffic generator assuming that Web object sizes form a Pareto distribution. The NS default Pareto shape_ parameter (or a) is 1.5. However, it is uncertain if that 1.5 makes sense for Web object size distribution.

To find a reasonable Pareto shape_ parameter for Web object sizes, one should sample Web object sizes during transmissions (or borrow the results other research), draw the pdf of the samples and compute the mean object size (i.e., E(X)). Since E(X) can be computed from the collected data, the theoretical f(x) should be:

f(x) = g(x, a)

Then, one should be able to find a that makes the f(x) closest to the pdf of the measured Web object samples.