Applications of Poisson Process

1. Applications of Poisson Process Wang C. Ng

2. Telephone traffic • Pure chance traffic: Independent random events (memoryless). • Stationary: Busy/peak hours only. • The number of calls follows the Poisson distribution.

3. Example: • On average, one call arrives every 5 seconds. During a period of 10 seconds, what is the probability that: • no call arrives? • one call arrives? • two calls arrive? • more than two calls arrive?

4. Solution

5. Telephone traffic • The interval between calls follows the exponential distribution. • The call duration also follows the exponential distribution.

6. Example: • Average call duration is 2 minutes. A call has already lasted 4 minutes. What is the probability that: • the call last at least 4 more minutes? • the call will end within the next 4 minutes?

7. Telephone traffic • The number of calls in progress, assuming infinite (large) number of trunks (circuits) carrying the call, also has a poisson distribution.

8. Example: • Average call duration is 2 minutes and the mean number of calls per minute is 3. What is the probability that • 2 calls are in progress? • More than 2 calls are in progress?

9. Solution

10. Poisson modeling • Poisson model has been used to study network traffic • It has attractive theoretical properties • It has been studied thoroughly • It has represented the telephone traffic well

11. The failure of the Poisson model • However, recent studies have shown that the Poisson model is inadequate for many types of internet traffic (see attached article) • In general, the Poisson model fails to represent the “bursty” nature of internet traffic • A new model has been proposed to replace the Poisson model

12. Self-similar process • In the internet traffic studies the properties of self-similarity has been observed. • This type of processes can be analyzed using the recently developed chaos and fractal theory