Applications of Poisson Process Wang C. Ng

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

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?

Solution

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

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?

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

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?

Solution

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

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

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