Applications of Poisson Process

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Applications of Poisson Process

• Wang C. Ng

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Telephone traffic

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

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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?

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Solution
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Telephone traffic

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

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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?

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Telephone traffic

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

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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?

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Solution
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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

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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

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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