Entry
Math: Model:Probability:Erlang:Support: Can you tell about distribution of telephone calls?[traffic]
Feb 19th, 2006 10:18
Knud van Eeden,
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--- Knud van Eeden --- 28 December 2004 - 03:39 pm -------------------
Math: Model:Probability:Erlang:Support: Can you tell about
distribution of telephone calls?[traffic]
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The frequency of incoming telephone calls can sometimes adequately
be described by:
-Probability distributions:
-exponential distribution
(a combination of the e power function)
-Poisson distribution
(a combination of a power function, an e power function and a
factorial function)
-Erlang distribution
(a combination of a 2 power functions, an e power function and a
factorial function)
1. Erlang B
2. Erlang C
-Queuing theory
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Note:
The exponential and Erlang distribution are special cases of the gamma
distribution (by choosing a specific constant parameter value)
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Exponential distribution:
In a Poisson process, the time between two successive arrivals has
exponential distribution.
If only 1 interval:
|------|---------|--------------|--------------------|-
call1 call2 call3 call4 call5
<--------------> -> time
time between calls
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Erlang distribution:
The time between an arrival and the Nth one following it has a Gamma
distribution, also called an Erlang distribution.
If more than 1 interval:
|------|---------|--------------|--------------------|-
call1 call2 call3 call4 call5
-> time
<-- time between calls ----------------------->
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The Erlang distribution is a special case of the gamma distribution
where the shape parameter is an integer.
It represents the sum of a series of exponential distributions (a sum
of horizontally translated scaled exponential distributions)
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Because an exponential distribution is just a special case of the more
general gamma function, you can thus also say that
it represents the sum of a series of gamma distributions (a sum of
horizontally translated scaled gamma distributions)
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Book: see also:
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[book: Childers, Donald G. - probability and random processes (using
MatLab with applications to continuous and discrete time systems) -
publisher: McGraw-Hill - ISBN 0-256-13361-1 - p. 152-156 'Poisson
random processes' (with good example relation telephone calls arriving
according to a Poisson distribution, and with exponential distribution
call length, with a formula workout and numeric example, in a network
and their arrival time, M/M/1 queue)]
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[book: Hastings, Kevin J. - Probability and statistics - ISBN 0-201-
59278-9 - p. 80 'Poisson distribution' with example, p. 82 'Poisson
probability mass function' (definition), p. 83 (comparison values
Binomial and Poisson distribution), p. 83 (example cumulative Poisson
distribution, for users wanting to access a server), p. 84 'Poisson
random processes' (rate parameter, solution of differential equation
gives Poisson formula), p. 215 'Poisson interarrival times', p.
237 'Generating functions' (superposition of Poisson arrival rates,
distribution of arrival times, about telephone calls arriving at a
switch board)]
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Internet: see also:
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Math: Traffic engineering: Can you give overview of links about the
distribution of telephone calls?
http://www.faqts.com/knowledge_base/view.phtml/aid/39736/fid/815
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