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Concept
Erlang distribution
Summary
The Erlang distribution is a two-parameter family of continuous probability distributions with support . The two parameters are:
a positive integer the "shape", and
a positive real number the "rate". The "scale", the reciprocal of the rate, is sometimes used instead.
The Erlang distribution is the distribution of a sum of independent exponential variables with mean each. Equivalently, it is the distribution of the time until the kth event of a Poisson process with a rate of . The Erlang and Poisson distributions are complementary, in that while the Poisson distribution counts the number of events that occur in a fixed amount of time, the Erlang distribution counts the amount of time until the occurrence of a fixed number of events. When , the distribution simplifies to the exponential distribution. The Erlang distribution is a special case of the gamma distribution wherein the shape of the distribution is discretised.
The Erlang distribution was developed by A. K. Erlang to examine the number of telephone calls which might be made at the same time to the operators of the switching stations. This work on telephone traffic engineering has been expanded to consider waiting times in queueing systems in general. The distribution is also used in the field of stochastic processes.
The probability density function of the Erlang distribution is
The parameter k is called the shape parameter, and the parameter is called the rate parameter.
An alternative, but equivalent, parametrization uses the scale parameter , which is the reciprocal of the rate parameter (i.e., ):
When the scale parameter equals 2, the distribution simplifies to the chi-squared distribution with 2k degrees of freedom. It can therefore be regarded as a generalized chi-squared distribution for even numbers of degrees of freedom.
The cumulative distribution function of the Erlang distribution is
where is the lower incomplete gamma function and is the lower regularized gamma function.
Official source
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