Median No simple closed form | ||
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Parameters k ∈ N {displaystyle scriptstyle k;in ;mathbb {N} } shape λ > 0 {displaystyle scriptstyle lambda ;>;0} , rate (real)alt.: μ = 1 λ > 0 {displaystyle scriptstyle mu ;=;{rac {1}{lambda }}>0,} scale (real) Support x ∈ [ 0 , ∞ ) {displaystyle scriptstyle x;in ;[0,,infty )!} PDF λ k x k − 1 e − λ x ( k − 1 ) ! {displaystyle scriptstyle {rac {lambda ^{k}x^{k-1}e^{-lambda x}}{(k-1)!,}}} CDF γ ( k , λ x ) ( k − 1 ) ! = 1 − ∑ n = 0 k − 1 1 n ! e − λ x ( λ x ) n {displaystyle scriptstyle {rac {gamma (k,,lambda x)}{(k,-,1)!}};=;1,-,sum _{n=0}^{k-1}{rac {1}{n!}}e^{-lambda x}(lambda x)^{n}} Mean k λ {displaystyle scriptstyle {rac {k}{lambda }},} |
The Erlang distribution is a two parameter family of continuous probability distributions with support
Contents
- Probability density function
- Cumulative distribution function CDF
- Properties
- Median
- Generating Erlang distributed random numbers
- Waiting times
- Stochastic processes
- Related distributions
- References
The Erlang distribution with shape parameter
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 now used in the fields of stochastic processes and of biomathematics.
Probability density function
The probability density function of the Erlang distribution is
The parameter k is called the shape parameter, and the parameter
An alternative, but equivalent, parametrization uses the scale parameter
When the scale parameter
Because of the factorial function in the denominator, the Erlang distribution is only defined when the parameter k is a positive integer. In fact, this distribution is sometimes called the Erlang-k distribution (e.g., an Erlang-2 distribution is an Erlang distribution with k = 2). The gamma distribution generalizes the Erlang distribution by allowing k to be any real number, using the gamma function instead of the factorial function.
Cumulative distribution function (CDF)
The cumulative distribution function of the Erlang distribution is
where
Properties
The Erlang distribution is a solution of the following differential equation:
with initial condition
Median
An asymptotic expansion is known for the median of an Erlang distribution, for which coefficients can be computed and bounds are known. An approximation is
Generating Erlang-distributed random numbers
Erlang-distributed random numbers can be generated from uniform distribution random numbers (
Waiting times
Events that occur independently with some average rate are modeled with a Poisson process. The waiting times between k occurrences of the event are Erlang distributed. (The related question of the number of events in a given amount of time is described by the Poisson distribution.)
The Erlang distribution, which measures the time between incoming calls, can be used in conjunction with the expected duration of incoming calls to produce information about the traffic load measured in erlangs. This can be used to determine the probability of packet loss or delay, according to various assumptions made about whether blocked calls are aborted (Erlang B formula) or queued until served (Erlang C formula). The Erlang-B and C formulae are still in everyday use for traffic modeling for applications such as the design of call centers.
It has also been used in business economics for describing interpurchase times.
There are also two other Erlang distributions, both used in modeling traffic:
Erlang B distribution: this is the easier of the two, and can be used, for example, in a call centre to calculate the number of trunks one need to carry a certain amount of phone traffic with a certain "target service".
Erlang C distribution: this formula is much more difficult and is often used, for example, to calculate how long callers will have to wait before being connected to a human in a call centre or similar situation.
Stochastic processes
The Erlang distribution is the distribution of the sum of k independent and identically distributed random variables each having an exponential distribution. The long-run rate at which events occur is the reciprocal of the expectation of