Erlang distribution

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Parameters Probability density function Cumulative distribution function $\scriptstyle k \;\in\; \mathbb{N}$ shape $\scriptstyle \lambda \;>\; 0$, rate (real) alt.: $\scriptstyle \mu \;=\; \frac{1}{\lambda} > 0\,$ scale (real) $\scriptstyle x \;\in\; [0,\, \infty)\!$ $\scriptstyle \frac{\lambda^k x^{k-1} e^{-\lambda x}}{(k-1)!\,}$ $\scriptstyle \frac{\gamma(k,\, \lambda x)}{(k \,-\, 1)!} \;=\; 1 \,-\, \sum_{n=0}^{k-1}\frac{1}{n!}e^{-\lambda x}(\lambda x)^{n}$ $\scriptstyle \frac{k}{\lambda}\,$ No simple closed form $\scriptstyle \frac{1}{\lambda}(k \,-\, 1)\,$ for $\scriptstyle k \;\geq\; 1\,$ $\scriptstyle \frac{k}{\lambda^2}\,$ $\scriptstyle \frac{2}{\sqrt{k}}$ $\scriptstyle \frac{6}{k}$ $\scriptstyle (1 \,-\, k)\psi(k) \,+\, \ln\left[\frac{\Gamma(k)}{\lambda}\right] \,+\, k$ $\scriptstyle \left(1 \,-\, \frac{t}{\lambda}\right)^{-k}\,$ for $\scriptstyle t \;<\; \lambda\,$ $\scriptstyle \left(1 \,-\, \frac{it}{\lambda}\right)^{-k}\,$

The Erlang distribution is a continuous probability distribution with wide applicability primarily due to its relation to the exponential and Gamma distributions. 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.

Overview

The distribution is a continuous distribution, which has a positive value for all real numbers greater than zero, and is given by two parameters: the shape $k$, which is a positive integer, and the rate $\lambda$, which is a positive real number. The distribution is sometimes defined using the inverse of the rate parameter, the scale $\mu$. It is the distribution of the sum of $k$ independent exponential variables with mean $\mu$.

When the shape parameter $k$ equals 1, the distribution simplifies to the exponential distribution. The Erlang distribution is a special case of the Gamma distribution where the shape parameter $k$ is an integer. In the Gamma distribution, this parameter is not restricted to the integers.

Characterization

Probability density function

The probability density function of the Erlang distribution is

$f(x; k,\lambda)={\lambda^k x^{k-1} e^{-\lambda x} \over (k-1)!}\quad\mbox{for }x, \lambda \geq 0,$

The parameter k is called the shape parameter, and the parameter $\lambda$ is called the rate parameter.

An alternative, but equivalent, parametrization uses the scale parameter $\mu$, which is the reciprocal of the rate parameter (i.e., $\mu = 1/\lambda$):

$f(x; k,\mu)=\frac{ x^{k-1} e^{-\frac{x}{\mu}} }{\mu^k (k-1)!}\quad\mbox{for }x, \mu \geq 0.$

When the scale parameter $\mu$ 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.

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

$F(x; k,\lambda) = \frac{\gamma(k, \lambda x)}{(k-1)!},$

where $\gamma()$ is the lower incomplete gamma function. The CDF may also be expressed as

$F(x; k,\lambda) = 1 - \sum_{n=0}^{k-1}\frac{1}{n!}e^{-\lambda x}(\lambda x)^n.$

Properties

The Erlang distribution is a solution of the following Differential equation[citation needed]:

$x f'(x) + ( \lambda x + 1 - k ) f(x) = 0$

with initial condition $f(1)=\frac{e^{-\lambda } \lambda^k}{\Gamma (k)}$ (the Poisson distribution).

Median

An asymptotic expansion is known for the median of an Erlang distribution,[1] for which coefficients can be computed and bounds are known.[2][3] An approximation is $\dfrac{k}{\lambda}\left(1-\dfrac{1}{3k+0.2}\right)$, i.e. below the mean $\tfrac{k}{\lambda}$.[4]

Generating Erlang-distributed random numbers

Erlang-distributed random numbers can be generated from uniform distribution random numbers ($U \in (0,1]$) using the following formula:[5]

$E(k,\lambda) \approx -\frac{1}\lambda \ln \prod_{i=1}^k U_{i}$

Occurrence

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 Erlang units. 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.[6]

A.K. Erlang worked a lot in traffic modeling. There are thus 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 $X$, that is $\lambda/k$. The (age specific event) rate of the Erlang distribution is, for $k>1$, monotonic in $x$, increasing from zero at $x=0$, to $\lambda$ as $x$ tends to infinity.[7]

Related distributions

• If $\scriptstyle X \;\sim\; \mathrm{Erlang}(k,\, \lambda)\,$ then $\scriptstyle a \cdot X \;\sim\; \mathrm{Erlang}\left(k,\, \frac{\lambda}{a}\right)\,$ with $\scriptstyle a \in \mathbb{R}$
• $\scriptstyle \lim_{k \to \infty}\frac{1}{\sigma_k}\left(\mathrm{Erlang}(k,\, \lambda) \,-\, \mu_k\right) \;\xrightarrow{d}\; N(0,\, 1) \,$ (normal distribution)
• If $\scriptstyle X \;\sim\; \mathrm{Erlang}(k_1,\, \lambda)\,$ and $\scriptstyle Y \;\sim\; \mathrm{Erlang}(k_2,\, \lambda)\,$ then $\scriptstyle X \,+\, Y \;\sim\; \mathrm{Erlang}(k_1 \,+\, k_2,\, \lambda)\,$
• If $\scriptstyle X_i \;\sim\; \mathrm{Exponential}(\lambda)\,$ then $\scriptstyle \sum_{i=1}^k{X_i} \;\sim\; \mathrm{Erlang}(k,\, \lambda)\,$
• Erlang distribution is a special case of type 3 Pearson distribution
• If $\scriptstyle X \;\sim\; \Gamma\left(k,\, \frac{1}{\lambda}\right) \,$ (gamma distribution) then $\scriptstyle X \;\sim\; \mathrm{Erlang}(k,\, \lambda)\,$
• If $\scriptstyle U \;\sim\; \mathrm{Exponential}(\lambda)\,$ and $\scriptstyle V \;\sim\; \mathrm{Erlang}(n,\, \lambda)\,$ then $\scriptstyle \frac{U}{V} \;\sim\; \mathrm{Pareto}(1,\, n)$