Editing Erlang distribution

{{short description|Family of continuous probability distributions}}
{{about|the mathematical / statistical distribution concept||Erlang (disambiguation){{!}}Erlang}}
{{Multiple issues|
{{more citations needed|date=June 2012}}
{{more footnotes|date=June 2012}}
}}
{{Probability distribution
|name       =Erlang
|type       =density
|pdf_image  =[[File:Erlang dist pdf2.svg|325px|Probability density plots of Erlang distributions]]
|cdf_image  =[[File:Erlang dist cdf2.svg|325px|Cumulative distribution plots of Erlang distributions]]
|parameters =<math> k \in \{1,2,3,\ldots\},</math> [[shape parameter|shape]] <br /><math> \lambda \in (0,\infty),</math> rate <br />alt.: <math> \beta = 1/\lambda,</math> [[scale parameter|scale]]
|support    =<math> x \in [0, \infty)</math>
|pdf        =<math> \frac{\lambda^k x^{k-1} e^{-\lambda x}}{(k-1)!}</math>
|cdf        =<math> P(k, \lambda x) = \frac{\gamma(k, \lambda x)}{(k - 1)!} = 1 - \sum_{n=0}^{k-1}\frac{1}{n!}e^{-\lambda x}(\lambda x)^{n}</math>
|mean       =<math> \frac{k}{\lambda}</math>
|mode       =<math> \frac{1}{\lambda}(k - 1)</math>
|variance   =<math> \frac{k}{\lambda^2}</math>
|median     =No simple closed form
|skewness   =<math> \frac{2}{\sqrt{k}}</math>
<!-- invalid parameter |scv        =<math> \frac{1}{k}</math> -->
|kurtosis   =<math> \frac{6}{k}</math>
|entropy    =<math> (1 - k)\psi(k) + \ln\left[\frac{\Gamma(k)}{\lambda}\right] + k</math>
|mgf        =<math> \left(1 - \frac{t}{\lambda}\right)^{-k}</math> for <math> t < \lambda</math>
|char       =<math> \left(1 - \frac{it}{\lambda}\right)^{-k}</math>|
}}
The '''Erlang distribution''' is a two-parameter family of continuous [[probability distribution]]s with [[Support (mathematics)|support]] <math> x \in [0, \infty)</math>. The two parameters are:
* a positive integer <math>k,</math> the "shape", and
* a positive real number <math>\lambda,</math> the "rate". The "scale", <math>\beta,</math> the reciprocal of the rate, is sometimes used instead.

The Erlang distribution is the distribution of a sum of <math>k</math> [[Independence (probability theory)|independent]] [[exponential distribution|exponential variables]] with mean <math>1/\lambda</math> each.  Equivalently, it is the distribution of the time until the ''k''th event of a [[Poisson process]] with a rate of <math>\lambda</math>.  The Erlang and Poisson distributions are complementary, in that while the Poisson distribution counts the 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 <math>k=1</math>, the distribution  simplifies to the [[exponential distribution]]. The Erlang distribution is a special case of the [[gamma distribution]] in which the shape of the distribution is discretized.

The Erlang distribution was developed by [[Agner Krarup Erlang|A. K. Erlang]] to examine the number of telephone calls that might be made at the same time to the operators of the switching stations. This work on telephone [[Teletraffic engineering|traffic engineering]] has been expanded to consider waiting times in [[queueing theory|queueing system]]s in general. The distribution is also used in the field of [[stochastic process]]es.

== Characterization ==

=== Probability density function ===

The [[probability density function]] of the Erlang distribution is

:<math>f(x; k,\lambda)={\lambda^k x^{k-1} e^{-\lambda x} \over (k-1)!}\quad\mbox{for }x, \lambda \geq 0,</math>

The parameter ''k'' is called the shape parameter, and the parameter <math>\lambda</math> is called the rate parameter.

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

:<math>f(x; k,\beta)=\frac{ x^{k-1} e^{-\frac{x}{\beta}} }{\beta^k (k-1)!}\quad\mbox{for }x, \beta \geq 0.</math>

When the scale parameter <math>\beta</math> equals 2, the distribution simplifies to the [[chi-squared distribution]] with 2''k'' degrees of freedom. It can therefore be regarded as a [[generalized chi-squared distribution]] for even numbers of degrees of freedom.

=== Cumulative distribution function (CDF) ===

The [[cumulative distribution function]] of the Erlang distribution is

:<math>F(x; k,\lambda) = P(k, \lambda x) = \frac{\gamma(k, \lambda x)}{\Gamma(k)} = \frac{\gamma(k, \lambda x)}{(k-1)!},</math>

where <math>\gamma</math> is the lower [[incomplete gamma function]] and <math>P</math> is the [[Incomplete gamma function#Regularized Gamma functions and Poisson random variables|lower regularized gamma function]].
The CDF may also be expressed as
:<math>F(x; k,\lambda) = 1 - \sum_{n=0}^{k-1}\frac{1}{n!}e^{-\lambda x}(\lambda x)^n.</math>

=== Erlang-''k'' ===
The Erlang-''k'' distribution (where ''k'' is a positive integer) <math>E_k(\lambda)</math> is defined by setting ''k'' in the PDF of the Erlang distribution.<ref>{{Cite web |title=h1.pdf |url=https://www.win.tue.nl/~iadan/sdp/h1.pdf}}</ref> For instance, the Erlang-2 distribution is <math>E_2(\lambda) ={\lambda^2 x} e^{-\lambda x} \quad\mbox{for }x, \lambda \geq 0</math>, which is the same as <math>f(x; 2,\lambda)</math>.

===Median===
An asymptotic expansion is known for the median of an Erlang distribution,<ref>{{Cite journal | last1 = Choi | first1 = K. P. | doi = 10.1090/S0002-9939-1994-1195477-8 | title = On the medians of gamma distributions and an equation of Ramanujan | journal = Proceedings of the American Mathematical Society | volume = 121 | pages = 245–251 | year = 1994 | issue = 1 | jstor = 2160389| doi-access =  }}</ref> for which coefficients can be computed and bounds are known.<ref>{{Cite journal | last1 = Adell | first1 = J. A. | last2 = Jodrá | first2 = P. | doi = 10.1090/S0002-9947-07-04411-X | title = On a Ramanujan equation connected with the median of the gamma distribution | journal = Transactions of the American Mathematical Society | volume = 360 | issue = 7 | pages = 3631 | year = 2010


 | doi-access = free }}</ref><ref>{{Cite journal | last1 = Jodrá | first1 = P. | title = Computing the Asymptotic Expansion of the Median of the Erlang Distribution | doi = 10.3846/13926292.2012.664571 | journal = Mathematical Modelling and Analysis | volume = 17 | issue = 2 | pages = 281–292 | year = 2012 | doi-access = free }}</ref> An approximation is <math>\frac{k}{\lambda}\left(1-\dfrac{1}{3k+0.2}\right),</math> i.e. below the mean <math>\frac{k}{\lambda}.</math><ref name=Banneheka2009>{{cite journal | last1 = Banneheka | first1 = BMSG | last2 = Ekanayake | first2 = GEMUPD | year = 2009 | title = A new point estimator for the median of gamma distribution | journal = Viyodaya J Science | volume = 14 | pages = 95–103 }}</ref>

== Generating Erlang-distributed random variates ==

Erlang-distributed random variates can be generated from uniformly distributed random numbers (<math>U \in [0,1]</math>) using the following formula:<ref>{{cite web|url=http://www.xycoon.com/erlang_random.htm|title=Statistical Distributions - Erlang Distribution - Random Number Generator|last=Resa|website=www.xycoon.com|access-date=4 April 2018}}</ref>

:<math>E(k,\lambda) = -\frac{1}\lambda \ln \prod_{i=1}^k U_{i} = -\frac{1}\lambda \sum_{i=1}^k \ln U_{i} </math>

== Applications ==

=== 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 [[Erlang unit#Erlang C formula|C]] formulae are still in everyday use for traffic modeling for applications such as the design of [[call center]]s.

===Other applications===

The age distribution of [[cancer]] [[Disease incidence|incidence]] often follows the Erlang distribution, whereas the shape and scale parameters predict, respectively, the number of [[Carcinogenesis|driver events]]  and the time interval between them.<ref>{{cite journal |last1=Belikov |first1=Aleksey V. |title=The number of key carcinogenic events can be predicted from cancer incidence |journal=Scientific Reports |date=22 September 2017 |volume=7 |issue=1 |page=12170 |doi=10.1038/s41598-017-12448-7|pmc=5610194 |pmid=28939880 |bibcode=2017NatSR...712170B }}</ref><ref>{{Cite journal|last1=Belikov|first1=Aleksey V.|last2=Vyatkin|first2=Alexey|last3=Leonov|first3=Sergey V.|date=2021-08-06|title=The Erlang distribution approximates the age distribution of incidence of childhood and young adulthood cancers|journal=PeerJ|language=en|volume=9|pages=e11976|pmid=34434669| doi=10.7717/peerj.11976| pmc=8351573|issn=2167-8359|doi-access=free}}</ref> More generally, the Erlang distribution has been suggested as good approximation of cell cycle time distribution, as result of multi-stage models.<ref>{{cite journal  |last1=Yates |first1=Christian A. |title=A Multi-stage Representation of Cell Proliferation as a Markov Process |journal=Bulletin of Mathematical Biology |date=21 April 2017 |volume=79 |issue=1 |doi=10.1007/s11538-017-0356-4 |pages=2905–2928|doi-access=free |pmid=29030804 |pmc=5709504 }}</ref><ref>{{cite journal  |last1=Gavagnin |first1=Enrico |title=The invasion speed of cell migration models with realistic cell cycle time distributions |journal=Journal of Theoretical Biology |date=21 November 2019 |volume=481 |doi=10.1016/j.jtbi.2018.09.010|arxiv=1806.03140 |pages=91–99 |pmid=30219568 |bibcode=2019JThBi.481...91G }}</ref>

The [[kinesin]] is a molecular machine with two "feet" that "walks" along a filament. The waiting time between each step is exponentially distributed. When [[green fluorescent protein]] is attached to a foot of the kinesin, then the green dot visibly moves with Erlang distribution of k = 2.<ref>{{Cite journal |last1=Yildiz |first1=Ahmet |last2=Forkey |first2=Joseph N. |last3=McKinney |first3=Sean A. |last4=Ha |first4=Taekjip |last5=Goldman |first5=Yale E. |last6=Selvin |first6=Paul R. |author1-link=Ahmet Yıldız (scientist) |author4-link=Taekjip Ha |author6-link=Paul R. Selvin  |date=2003-06-27 |title=Myosin V Walks Hand-Over-Hand: Single Fluorophore Imaging with 1.5-nm Localization |url=https://www.science.org/doi/10.1126/science.1084398 |journal=Science |language=en |volume=300 |issue=5628 |pages=2061–2065 |doi=10.1126/science.1084398 |pmid=12791999 |bibcode=2003Sci...300.2061Y |issn=0036-8075}}</ref>

It has also been used in marketing for describing interpurchase times.<ref>{{cite journal |first1=C. |last1=Chatfield |first2=G.J. |last2=Goodhardt |title=A Consumer Purchasing Model with Erlang Interpurchase Times |journal=Journal of the American Statistical Association |date=December 1973 |volume=68 |issue=344 |pages=828–835 |doi=10.1080/01621459.1973.10481432 }}</ref>

==Properties==
*If <math> X \sim \operatorname{Erlang}(k, \lambda)</math> then <math> a \cdot X \sim \operatorname{Erlang}\left(k, \frac{\lambda}{a}\right)</math> with <math> a \in \mathbb{R}</math>
*If <math> X \sim \operatorname{Erlang}(k_1, \lambda)</math> and <math> Y \sim \operatorname{Erlang}(k_2, \lambda)</math> then <math> X + Y \sim \operatorname{Erlang}(k_1 + k_2, \lambda)</math> if <math> X, Y </math> are independent

==Related distributions==

* 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 <math>X,</math> that is, <math>\lambda/k.</math> The (age specific event) rate of the Erlang distribution is, for <math>k>1,</math> monotonic in <math>x,</math> increasing from 0 at <math>x=0,</math> to <math>\lambda</math> as <math>x</math> tends to infinity.<ref>Cox, D.R. (1967) ''Renewal Theory'', p20, Methuen.</ref> 
** That is: if <math> X_i \sim \operatorname{Exponential}(\lambda),</math> then <math display="block"> \sum_{i=1}^k{X_i} \sim \operatorname{Erlang}(k, \lambda)</math>
* Because of the factorial function in the denominator of the [[#Probability density function|PDF]] and [[#Cumulative distribution function (CDF)|CDF]], 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 <math>k=2</math>). The [[gamma distribution]] generalizes the Erlang distribution by allowing ''k'' to be any positive real number, using the [[gamma function]] instead of the factorial function.
** That is: if k is an [[integer]] and <math> X \sim \operatorname{Gamma}(k, \lambda),</math> then <math> X \sim \operatorname{Erlang}(k, \lambda)</math>
*If <math> U \sim \operatorname{Exponential}(\lambda)</math> and <math> V \sim \operatorname{Erlang}(n, \lambda)</math> then <math> \frac{U}{V}+1 \sim \operatorname{Pareto}(1, n)</math>
*The Erlang distribution is a special case of the [[Pearson distribution|Pearson type III distribution]]{{citation needed|date=March 2016}}
*The Erlang distribution is related to the [[chi-squared distribution]]. If <math> X \sim \operatorname{Erlang}(k,\lambda),</math> then <math> 2\lambda X\sim \chi^2_{2k}.</math>{{citation needed|date=March 2016}}
*The Erlang distribution is related to the [[Poisson distribution]] by the [[Poisson process]]: If <math> S_n = \sum_{i=1}^n X_i</math> such that <math> X_i \sim \operatorname{Exponential}(\lambda),</math> then <math display="block"> S_n \sim \operatorname{Erlang}(n, \lambda)</math> and <math display="block"> \operatorname{Pr}(N(x) \leq n - 1) = \operatorname{Pr}(S_n > x) = 1 - F_X(x; n, \lambda) = \sum_{k=0}^{n-1} \frac{1}{k!}e^{-\lambda x} (\lambda x)^k.</math> Taking the differences over <math>n</math> gives the Poisson distribution.

==See also==
* [[phase-type distribution#Coxian distribution|Coxian distribution]]
* [[Engset calculation]]
* [[Erlang B]] formula
* [[Erlang unit]]
* [[Phase-type distribution]]
* [[Traffic generation model]]

==Notes==
{{Reflist}}

==References==

* Ian Angus [http://www.tarrani.net/linda/ErlangBandC.pdf "An Introduction to Erlang B and Erlang C"], Telemanagement #187 (PDF Document - Has terms and formulae plus short biography)
* Stuart Harris [https://portagecommunications.com/a-primer-on-two-call-center-staffing-methods/  "Erlang Calculations vs. Simulation"]

== External links ==
*[http://www.xycoon.com/erlang.htm Erlang Distribution]
*[http://www.eventhelix.com/RealtimeMantra/CongestionControl/resource_dimensioning_erlang_b_c.htm Resource Dimensioning Using Erlang-B and Erlang-C]

{{ProbDistributions|continuous-semi-infinite}}

{{DEFAULTSORT:Erlang Distribution}}
[[Category:Continuous distributions]]
[[Category:Exponential family distributions]]
[[Category:Infinitely divisible probability distributions]]