Erlang

Probability density function
Cumulative distribution function
Parameters	${\displaystyle k\in \{1,2,3,\ldots \},}$ shape ${\displaystyle \lambda \in (0,\infty ),}$ rate alt.: ${\displaystyle \beta =1/\lambda ,}$ scale
Support	${\displaystyle x\in [0,\infty )}$
PDF	${\displaystyle {\frac {\lambda ^{k}x^{k-1}e^{-\lambda x}}{(k-1)!}}}$
CDF	${\displaystyle P(k,\lambda x)={\frac {\gamma (k,\lambda x)}{(k-1)!}}=1-\sum _{n=0}^{k-1}{\frac {1}{n!}}e^{-\lambda x}(\lambda x)^{n}}$
Mean	${\displaystyle {\frac {k}{\lambda }}}$
Median	No simple closed form
Mode	${\displaystyle {\frac {1}{\lambda }}(k-1)}$
Variance	${\displaystyle {\frac {k}{\lambda ^{2}}}}$
Skewness	${\displaystyle {\frac {2}{\sqrt {k}}}}$
Ex. kurtosis	${\displaystyle {\frac {6}{k}}}$
Entropy	${\displaystyle (1-k)\psi (k)+\ln \left[{\frac {\Gamma (k)}{\lambda }}\right]+k}$
MGF	${\displaystyle \left(1-{\frac {t}{\lambda }}\right)^{-k}}$ for ${\displaystyle t<\lambda }$
CF	${\displaystyle \left(1-{\frac {it}{\lambda }}\right)^{-k}}$

The Erlang distribution is a two-parameter family of continuous probability distributions with support ${\displaystyle x\in [0,\infty )}$. The two parameters are:

• a positive integer ${\displaystyle k,}$ the "shape", and
• a positive real number ${\displaystyle \lambda ,}$ the "rate". The "scale", ${\displaystyle \beta ,}$ the reciprocal of the rate, is sometimes used instead.

The Erlang distribution is the distribution of a sum of ${\displaystyle k}$ independent exponential variables with mean ${\displaystyle 1/\lambda }$ each. Equivalently, it is the distribution of the time until the kth event of a Poisson process with a rate of ${\displaystyle \lambda }$. 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 ${\displaystyle k=1}$, 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.

Characterization

Generating Erlang-distributed random variates

Erlang-distributed random variates can be generated from uniformly distributed random numbers (${\displaystyle U\in [0,1]}$) using the following formula:^[6]

${\displaystyle E(k,\lambda )=-{\frac {1}{\lambda }}\ln \prod _{i=1}^{k}U_{i}=-{\frac {1}{\lambda }}\sum _{i=1}^{k}\ln U_{i}}$

Applications

Properties

• If ${\displaystyle X\sim \operatorname {Erlang} (k,\lambda )}$ then ${\displaystyle a\cdot X\sim \operatorname {Erlang} \left(k,{\frac {\lambda }{a}}\right)}$ with ${\displaystyle a\in \mathbb {R} }$
• If ${\displaystyle X\sim \operatorname {Erlang} (k_{1},\lambda )}$ and ${\displaystyle Y\sim \operatorname {Erlang} (k_{2},\lambda )}$ then ${\displaystyle X+Y\sim \operatorname {Erlang} (k_{1}+k_{2},\lambda )}$ if ${\displaystyle X,Y}$ 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 ${\displaystyle X,}$ that is, ${\displaystyle \lambda /k.}$ The (age specific event) rate of the Erlang distribution is, for ${\displaystyle k>1,}$ monotonic in ${\displaystyle x,}$ increasing from 0 at ${\displaystyle x=0,}$ to ${\displaystyle \lambda }$ as ${\displaystyle x}$ tends to infinity.^[12]
  • That is: if ${\displaystyle X_{i}\sim \operatorname {Exponential} (\lambda ),}$ then
    $${\displaystyle \sum _{i=1}^{k}{X_{i}}\sim \operatorname {Erlang} (k,\lambda )}$$

    
• Because of the factorial function in the denominator of the PDF and 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 ${\displaystyle k=2}$). 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 ${\displaystyle X\sim \operatorname {Gamma} (k,\lambda ),}$ then ${\displaystyle X\sim \operatorname {Erlang} (k,\lambda )}$
• If ${\displaystyle U\sim \operatorname {Exponential} (\lambda )}$ and ${\displaystyle V\sim \operatorname {Erlang} (n,\lambda )}$ then ${\displaystyle {\frac {U}{V}}+1\sim \operatorname {Pareto} (1,n)}$
• The Erlang distribution is a special case of the Pearson type III distribution
• The Erlang distribution is related to the chi-squared distribution. If ${\displaystyle X\sim \operatorname {Erlang} (k,\lambda ),}$ then ${\displaystyle 2\lambda X\sim \chi _{2k}^{2}.}$
• The Erlang distribution is related to the Poisson distribution by the Poisson process: If ${\displaystyle S_{n}=\sum _{i=1}^{n}X_{i}}$ such that ${\displaystyle X_{i}\sim \operatorname {Exponential} (\lambda ),}$ then
  $${\displaystyle S_{n}\sim \operatorname {Erlang} (n,\lambda )}$$

  
  and
  $${\displaystyle \operatorname {Pr} (N(x)\leq n-1)=\operatorname {Pr} (S_{n}>x)=1-F_{X}(x;n,\lambda )=\sum _{k=0}^{n-1}{\frac {1}{k!}}e^{-\lambda x}(\lambda x)^{k}.}$$

  
  Taking the differences over ${\displaystyle n}$ gives the Poisson distribution.

See also

• Coxian distribution
• Engset calculation
• Erlang B formula
• Erlang unit
• Phase-type distribution
• Traffic generation model

Notes

References

• Ian Angus "An Introduction to Erlang B and Erlang C", Telemanagement #187 (PDF Document - Has terms and formulae plus short biography)
• Stuart Harris "Erlang Calculations vs. Simulation"

External links

• Erlang Distribution
• Resource Dimensioning Using Erlang-B and Erlang-C