Erlang Distribution

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The Erlang distribution is a two-parameter family of continuous

probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...

s with

support Support may refer to: Arts, entertainment, and media * Supporting character Business and finance * Support (technical analysis) * Child support * Customer support * Income Support Construction * Support (structure), or lateral support, a ...

x \in

\lambda,_the_"rate"._The_"scale",_

\beta,

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\beta,

_the_reciprocal_of_the_rate,_is_sometimes_used_instead. The_Erlang_distribution_is_the_distribution_of_a_sum_of_

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_Independence_(probability_theory)">independent_ Independent_or_Independents_may_refer_to: _Arts,_entertainment,_and_media_Artist_groups *__Independents_(artist_group),_a_group_of_modernist_painters_based_in_the_New_Hope,_Pennsylvania,_area_of_the_United_States_during_the_early_1930s *__Independ_...

_exponential_distribution.html" ;"title="Independence_(probability_theory).html" "title=", \infty). The two parameters are: * a positive integer

k,

the "shape", and * a positive real number

\lambda,

the "rate". The "scale",

\beta,

the reciprocal of the rate, is sometimes used instead. The Erlang distribution is the distribution of a sum of

k

Independence (probability theory)">independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independ ...

exponential distribution">exponential variables with mean

1/\lambda

each. Equivalently, it is the distribution of the time until the ''k''th event of a

Poisson process In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one ...

with a rate of

\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

k=1

, the distribution simplifies to the

exponential distribution In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average ...

. The Erlang distribution is a special case of the

gamma distribution In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma distri ...

wherein the shape of the distribution is discretised. The Erlang distribution was developed by

A. K. Erlang Agner Krarup Erlang (1 January 1878 – 3 February 1929) was a Denmark, Danish mathematician, statistician and engineer, who invented the fields of teletraffic engineering, traffic engineering and queueing theory. By the time of his relatively ...

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 process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...

es.

Characterization

Probability density function

The

probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...

of the Erlang distribution is :

f(x; k,\lambda)=\quad\mboxx, \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

\beta

, which is the reciprocal of the rate parameter (i.e.,

\beta = 1/\lambda

): :

f(x; k,\beta)=\frac\quad\mboxx, \beta \geq 0.

When the scale parameter

\beta

equals 2, the distribution simplifies to the

chi-squared distribution In probability theory and statistics, the chi-squared distribution (also chi-square or \chi^2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-squa ...

with 2''k'' degrees of freedom. It can therefore be regarded as a

generalized chi-squared distribution In probability theory and statistics, the generalized chi-squared distribution (or generalized chi-square distribution) is the distribution of a quadratic form of a multinormal variable (normal vector), or a linear combination of different no ...

for even numbers of degrees of freedom.

Cumulative distribution function (CDF)

The

cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ev ...

of the Erlang distribution is :

F(x; k,\lambda) = P(k, \lambda x) = \frac = \frac,

where

\gamma

is the lower

incomplete gamma function In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to various mathematical problems such as certain integrals. Their respective names stem from their integral definitions, which ...

and

P

is the lower regularized gamma function. The CDF may also be expressed as :

F(x; k,\lambda) = 1 - \sum_^\frace^(\lambda x)^n.

Erlang-''k''

The Erlang-''k'' distribution (where ''k'' is a positive integer)

E_k(\lambda)

is defined by setting ''k'' in the PDF of the Erlang distribution. For instance, the Erlang-2 distribution is

E_2(\lambda) = e^ \quad\mboxx, \lambda \geq 0

, which is the same as

f(x; 2,\lambda)

.

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

\frac\left(1-\dfrac\right),

i.e. below the mean

\frac.

Generating Erlang-distributed random variates

Erlang-distributed random variates can be generated from uniformly distributed random numbers (

U \in,1 /math>) using the following formula:

: E(k,\lambda) = -\frac\lambda \ln \prod_^k U_ = -\frac\lambda \sum_^k \ln U_

Applications

Waiting times

Events that occur independently with some average rate are modeled with a

Poisson process In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one ...

. 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 In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known co ...

.) 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 The erlang (symbol E) is a dimensionless unit that is used in telephony as a measure of offered load or carried load on service-providing elements such as telephone circuits or telephone switching equipment. A single cord circuit has the capaci ...

and C formulae are still in everyday use for traffic modeling for applications such as the design of

call center A call centre ( Commonwealth spelling) or call center (American spelling; see spelling differences) is a managed capability that can be centralised or remote that is used for receiving or transmitting a large volume of enquiries by telephone. ...

s.

Other applications

The age distribution of

cancer Cancer is a group of diseases involving abnormal cell growth with the potential to invade or spread to other parts of the body. These contrast with benign tumors, which do not spread. Possible signs and symptoms include a lump, abnormal b ...

incidence often follows the Erlang distribution, whereas the shape and scale parameters predict, respectively, the number of driver events and the time interval between them. More generally, the Erlang distribution has been suggested as good approximation of cell cycle time distribution, as result of multi-stage models. It has also been used in business economics for describing interpurchase times.

Properties

*If

X \sim \operatorname(k, \lambda)

then

a \cdot X \sim \operatorname\left(k, \frac\right)

with

a \in \mathbb

*If

X \sim \operatorname(k_1, \lambda)

and

Y \sim \operatorname(k_2, \lambda)

then

X + Y \sim \operatorname(k_1 + k_2, \lambda)

if

X, Y

are independent

Related distributions

* The Erlang distribution is the distribution of the sum of ''k''

independent and identically distributed random variables In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is us ...

, each having an

exponential distribution In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average ...

. 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 0 at

x=0,

to

\lambda

as

x

tends to infinity.Cox, D.R. (1967) ''Renewal Theory'', p20, Methuen. ** That is: if

X_i \sim \operatorname(\lambda),

then

\sum_^k \sim \operatorname(k, \lambda)

* Because of the factorial function in the denominator of the

PDF Portable Document Format (PDF), standardized as ISO 32000, is a file format developed by Adobe in 1992 to present documents, including text formatting and images, in a manner independent of application software, hardware, and operating systems. ...

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

k=2

). The

gamma distribution In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma distri ...

generalizes the Erlang distribution by allowing ''k'' to be any positive real number, using the

gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...

instead of the factorial function. ** That is: if k is an

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

and

X \sim \operatorname(k, \lambda),

then

X \sim \operatorname(k, \lambda)

*If

U \sim \operatorname(\lambda)

and

V \sim \operatorname(n, \lambda)

then

\frac+1 \sim \operatorname(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 In probability theory and statistics, the chi-squared distribution (also chi-square or \chi^2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-squa ...

. If

X \sim \operatorname(k,\lambda),

then

2\lambda X\sim \chi^2_.

*The Erlang distribution is related to the

Poisson distribution In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known co ...

by the

Poisson process In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one ...

: If

S_n = \sum_^n X_i

such that

X_i \sim \operatorname(\lambda),

then

S_n \sim \operatorname(n, \lambda)

and

\operatorname(N(x) \leq n - 1) = \operatorname(S_n > x) = 1 - F_X(x; n, \lambda) = \sum_^ \frace^ (\lambda x)^k.

Taking the differences over

n

gives the Poisson distribution.

See also

* Coxian distribution *

Engset calculation In queueing theory, the Engset formula is used to determine the blocking probability of an M/M/c/c/N queue (in Kendall's notation). The formula is named after its developer, T. O. Engset. Example application Consider a fleet of c vehicles and N ...

*

Erlang B The erlang (symbol E) is a dimensionless unit that is used in telephony as a measure of offered load or carried load on service-providing elements such as telephone circuits or telephone switching equipment. A single cord circuit has the capa ...

formula *

Erlang unit The erlang (symbol E) is a dimensionless unit that is used in telephony as a measure of offered load or carried load on service-providing elements such as telephone circuits or telephone switching equipment. A single cord circuit has the capaci ...

*

Phase-type distribution A phase-type distribution is a probability distribution constructed by a convolution or mixture of exponential distributions. It results from a system of one or more inter-related Poisson processes occurring in sequence, or phases. The sequence ...

*

Traffic generation model A traffic generation model is a stochastic model of the traffic flows or data sources in a communication network, for example a cellular network or a computer network. A packet generation model is a traffic generation model of the packet flows or ...

Notes

References

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

External links

{{DEFAULTSORT:Erlang Distribution Continuous distributions Exponential family distributions Infinitely divisible probability distributions