Erlang

Probability density function
Cumulative distribution function
Parameters	shape (integer) rate (real) alt.: scale (real)
Support
Probability density function (pdf)
Cumulative distribution function (cdf)
Mean
Median
Mode	for
Variance
Skewness
Kurtosis
Entropy
mgf	for
Char. func.

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 field of stochastic processes. Download high resolution version (1300x975, 158 KB) See the image on the commons for gnuplot source. ... Image File history File links Download high resolution version (1300x975, 174 KB) Please see the file description page for further information. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In mathematics, the set of real numbers, denoted R, or in blackboard bold , is the set of all rational and irrational numbers. ... In statistics, if a family of probabiblity densities parametrized by a parameter s is of the form fs(x) = f(sx)/s then s is called a scale parameter, since its value determines the scale of the probability distribution. ... The word real has many different meanings. ... In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than... In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... In probability theory and statistics, a median is a number dividing the higher half of a sample, a population, or a probability distribution from the lower half. ... In statistics, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. ... In probability theory and statistics, the variance of a random variable (or equivalently, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ... In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ... In probability theory and statistics, kurtosis is a measure of the peakedness of the probability distribution of a real-valued random variable. ... Entropy of a Bernoulli trial as a function of success probability, often called the binary entropy function. ... In probability theory and statistics, the moment-generating function of a random variable X is wherever this expectation exists. ... In probability theory, the characteristic function of any random variable completely defines its probability distribution. ... In mathematics and statistics, a probability distribution, more properly called a probability density, assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... In probability theory and statistics, the exponential distributions are a class of continuous probability distribution. ... In probability theory and statistics, the gamma distribution is a continuous probability distribution. ... Agner Krarup Erlang (January 1, 1878–February 3, 1929) was a Danish mathematician, statistician, and engineer who invented the fields of queueing theory and traffic engineering. ... For another meaning of the term traffic engineering, please see transport traffic engineering. ... Queueing theory (also commonly spelled queuing theory) is the mathematical study of waiting lines (or queues). ... In the mathematics of probability, a stochastic process is a random function. ...

Overview

The Erlang 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 an integer, and the rate λ, which is a real. The distribution is sometimes defined using the inverse of the rate parameter, the scale θ.

When the shape parameter k equals 1, the distribution simplifies to the exponential distribution. In probability theory and statistics, the exponential distributions are a class of continuous probability 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 a real. In probability theory and statistics, the gamma distribution is a continuous probability distribution. ...

Specification of the Erlang distribution

Probability density function

The probability density function of the Erlang distribution is In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ...

where e is the base of the natural logarithm and ! is the factorial function. The parameter k is called the shape parameter and the parameter λ is called the rate parameter. An alternative, but equivalent, parametrization uses the scale parameter θ which is simply the inverse of the rate parameter (i.e. θ = 1 / λ): e is the unique number such that the value of the derivative (slope) of f(x)=ex for any value of x is equal to the value of f(x). ... The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is equal to 2. ... In mathematics, the factorial of a natural number n is the product of the positive integers less than or equal to n. ...

Because of the factorial function in the denominator, the Erlang distribution is only defined when the parameter k is a positive integer. The Gamma distribution generalizes the Erlang by allowing its first parameter to be a real, using the gamma function instead of the factorial function. In probability theory and statistics, the gamma distribution is a continuous probability distribution. ... The Gamma function along part of the real axis In mathematics, the Gamma function extends the factorial function to complex and non integer numbers (it is already defined on the naturals, and has simple poles at the negative integers). ...

Cumulative distribution function

The cumulative distribution function of the Erlang distribution is In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than...

where γ() is the incomplete gamma function. In mathematics, the gamma function is defined by a definite integral. ...

Occurrence

Waiting times

Events which 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.) A simple Poisson process, named after the French mathematician Siméon-Denis Poisson (1781 - 1840), is a stochastic process which is defined in terms of the occurrences of events. ... In probability theory and statistics, the Poisson distribution is a discrete probability 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 determined 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 formulas are still in everyday use for traffic modelling for applications such as the design of call centers. The dimensionless unit named the Erlang is a statistical measure of telecommunications traffic used in telephony. ... A call centre (Commonwealth English) or call center (AmE) is a centralized office of a company that answers incoming telephone calls from customers(often for the purposes of product support) , or that makes outgoing telephone calls to customers (telemarketing). ...

Compartment models

The Erlang distribution also occurs as a description of the rate of transition of elements through a system of compartments. Such systems are widely used in biology and ecology.

Stochastic processes

The Erlang distribution is the distribution of the sum of k independent identically distributed random variables each having an exponential distribution. In probability theory and statistics, the exponential distributions are a class of continuous probability distribution. ...

See also

• Exponential distribution
• Gamma distribution
• Poisson distribution
• Coxian distribution
• Poisson process
• Erlang unit
• Engset calculation
• Phase-type distribution

	Probability distributions [ view • talk • edit ]
	Univariate	Multivariate
Discrete:	Bernoulli • binomial • Boltzmann • compound Poisson • degenerate • degree • Gauss-Kuzmin • geometric • hypergeometric • logarithmic • negative binomial • parabolic fractal • Poisson • Rademacher • Skellam • uniform • Yule-Simon • zeta • Zipf • Zipf-Mandelbrot	Ewens • multinomial
Continuous:	Beta • Beta prime • Cauchy • chi-square • Dirac delta function • Erlang • exponential • exponential power • F • fading • Fisher's z • Fisher-Tippett • Gamma • generalized extreme value • generalized hyperbolic • generalized inverse Gaussian • Half-Logistic • Hotelling's T-square • hyperbolic secant • hyper-exponential • hypoexponential • inverse chi-square • inverse gaussian • inverse gamma • Kumaraswamy • Landau • Laplace • Lévy • Lévy skew alpha-stable • logistic • log-normal • Maxwell-Boltzmann • Maxwell speed • normal (Gaussian) • Pareto • Pearson • polar • raised cosine • Rayleigh • relativistic Breit-Wigner • Rice • Student's t • triangular • type-1 Gumbel • type-2 Gumbel • uniform • Voigt • von Mises • Weibull • Wigner semicircle	Dirichlet • Kent • matrix normal • multivariate normal • von Mises-Fisher • Wigner quasi • Wishart
Miscellaneous:	Cantor • conditional • exponential family • infinitely divisible • location-scale family • marginal • maximum entropy • phase-type • posterior • prior • quasi • sampling

In probability theory and statistics, the exponential distributions are a class of continuous probability distribution. ... In probability theory and statistics, the gamma distribution is a continuous probability distribution. ... In probability theory and statistics, the Poisson distribution is a discrete probability distribution. ... A simple Poisson process, named after the French mathematician Siméon-Denis Poisson (1781 - 1840), is a stochastic process which is defined in terms of the occurrences of events. ... The dimensionless unit named the Erlang is a statistical measure of telecommunications traffic used in telephony. ... This article is one of a group being considered for deletion in accordance with Wikipedias deletion policy. ... In probability theory, a phase-type distribution is a probability distribution that can be represented as the time to absorption in a continuous-time Markov chain with m transient states i = 1, 2, ..., m and one absorbing state 0. ... Image File history File links Bvn-small. ... In mathematics and statistics, a probability distribution, more properly called a probability density, assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... A multivariate random variable or random vector is a vector X = (X1, ..., Xn) whose components are scalar-valued random variables on the same probability space (Ω, P). ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... In probability theory and statistics, the Bernoulli distribution, named after Swiss scientist James Bernoulli, is a discrete probability distribution, which takes value 1 with success probability and value 0 with failure probability . ... In probability theory and statistics, the binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. ... The Maxwell-Boltzmann distribution is a probability distribution with applications in physics and chemistry. ... In probability theory, a compound Poisson distribution is the probability distribution of a Poisson-distibuted number of independent identically-distributed random variables. ... In mathematics, a degenerate distribution is the probability distribution of a random variable which always has the same value. ... In the mathematical field of graph theory the degree distribution of a graph is a function describing the total number of vertices in a graph with a given degree (number of connections to other vertices). ... In mathematics, the Gauss-Kuzmin distribution gives the probability distribution of the occurrence of a given integer in the continued fraction expansion of an arbitrary real number. ... In probability theory and statistics, the geometric distribution is either of two discrete probability distributions: the probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set { 1, 2, 3, ...}, or the probability distribution of the number Y = X âˆ’ 1 of failures before... In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the number of successes in a sequence of n draws from a finite population without replacement. ... In probability and statistics, the logarithmic distribution (also known as the logarithmic series distribution) is a discrete probability distribution. ... In probability and statistics the negative binomial distribution is a discrete probability distribution. ... In the parabolic fractal distribution, the logarithm of the frequency or size of entities in a population is a quadratic polynomial of the logarithm of the rank. ... In probability theory and statistics, the Poisson distribution is a discrete probability distribution. ... In probability theory and statistics, the Rademacher distribution is a discrete probability distribution. ... The Skellam distribution is the discrete probability distribution of the difference N1 − N2 of two correlated or uncorrelated random variables N1 and N2 having Poisson distributions with different expected values μ1 and μ2. ... In probability theory and statistics, the discrete uniform distribution is a discrete probability distribution that can be characterized by saying that all values of a finite set of possible values are equally probable. ... In probability and statistics, the Yule-Simon distribution is a discrete probability distribution. ... In probability theory and statistics, the zeta distribution is a discrete probability distribution. ... Originally, Zipfs law stated that, in a corpus of natural language utterances, the frequency of any word is roughly inversely proportional to its rank in the frequency table. ... In probability theory and statistics, the Zipf-Mandelbrot law is a discrete probability distribution. ... In population genetics, Ewenss sampling formula, introduced by Warren Ewens, states that under certain conditions (specified below), if a random sample of n gametes is taken from a population and classified according to the gene at a particular locus then the probability that there are a1 alleles represented once... In probability theory, the multinomial distribution is a generalization of the binomial distribution. ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... In probability theory and statistics, the beta distribution is a continuous probability distribution with the probability density function (pdf) defined on the interval [0, 1]: where α and β are parameters that must be greater than zero and B is the beta function. ... A Beta Prime Distribution is a distribution with probability function: where is a Beta function. ... The Cauchy-Lorentz distribution, named after Augustin Cauchy, is a continuous probability distribution with probability density function where x0 is the location parameter, specifying the location of the peak of the distribution, and γ is the scale parameter which specifies the half-width at half-maximum (HWHM). ... In probability theory and statistics, the chi-square distribution (also chi-squared distribution), or χ2  distribution, is one of the theoretical probability distributions most widely used in inferential statistics, i. ... The Dirac delta function, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere. ... In probability theory and statistics, the exponential distributions are a class of continuous probability distribution. ... The exponential power distribution, also known as the generalized error distribution, takes a scale parameter a and exponent b. ... In statistics and probability, the F-distribution is a continuous probability distribution. ... In telecommunication, a fading distribution is the probability distribution that signal fading will exceed a given value relative to a specified reference level. ... Fishers z-distribution is the distribution of half the logarithm of a F distribution variate: It was first described by Ronald Fisher in a paper delivered at the International Mathematical Congress of 1924 in Toronto, entitled On a distribution yielding the error functions of several well-known statistics. Nowadays... In probability theory and statistics the Gumbel distribution (named after Emil Julius Gumbel (1891–1966)) is used to find the minimum (or the maximum) of a number of samples of various distributions. ... In probability theory and statistics, the gamma distribution is a continuous probability distribution. ... In probability theory and statistics, the generalized extreme value distribution (GEV) is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel, Fréchet and Weibull families also known as type I, II and III extreme value distributions. ... The generalised hyperbolic distribution is a continuous probability distribution defined by the probability density function where is the modified Bessel function of the second kind. ... In probability theory, the Generalized inverse Gaussian distribution (GIG) is a probability distribution with probability density function It is used extensively in geostatistics, statistical linguistics, finance, etc. ... In probability theory and statistics, the half-logistic distribution is a continuous probability distribution—the distribution of the absolute value of a random variable following the logistic distribution. ... In statistics, Hotellings T-square statistic, named for Harold Hotelling, is a generalization of Students t statistic that is used in multivariate hypothesis testing. ... In probability theory and statistics, the hyperbolic secant distribution is a continuous probability distribution whose probability density function and characteristic function are proportional to the hyperbolic secant function. ... In probability theory, a hyper-exponential distribution is a continuous distribution such that the probability density function of the random variable X is given by: Where is an exponentially distributed random variable with rate parameter , and is the probability that X will take on the form of the exponential distribution... The hypoexponential distribution is a generalization of Erlang distribution in the sense that the n exponential distributions may have different rates. ... In probability and statistics, the inverse-chi-square distribution is the probability distribution of a random variable whose inverse has a chi-square distribution. ... Edit ... The inverse gamma distribution has the probability density function over the support with shape parameter and scale parameter . ... In probability theory and statistics, Kumaraswamys double bounded distribution is as versatile as the Beta distribution, but much simpler to use especially in simulation studies as it has a simple closed form solution for both its pdf and cdf. ... The probability distribution for Landau random variates is defined analytically by the complex integral, For numerical purposes it is more convenient to use the following equivalent form of the integral, From GSL manual, used under GFDL. ... In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. ... In probability theory and statistics, the Lévy distribution, named after Paul Pierre Lévy, is one of the few distributions that are stable and that have probability density functions that are analytically expressible. ... In probability theory, a Lévy skew alpha-stable distribution or just stable distribution, developed by Paul Lévy, is a probability distribution where sums of independent identically distributed random variables have the same distribution as the original. ... In probability theory and statistics, the logistic distribution is a continuous probability distribution. ... In probability and statistics, the log-normal distribution is the probability distribution of any random variable whose logarithm is normally distributed (the base of the logarithmic function is immaterial in that loga X is normally distributed if and only if logb X is normally distributed). ... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... The normal distribution, also called Gaussian distribution (although Gauss was not the first to work with it), is an extremely important probability distribution in many fields. ... The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution found in a large number of real-world situations. ... The Pearson distribution is a family of probability distributions that are a generalisation of the normal distribution. ... In probability theory, the polar distribution is the probability distribution of angles occurring in a set of two-dimensional vectors, denoted by It is usually graphically represented as a closed curve , where the radius equals the probability . ... In probability theory and statistics, the raised cosine distribution is a probability distribution supported on the interval []. The probability density function is for and zero otherwise. ... In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution. ... NB: The information in this article should be reviewed. ... In probability theory and statistics, the Rice distribution distribution is a continuous probability distribution. ... In probability and statistics, the t-distribution or Students t-distribution is a probability distribution that arises in the problem of estimating the mean of a normally distributed population when the sample size is small. ... In probability theory and statistics, the triangular distribution is a continuous probability distribution with lower limit a, mode c and upper limit b. ... In probability theory, the Type-1 Gumbel distribution function is for . Reference Taken from the gsl-ref_19. ... In probability theory, the Type-2 Gumbel distribution function is for . Based on gsl-ref_19. ... In mathematics, the continuous uniform distributions are probability distributions such that all intervals of the same length are equally probable. ... In spectroscopy, the Voigt profile is a spectral line profile named after Woldemar Voigt and found in all branches of spectroscopy in which a spectral line is broadened by two types of mechanisms, one of which alone would produce a Doppler profile, and the other of which would produce a... In probability theory and statistics, the von Mises distribution is a continuous probability distribution. ... In probability theory and statistics, the Weibull distribution (named after Waloddi Weibull) is a continuous probability distribution with the probability density function where and is the shape parameter and is the scale parameter of the distribution. ... The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution supported on the interval [−R, R] the graph of whose probability density function f is a semicircle of radius R centered at (0, 0) and then suitably normalized (so that it is really a semi-ellipse... In probability and statistics, the Dirichlet distribution (after Johann Peter Gustav Lejeune Dirichlet) is a continuous multivariate probability distribution. ... The 5-parameter Fisher-Bingham distribution or Kent distribution is a probability distribution on the three-dimensional sphere. ... The matrix normal distribution is a probability distribution that is a generalization of the normal distribution. ... In probability theory and statistics, a multivariate normal distribution, also sometimes called a multivariate Gaussian distribution, is a specific probability distribution, which can be thought of as a generalization to higher dimensions of the one-dimensional normal distribution (also called a Gaussian distribution). ... Points sampled from three von Mises-Fisher distributions on the sphere (blue: , green: , red: ). The mean directions are shown with arrows. ... The Wigner quasi-probability distribution was introduced by Eugene Wigner in 1932 to study quantum corrections to classical statistical mechanics. ... In statistics, the Wishart distribution, named in honor of John Wishart, is any of a family of probability distributions for nonnegative-definite matrix-valued random variables (random matrices). These distributions are of great importance in the estimation of covariance matrices in multivariate statistics. ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... The Cantor distribution is the probability distribution whose cumulative distribution function is the Cantor function. ... This article defines some terms which characterize probability distributions of two or more variables. ... In probability and statistics, an exponential family is any class of probability distributions having a certain form. ... The concept of infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). ... In probability theory, especially as that field is used in statistics, a location-scale family is a set of probability distributions on the real line parametrized by a location parameter μ and a scale parameter σ â‰¥ 0; if X is any random variable whose probability distribution belongs to such a family, then... In probability theory, given two jointly distributed random variables X and Y, the marginal distribution of X is simply the probability distribution of X ignoring information about Y, typically calculated by summing or integrating the joint probability distribution over Y. For discrete random variables, the marginal probability mass function can... In statistics and information theory, a maximum entropy probability distribution is a probability distribution whose entropy is larger than (or equal to) that of all other members of a specified class of distributions. ... In probability theory, a phase-type distribution is a probability distribution that can be represented as the time to absorption in a continuous-time Markov chain with m transient states i = 1, 2, ..., m and one absorbing state 0. ... The posterior probability can be calculated by Bayes theorem from the prior probability and the likelihood function. ... A prior probability is a marginal probability, interpreted as a description of what is known about a variable in the absence of some evidence. ... // Introduction In the most general form, the dynamics of a quantum mechanical system are determined by a master equation - an equation of motion for the density operator (usually written ) of the system. ... In statistics, a sampling distribution is the probability distribution, under repeated sampling of the population, of a given statistic (a numerical quantity calculated from the data values in a sample). ...

External links

• Erlang Distribution
• An Introduction to Erlang B and Erlang C by Ian Angus (PDF Document - Has terms and formulae plus biography)
• Resource Dimensioning Using Erlang-B and Erlang-C
• Erlang-C