The Erlang distribution is a two parameter family of continuous
probability distributions with support . The two parameters are:
a positive integer 'shape' a positive real 'rate' ; sometimes the
scale , the inverse of the rate is used. The Erlang distribution with shape
parameter equal to 1 simplifies to the exponential distribution. It is a
special case of the Gamma distribution. It is the distribution of a sum of
independent exponential variables with mean .
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. Characterization
= Probability density function= The probability density function of the
Erlang distribution is The parameter k is called the shape
parameter, and the parameter is called the rate parameter.
An alternative, but equivalent, parametrization uses the scale parameter
, which is the reciprocal of the rate parameter:
When the scale parameter 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. 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=
The cumulative distribution function of the Erlang distribution is
where is the lower incomplete gamma function. The CDF may also be expressed
as Properties
The Erlang distribution is a solution of the following differential equation:
with initial condition . = 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 ,
i.e. below the mean . Generating Erlang-distributed random
numbers Erlang-distributed random numbers can be
generated from uniform distribution random numbers using the following
formula: 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 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 or queued until
served. 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. There are also 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 , that is . The rate of
the Erlang distribution is, for , monotonic in , increasing from zero at ,
to as tends to infinity. Related distributions
If then with If and then
If Exponential(λ) then Erlang distribution is a special case of
type 3 Pearson distribution If k is an integer: then
If and then See also
Erlang B formula Exponential distribution
Gamma distribution Poisson distribution
Coxian distribution Poisson process
Erlang unit Engset calculation
Phase-type distribution Traffic generation model
Notes ^ Choi, K. P.. "On the medians of gamma
distributions and an equation of Ramanujan". Proceedings of the American
Mathematical Society 121: 245–251. doi:10.1090/S0002-9939-1994-1195477-8.
JSTOR 2160389. ^ Adell, J. A.; Jodrá, P.. "On a
Ramanujan equation connected with the median of the gamma distribution".
Transactions of the American Mathematical Society 360: 3631.
doi:10.1090/S0002-9947-07-04411-X. ^ Jodrá, P.. "Computing the Asymptotic
Expansion of the Median of the Erlang Distribution". Mathematical Modelling
and Analysis 17: 281–292. doi:10.3846/13926292.2012.664571.
^ Banneheka BMSG, Ekanayake GEMUPD "A new point estimator for the median of
gamma distribution". Viyodaya J Science, 14:95-103
^ http:www.xycoon.com/erlang_random.htm ^ C. Chatfield and G.J. Goodhardt: “A
Consumer Purchasing Model with Erlang Interpurchase Times”; Journal of the
American Statistical Association, Dec. 1973, Vol.68, pp.828-835
^ Cox, D.R. Renewal Theory, p20, Methuen.
References Ian Angus "An Introduction to Erlang B
and Erlang C", Telemanagement #187 External links
Erlang Distribution Resource Dimensioning Using Erlang-B and
Erlang-C Erlang-C