Erlang distribution

In probability theory and statistics, the Erlang distribution is a continuous probability distribution with two parameters given by

• n{displaystyle n} the form factor of distribution.
• λ λ {displaystyle lambda } the distribution ratio factor.

Definition

Density Function

A continuous random variable X{displaystyle X} has Erlang distribution with parameters n한 한 Z+{displaystyle nin mathbb {Z} ^{+} and 0}" xmlns="http://www.w3.org/1998/Math/MathML">λ λ ▪0{displaystyle lambda 한0}0}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eea25afc0351140f919cf791c49c1964b8b081de" style="vertical-align: -0.338ex; width:5.616ex; height:2.176ex;"/> and write X♥ ♥ Erlang (n,λ λ ){displaystyle Xsim operatorname {Erlang} (n,lambda)} if the density function for values 0}" xmlns="http://www.w3.org/1998/Math/MathML">x▪0{displaystyle x 2005}0" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80d24be5f0eb4a9173da6038badc8659546021d0" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;"/> That's it.

f(x)=λ λ (λ λ x)n− − 1e− − λ λ x(n− − 1)!{displaystyle f(x)={frac {lambda (lambda x)^{n-1}e^{-lambda x}{(n-1)}}}}}}

This distribution is used to describe the waiting time until the event number n{displaystyle n} in a Poisson process.

This distribution is named after the Danish mathematician and engineer Agner Krarup Erlang, who introduced it in 1909 to examine the number of telephone calls that can be simultaneously assigned to switchboard operators. This telephone traffic engineering work has been extended to consider waiting times in queuing systems in general. The distribution is also used in the field of stochastic processes.

Distribution Function

Yeah. X♥ ♥ Erlang (n,λ λ ){displaystyle Xsim operatorname {Erlang} (n,lambda)} then its accumulated distribution function is given by

F(x)=␡ ␡ k=n∞ ∞ (λ λ x)ke− − λ λ xk!=1− − ␡ ␡ k=0n− − 1(λ λ x)ke− − λ λ xk!{display {begin{aligned}F(x) style=sum _{k=n}{infty }{frac {(lambda x){k}e^{-lambda x}{k}{k}{k}{k!}{k=1-sum _{k=0^}{n-1}{lambda x}{

Properties

Yeah. X♥ ♥ Erlang (n,λ λ ){displaystyle Xsim operatorname {Erlang} (n,lambda)} then.

• The hope of the random variable X{displaystyle X} That's it.

E [chuckles]X]=nλ λ {displaystyle operatorname {E} [X]={frac {n}{lambda }}}}}

• Variance of the random variable X{displaystyle X} That's it.

Var (X)=nλ λ 2{displaystyle operatorname {Var} (X)={frac {n}{lambda ^{2}}}}}

• The moment-generating function is given by

MX(t)=(λ λ λ λ − − t)n{displaystyle M_{X}(t)=left({frac {lambda }{lambda -t}right)^{n}}

Related Distributions

• Erlang's distribution is a particular case of Gamma distribution as if X♥ ♥ Interpreter Interpreter (α α ,λ λ ){displaystyle Xsim Gamma (alphalambda)} with α α =n한 한 N{displaystyle alpha =nin mathbb {N} } then. X♥ ♥ Erlang (n,λ λ ){displaystyle Xsim operatorname {Erlang} (n,lambda)}.
• Yeah. X♥ ♥ Erlang (1,λ λ ){displaystyle Xsim operatorname {Erlang} (1,lambda)} then. X♥ ♥ Exponencial (λ λ ){displaystyle Xsim operatorname {Exponencial} (lambda)}.

Applications

Timeouts

The original implementation of the distribution model occurred in the telephony area in large concentrations. In this case, the events that occur independently with an average rate are modeled with a Poisson process. The wait times between k occurrences of the event are distributed in an Erlang. (As a related relevant factor, the number of events in a given period of time is described by the Poisson distribution)

The Erlang distribution, when used to measure the time between incoming calls, can be used together with the expected duration of calls to produce information about the traffic load measured in erlangs. This can be used to determine the probability of event loss or delay (in telephony abandon or long hold), based on various assumptions about whether blocked calls are aborted (Erlang B formula) or queued until they are called. attended (Erlang C formula). The Erlang-B and C formulas are still used daily for traffic modeling for applications such as call center design.

Other applications

Cancer incidence, distributed by age, often follows the Erlang distribution, while shape and scale parameters predict, respectively, the number of causal events and the time interval between them. More In general, the Erlang distribution has been suggested as a good approximation of the cell cycle time distribution, as a result of multi-stage models.

It has also been used in business economics to describe times between purchases.