The Kaniadakis Erlang distribution (or κ-Erlang Gamma distribution) is a family of continuous statistical distributions, which is a particular case of the κ-Gamma distribution, when

positive integer.

[1] The first member of this family is the κ-exponential distribution of Type I.

The κ-Erlang is a κ-deformed version of the Erlang distribution.

It is one example of a Kaniadakis distribution.

The Kaniadakis κ-Erlang distribution has the following probability density function:[1] valid for

{\displaystyle n={\textrm {positive}}\,\,{\textrm {integer}}}

is the entropic index associated with the Kaniadakis entropy.

The ordinary Erlang Distribution is recovered as

The cumulative distribution function of κ-Erlang distribution assumes the form:[1] valid for

The cumulative Erlang distribution is recovered in the classical limit

The survival function of the κ-Erlang distribution is given by:

exp

}}\prod _{m=0}^{n}\left[1+(2m-n)\kappa \right]\int _{0}^{x}z^{n-1}\exp _{\kappa }(-z)dz}

The survival function of the κ-Erlang distribution enables the determination of hazard functions in closed form through the solution of the κ-rate equation:

is the hazard function.

A family of κ-distributions arises from the κ-Erlang distribution, each associated with a specific value of

Such members are determined from the κ-Erlang cumulative distribution, which can be rewritten as: where with The first member (

) of the κ-Erlang family is the κ-Exponential distribution of type I, in which the probability density function and the cumulative distribution function are defined as: The second member (

) of the κ-Erlang family has the probability density function and the cumulative distribution function defined as: The second member (

) has the probability density function and the cumulative distribution function defined as: