Closed Form Expressions for the Quantile Function of the Erlang Distribution Used in Engineering Models

Quantile function is heavily utilized in modeling, simulation, reliability analysis and random number generation. The use is often limited if the inversion method fails to estimate it from the cumulative distribution function (CDF). As a result, approximation becomes the other option. The failure of the inversion method is often due to the intractable nature of the CDF of the distribution. Erlang distribution belongs to those classes of distributions. The distribution is a particular case of the gamma distribution. Little is known about the quantile approximation of the Erlang distribution. This is due to the fact that researchers prefer to work with the gamma distribution of which the Erlang is a particular case. This work applied the quantile mechanics approach, power series method and cubic spline interpolation to obtain the approximate of the quantile function of the Erlang distribution for degrees of freedom from one to two. The approximate values compares favorably with the exact ones. Consequently, the result in this paper improved the existing results on the extreme tails of the distribution. The closed form expression for the quantile function obtained here is very useful in modeling physical and engineering systems that are completely described by or fitted with the Erlang distribution.