Fundamentals of Applied Probability and Random Processes, 2nd Edition

4.9 Erlang Distribution

The Erlang distribution is a generalization of the exponential distribution. While the exponential random variable describes the time between adjacent events, the Erlang random variable of order k describes the time interval between any event and the kth following event. Thus, the Erlang random variable is to the exponential random variable as the Pascal random variable is to the geometric random variable.

A random variable X_k is referred to as a kth-order Erlang (or Erlang-k) random variable with parameter λ if its PDF is given by

${\mathit{f}}_{{\mathit{X}}_{\mathit{k}}}\left(\mathit{x}\right)=\left\{\begin{array}{l}\frac{{\mathit{\lambda }}^{\mathit{k}}{\mathit{x}}^{\mathit{k}-1}{\mathit{e}}^{-\mathit{\lambda x}}}{\left(\mathit{k}-1\right)!}\mathit{k}=1,2,3,\dots ;\mathit{x}\ge 0\\ 0\mathit{x}<0\end{array}\right.$

  (4.36)

This PDF can be obtained by considering