If the time between successive events is independent each with an exponential distribution with \(\lambda\), then the total time \(X\) that elapses before all of the next \(n\) events occur is given by the Erlang distribution.
Probability Density Function
Let \(\alpha=n,\beta=\frac{1}{\lambda}\) in the Gamma distribution. \(n\) is a positive integer.
\begin{equation*}
f(x;\lambda,n)=\frac{\lambda\left(x\lambda\right)^{n-1}e^{-\lambda x}}{(n-1)!}
\end{equation*}
for \(x\ge0\)
Mean and Variance
Mean: \(\frac{n}{\lambda}\)
Variance: \(\frac{n}{\lambda^{2}}\)