= Equilibrium delay of an Erlang renewal process
For inter-renewal times with an <Erlang distribution> of integer shape $r$ and rate $\lambda$, the mean is $r/\lambda$ and survival function is $e^{-\lambda t}\sum_{j=0}^{r-1}(\lambda t)^j/j!$. The equilibrium first-delay density is survival divided by the mean,
$$
f_e(t)=\frac\lambda r e^{-\lambda t}\sum_{j=0}^{r-1}\frac{(\lambda t)^j}{j!},\qquad t>0.
$$
This is the equal mixture of Erlang distributions of shapes $1,\ldots,r$. It can be constructed by observing a Poisson clock with uniformly randomized phase modulo $r$, whose every $r$th event is a renewal.
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