A paper[1] points out the fact that:
If we model the Email activity pattern as a renewal process [2] with interevent time distribution $P(\tau)$ then the generation time is the residual waiting time and is characterized by the probability density function [2] $$g(\tau)=\frac{1}{\langle\tau\rangle} \int_{\tau}^{\infty} P(x) d x.$$ Where $\langle\tau\rangle=\int_0^{\infty} \tau P(\tau) d\tau$ is the mean interevent time.
I don't understand how this formula was obtained, and I didn't find an answer for it in the book [2]. Does this formula contradict the classic "waiting time paradox" or does it have anything to do with it?
Thanks a lot for any comments!
[1] V. Alexei, et al, Phys. Rev. Lett. 98, 158702(2007). https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.98.158702
[2] W. Feller, An Introduction to Probability Theory and its Applications (Wiley, New York, 1966), Vol. II.