Frog Model Wakeup Time on the Complete Graph

The frog model is a system of random walks where active particles set sleeping particles in motion. On the complete graph with n vertices it is equivalent to a well-understood rumor spreading model. We given an alternate and elementary proof that the wakeup time, that is, the expected time for every particle to be activated, is &Theta(log n). Additionally, we give an explicit distributional equation for the wakeup time as a mixture of geometric random variables