In an effort to mathematically optimize our evil-vanquishing performance, I and some fellow WoW players are grappling with the following...and I've gotten to the limits of my scant statistical skills. Here's the problem:
When you hit a bad guy with the Seal of Vengeance, there's a probability (P) that the target is bathed in holy flames; they get an effect put on them called Holy Vengeance  which burns them for the next fifteen seconds.
If you hit them while Holy Vengeance  is on, there's the same probability that the effect will be applied again. When that happens it 'stacks', turning into Holy Vengeance , and the 15-second countdown starts all over again. The highest stack possible is Holy Vengeance ; if you get another application when the stack is already at 5, you reset the 15-second timer but the stack doesn't get any bigger.
If you ever go 15 seconds without another application, the entire stack disappears and you have to start all over again.
How often you hit the target is based on the speed of the weapon you're using, usually called WS, and is a value somewhere between 1.5 and 2.7 seconds. (1.8 is very common, if you want to do examples.) P depends on a bunch of things but is in the neighborhood of 0.5.
So - given all this and assuming a fight of infinite length, how can we predict the mean stack size?
Bonus questions: given all this and assuming a fight of length t with the stack starting at zero, how can we predict the mean stack size?
I've gotten plausible answers to this using Monte Carlo techniques, but we'd like to generate a probability distribution function so we can really pick it apart.