With trinket procs not being assessed correctly for now, I want to modify Grim Toll for my Cat druid model to reflect its average proc benefit.
It procs 612 ArP for 10s, with a 15% proc rate and a 45s internal CD.
The question is: how to compute the long run up time of this proc?
More generally, how to compute the average long run up time of a proc a duration t, with procrate p and internal CD c?
My approach (I use Grim Toll numbers to make it more concrete). I'd appreciate some feedback!
* upper bound: if it procs right after CD expiration, it procs every 45s for 10s, for an uptime of 10s per 45s. uptime: 10/45 ≈ 22.2%.
This assumes the CD starts at the beginning of the proc, and not at the end. Is this correct?
* expected time to proc, when off CD:
let N+1 be the number of strikes needed for the trinket to proc.
(N is the number of strikes which failed to proc the trinket before the proc).
N has a geometric distribution : P[N=k] = (1p)^k * p
The expected value of N is E[N] = (1p)/p = .85/.15 ≈ 5.67
(see http://en.wikipedia.org/wiki/Geometric_distribution)
So the trinket can be expected to proc after 5.67+1=6.67 blows.
Now what is the average time between two strikes? For us cats, it's between 1s (auto attacks), and 1.5s (yellow damage), but it's affected by our haste, as well as by our miss rate. Therefore, we need to take into consideration our specific stats. I'll
ignore my miss rate on the basis that it's not significant at less than 10% (this assumption is questionable).
I could not work out a reliable way to estimate the proportion of white attacks vs yellow attacks. I'll suppose it's about 50/50. Is there any better estimate? With this estimate, the average time between two attacks is 1.25 prehaste.
Rawr tells me my toon has an attack speed of 0.755s. That's a haste factor of 0.755. The average time between two attacks posthaste is therefore 1.25*.755 ≈ 0.94s.
Finally the expected time for the trinket to proc when off CD is 6.67*.94 ≈ 6.3s.
Taking the CD into account, I expect the trinket to proc every 45+6.3 = 51.3s.
The average uptime should thus be 10s/51.3s ≈ 19.49%.
(With the decimal part probably not being significant at all.).
And I will assign an average ArP for Grim Toll of 612*19.49 = 119.
Is this correct? Is this realistic?
Note: this ignores the fact that for ArP, it's better to have a lot for a little while, than some for a long time. I don't know how to take that into account.
