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Von Kaiser
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I ran about four hours of Dragonspine testing on the 2.4 PTR, and saw several procs occuring about 20 seconds apart (from start to start, not from fade to start). I don't have any screenshots, but here's an excerpt from the combat log file for the shortest start-to-start time:

3/2 14:44:33.359 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,154,1,nil,nil,nil,nil,nil,nil
3/2 14:44:33.593 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,84,1,nil,nil,nil,nil,nil,nil
3/2 14:44:34.234 SWING_MISSED,0xF130001DF5000795,Servant of Grol,0x10a48,0x000000000058242D,Rezarel,0x511,PARRY
3/2 14:44:34.593 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,156,1,nil,nil,nil,nil,nil,nil
3/2 14:44:35.312 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,79,1,nil,nil,nil,nil,nil,nil
3/2 14:44:35.406 SPELL_AURA_APPLIED,0x0000000000000000,nil,0x80000000,0x000000000058242D,Rezarel,0x511,34775,Haste,0x1,BUFF
3/2 14:44:36.078 SWING_MISSED,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,MISS
3/2 14:44:36.421 SWING_MISSED,0xF130001DF5000795,Servant of Grol,0x10a48,0x000000000058242D,Rezarel,0x511,PARRY
3/2 14:44:36.625 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,76,1,nil,nil,nil,nil,nil,nil
3/2 14:44:37.015 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,144,1,nil,nil,nil,nil,nil,nil
3/2 14:44:38.234 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,76,1,nil,nil,nil,nil,nil,nil
3/2 14:44:38.234 SWING_MISSED,0xF130001DF5000795,Servant of Grol,0x10a48,0x000000000058242D,Rezarel,0x511,DODGE
3/2 14:44:38.546 SWING_MISSED,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,MISS
3/2 14:44:39.125 SPELL_HEAL,0x000000000058242D,Rezarel,0x511,0x000000000058242D,Rezarel,0x511,23682,Heroism,0x2,147,nil
3/2 14:44:39.625 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,171,1,nil,nil,nil,1,nil,nil
3/2 14:44:40.046 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,144,1,nil,nil,nil,nil,nil,nil
3/2 14:44:40.250 SWING_MISSED,0xF130001DF5000795,Servant of Grol,0x10a48,0x000000000058242D,Rezarel,0x511,PARRY
3/2 14:44:40.953 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,161,1,nil,nil,nil,nil,nil,nil
3/2 14:44:41.125 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,86,1,nil,nil,nil,nil,nil,nil
3/2 14:44:42.578 SWING_MISSED,0xF130001DF5000795,Servant of Grol,0x10a48,0x000000000058242D,Rezarel,0x511,DODGE
3/2 14:44:42.578 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,304,1,nil,nil,nil,1,nil,nil
3/2 14:44:42.703 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,79,1,nil,nil,nil,nil,nil,nil
3/2 14:44:44.062 SWING_MISSED,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,MISS
3/2 14:44:44.140 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,83,1,nil,nil,nil,nil,nil,nil
3/2 14:44:44.359 SWING_DAMAGE,0xF130001DF5000795,Servant of Grol,0x10a48,0x000000000058242D,Rezarel,0x511,35,1,nil,nil,nil,nil,nil,nil
3/2 14:44:45.484 SPELL_AURA_REMOVED,0x0000000000000000,nil,0x80000000,0x000000000058242D,Rezarel,0x511,34775,Haste,0x1,BUFF
3/2 14:44:45.484 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,145,1,nil,nil,nil,nil,nil,nil
3/2 14:44:45.765 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,151,1,nil,nil,nil,1,nil,nil
3/2 14:44:46.203 SPELL_HEAL,0x000000000058242D,Rezarel,0x511,0x000000000058242D,Rezarel,0x511,23682,Heroism,0x2,165,nil
3/2 14:44:46.203 SWING_DAMAGE,0xF130001DF5000795,Servant of Grol,0x10a48,0x000000000058242D,Rezarel,0x511,41,1,nil,nil,nil,nil,nil,nil
3/2 14:44:47.281 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,161,1,nil,nil,nil,nil,nil,nil
3/2 14:44:47.609 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,74,1,nil,nil,nil,nil,nil,nil
3/2 14:44:48.296 SPELL_HEAL,0x000000000058242D,Rezarel,0x511,0x000000000058242D,Rezarel,0x511,23682,Heroism,0x2,121,nil
3/2 14:44:48.296 SWING_DAMAGE,0xF130001DF5000795,Servant of Grol,0x10a48,0x000000000058242D,Rezarel,0x511,42,1,nil,nil,nil,nil,nil,nil
3/2 14:44:49.390 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,153,1,nil,nil,nil,nil,nil,nil
3/2 14:44:49.390 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,80,1,nil,nil,nil,nil,nil,nil
3/2 14:44:50.281 SWING_DAMAGE,0xF130001DF5000795,Servant of Grol,0x10a48,0x000000000058242D,Rezarel,0x511,42,1,nil,nil,nil,nil,nil,nil
3/2 14:44:50.890 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,153,1,nil,nil,nil,nil,nil,nil
3/2 14:44:51.187 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,74,1,nil,nil,nil,nil,nil,nil
3/2 14:44:52.234 SWING_DAMAGE,0xF130001DF5000795,Servant of Grol,0x10a48,0x000000000058242D,Rezarel,0x511,43,1,nil,nil,nil,nil,nil,nil
3/2 14:44:52.828 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,145,1,nil,nil,nil,nil,nil,nil
3/2 14:44:52.953 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,81,1,nil,nil,nil,nil,nil,nil
3/2 14:44:54.437 SWING_DAMAGE,0xF130001DF5000795,Servant of Grol,0x10a48,0x000000000058242D,Rezarel,0x511,40,1,nil,nil,nil,nil,nil,nil
3/2 14:44:54.546 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,147,1,nil,nil,nil,nil,nil,nil
3/2 14:44:54.843 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,82,1,nil,nil,nil,nil,nil,nil
3/2 14:44:55.421 SPELL_AURA_APPLIED,0x0000000000000000,nil,0x80000000,0x000000000058242D,Rezarel,0x511,34775,Haste,0x1,BUFF
3/2 14:44:56.140 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,142,1,nil,nil,nil,nil,nil,nil
3/2 14:44:56.343 SWING_MISSED,0xF130001DF5000795,Servant of Grol,0x10a48,0x000000000058242D,Rezarel,0x511,DODGE
3/2 14:44:56.343 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,85,1,nil,nil,nil,nil,nil,nil
3/2 14:44:57.531 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,153,1,nil,nil,nil,nil,nil,nil
3/2 14:44:57.843 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,74,1,nil,nil,nil,nil,nil,nil
3/2 14:44:58.437 SWING_MISSED,0xF130001DF5000795,Servant of Grol,0x10a48,0x000000000058242D,Rezarel,0x511,DODGE
3/2 14:44:59.062 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,173,1,nil,nil,nil,nil,nil,nil
3/2 14:44:59.375 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,82,1,nil,nil,nil,nil,nil,nil
3/2 14:45:00.265 SWING_DAMAGE,0xF130001DF5000795,Servant of Grol,0x10a48,0x000000000058242D,Rezarel,0x511,39,1,nil,nil,nil,nil,nil,nil
3/2 14:45:00.546 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,156,1,nil,nil,nil,nil,nil,nil
3/2 14:45:00.859 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,151,1,nil,nil,nil,1,nil,nil
3/2 14:45:02.031 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,140,1,nil,nil,nil,nil,nil,nil
3/2 14:45:02.296 SWING_DAMAGE,0xF130001DF5000795,Servant of Grol,0x10a48,0x000000000058242D,Rezarel,0x511,43,1,nil,nil,nil,nil,nil,nil
3/2 14:45:02.296 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,71,1,nil,nil,nil,nil,nil,nil
3/2 14:45:03.531 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,150,1,nil,nil,nil,nil,nil,nil
3/2 14:45:03.890 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,79,1,nil,nil,nil,nil,nil,nil
3/2 14:45:04.484 SWING_MISSED,0xF130001DF5000795,Servant of Grol,0x10a48,0x000000000058242D,Rezarel,0x511,PARRY
3/2 14:45:04.578 SWING_DAMAGE,0x000000000058242D,Rezarel,0x511,0xF130001DF5000795,Servant of Grol,0x10a48,148,1,nil,nil,nil,nil,nil,nil
3/2 14:45:05.390 SPELL_AURA_REMOVED,0x0000000000000000,nil,0x80000000,0x000000000058242D,Rezarel,0x511,34775,Haste,0x1,BUFF
The buff comes up at 14:44:35.406, fades at 14:44:45.484, and comes back up again at 14:44:55.421, for a start-to-start time of 20.015 seconds.
If I counted all the eligible swings (ie, all white strikes greater than 20 seconds since the last proc) and all the procs that occured, I can calculate the proc chance for an individual eligible attack, and convert that to a PPM for the trinket. This gives something pretty close to 1 PPM. So, if you were autoattacking with a single weapon and never missed, you'd get on average 10 seconds of uptime for every 20 + 60 seconds.
Here's how I model uptime for a real life scenario:
The chance per second that the MH will proc is
MH_PPS = 1 PPM / 60 seconds * (1 - %miss - %dodge).
since the proc chance adjusts for hastes to maintain a constant PPM. Same thing for OH:
OH_PPS = 1 PPM / 60 seconds * (1 - %miss - %dodge).
Specials (Bloodthirst, WW, etc... maybe Windfury too, I haven't tested) use the MH base weapon speed to calculate proc chance (I'm ignoring offhand hits from whirlwind - I haven't done any testing on that either).
Special_Proc_Chance = 1 PPM * MH_Base_Weapon_Speed / 60 seconds
Special_PPS = Special_Proc_Chance * Specials_Per_Second * (1 - %miss - %dodge)
Then all three PPS values add to give the total chance that the trinket will proc in any given second:
PPS = MH_PPS + OH_PPS + Special_PPS
The average time to proc after the internal cooldown comes up is then 1 / PPS. So, our uptime can be calculated as
Uptime = 10 / (20 + 1 / PPS)
using 10 seconds up with a 20 second internal cooldown.
I guess there's a few points in there where I made some approximations -- using a proc per second instead of by swing, and just adding the three PPS values -- but the individual proc chances are small so I think that's a valid simplification.
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