28 April 2010

The non-linearity of the ICC zone-buff

Today (or yesterday, depending on which continent your WoW servers are) the ICC zone-buff was increased from 10% to 15%. That's an increase of your DPS, HPS and HP by 4.5%. I think that the main benefit of this buff is the DPS component, not only does it affect more people, but 4.5% more HP or HPS doesn't drastically alter your ability to survive poorly handled boss-abilities. Sure, there may be borderline cases where those few percent are enough to save you, but in general that Malleable Goo or Vengeful Shade will kill you just as hard today as it did last week.

So why then does this not apply to DPS in the same way? Knocking a few percent of the duration of a 5 minute fight is not a major difference. Except for two things. The first is the enrage timer. On heroic modes, enrage timers on some bosses are a real threat. Especially since we typically have someone die here and there, bosses like Festergut and Blood Queen can end up being close calls, especially on earlier kills.

The time-based add-spawn mechanic
The second effect is far more interesting. Quite a few boss-encounters involve some type of spawns or secondary targets that need to be killed. These spawns are almost always on a fixed timer. In ICC we have Marrowgar (bonespikes), Deathwhisper (cultists), Saurfang (blood beasts), Putricide (oozes) and the Lich King (val'kyr, vile spirits). I've skipped over Gunship and Valithria because the "boss-kill" mechanics are very different from the regular boss-encounter.

Now, with this type of add-spawning, you will have to divert your DPS to the adds before going back to the boss. And this is where the fun kicks in: Higher DPS means the adds go down more quickly and you spend more time on the boss with your higher DPS. The effect of a DPS-increase on your effective boss-DPS (which is what determines the length of the fight) is amplified by the fact that there are periodic adds that need to be killed.

Enough talk, let there be math!
Assume a boss with B HP. Let R denote the raid DPS on the boss and f R the raid DPS on the adds (In case of single-target adds 0 < f < 1, as you lose DPS by swapping to an unbuffed target and having to ramp up your rotation. For AoE adds, f can be larger than 1.). We have adds with a total of A HP spawning every Ta seconds.

To kill the adds, we need A / (f R) seconds, which leaves us with Tb = Ta - A / (f R) seconds to DPS the boss per wave of adds (Pro-tip: If Tb < 0 then you're doing it wrong). Divide by the interval between two waves, Ta and we obtain the fraction of time we can spend on the boss:
1 - A / (f R Ta)
Multiply by raid DPS on the boss to find the effective raid DPS on the boss:
R - R A / (f R Ta) = R - A / f Ta

Not a big surprise, eh? If you have adds with a total of 300K HP spawn every 30 seconds, your effective DPS on the boss is lowered by 10K, divided by the DPS-correction-factor caused by the difference in DPS between boss and adds. Could've worked that one out without the whole derivation...

Buffing your DPS
So now we get 5% more DPS from a new ICC buff (Yeah, I know it's actually 4.5% for this weeks change as I said before, but I'll take 5% for simplicity). How does that change our effective boss DPS?
1.05 R - A / f Ta
So the term that adds a positive contribution to effective boss DPS is increased, while the term that adds a negative contribution remains unchanged. The net effect will be a greater positive contribution on effective boss DPS.

An example
Lady Deathwhisper has 6.7M HP on 10-heroic. In phase 2, she will summon a single add every 40 seconds. The add has 176K HP (averaged between a Fanatic and an Adherent). Lets assume that raid DPS is 30K on the boss and 20K on the adds (So f = 0.667). Your effective boss DPS will be 16.8K. Now, add 5% raid DPS (31.5K on boss, 21K on adds). Your effective boss DPS will increase to 18.3K, an increase of about 9%.

This effect becomes more noticable, the more time you spend on adds. The most notable example was pre-30% nerf M'uru (which I never had the pleasure of fighting, unfortunately), where the tuning was so tight that there was very little time to DPS the boss. Small increases in raid DPS had large effects on the time it took to burn the boss down.

Conclusion and take-home message
We've seen a quick analysis of how the commonly used add-spawn mechanic can amplify the effect a DPS increase has on your raids performance. The effect becomes stronger the longer you take to kill the adds. Obviously, there are all kinds of other benefits to having the adds down more quickly (preventing the Val'kyr from dropping your buddies off the ledge in the Lich King encounter is one such benefit), but from a mathematical point of view, the boss DPS effect is the most interesting.

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