Elitist Jerks Bloodlust & the "Execute" Talents

# Bloodlust & the "Execute" Talents

Posted 02/10/09 at 10:28 PM by Binkenstein
Updated 03/05/09 at 9:49 PM by Binkenstein
Last week I asked a question in the Benefactor's Bar to see if anyone had done any math/testing of the common assumption that Bloodlust/Heroism should be used at 35%, or about 55 seconds before the fight ends. Not surprisingly, a few people said I was silly for asking such a question, as obviously stacking the extra 12% damage on Fire/FFB Mages & Affliction Locks with bloodlust was superior than not doing so.

However, two mathematically inclined individuals took up the challenge, namely Ichal and Wraithlin (I have never been good at complex algebra). Thusly, I am reposting their work here so that everyone can see it, rather than keeping it tucked away on a private forum. Please note that unless you are a forum Benefactor, you will not be able to see the original posts, nor the 100-150 post discussion around it.

The TL;DR version for those of you that don't want to read a large mathy post.
Quote:
Originally Posted by Wraithlin
1) If there is no other factor, use BL early to maximize the number of people and cooldowns available.
2) If one particular phase needs to be burnt through in the shortest possible time, use BL then, and match cooldowns to BL.
And now, on to the math!

Quote:
Originally Posted by Ichal

x: dps above 35%
y: dps below 35% (so y > x)
H: Boss health
A: Multiplicative modifier from popping cooldowns (example: Bloodlust/Heroism)
B: Additive modifier from popping cooldowns (example: Clickable trinket with +damage or AP)
T: Time to kill without cooldowns
T_1: Time to kill with early cooldowns
T_2: Time to kill with late cooldowns

I'm assuming that cooldowns will always work for the full duration, which I've set to 40 sec.

Using no cooldowns you get:

$T= \frac{0.65H}{x} + \frac{0.35H}{y}$

Assuming purely multiplicative cooldowns:

$T_1= 40 + \frac{0.65H - 40Ax}{x} + \frac{0.35H}{y}$

$T_2= 40 + \frac{0.65H}{x} + \frac{0.35H - 40Ay}{y}$

With a bit of elimination, it's clear that in this case $T_1= T_2$.

Assuming both types of coolodowns:

$T_1= 40 + \frac{0.65H - 40A(x+B)}{x} + \frac{0.35H}{y}$

$T_2= 40 + \frac{0.65H}{x} + \frac{0.35H - 40A(y+B)}{y}$

Unless I messed up (not unlikely) this eliminates to: $T_1 < T_2$ (i.e. it's better to pop cooldowns early).
Quote:
Originally Posted by Wraithlin
Same result, different method:
$X=DPS$

$B= \text{BL}$

$T= \text{Trinkets}$

$E= \text{Execute Bonus}$

$A= \text{Time Above } 35\%$

$U= \text{Time Under } 35\%$

Now in both scenarios, the damage dealt above 35% must be equal to 65% of the boss health, and the damage dealt below 35% must be equal 35% of the boss health.

$40XBT + (A_{1}-40)X= XA_{2} = 0.65H, \qquad XE= XE(U_{2}-40) + 40BTXE =0.35H$

Rearranging:
$A_{1}= \frac{(XA_{2} - 40XBT)+ 40X}{X}, \qquad U_{1} = \frac{XE(U_{2}-40) + 40BTXE}{XE}$

So:
$U_{1}+A_{1}=$

$=\frac{1}{XE}\left( E(XA_{2} - 40XBT)+ 40XE + XE(U_{2}-40) + 40BTXE\right)$

$=\frac{1}{XE}\left( XEA_{2} + XEU_{2}- 40BXTE+ 40BTXE + 40XE -40XE\right)$

$=\frac{1}{XE}\left( XE(A_{2} + U_{2})\right)$

$=U_{2}+A_{2}$

This would mean that, for multiplicative effects:
1) It does not matter when you pop blood lust (Above or below 35%)
2) It also does not matter when you pop trinkets (Above or below 35%)
3) It MAY not matter whether or not you line up trinkets with bloodlust.

Interesting results all round.
In response to point #3, Ichal goes even further.
Quote:
Originally Posted by Ichal
It does in general. Multiplicative buffs, linked:

$T_1= \frac{H}{xAB}$

$T_2= \frac{H_1}{xA} + \frac{H_2}{xB}$

$T_2 - T_1 = \frac{BH_1 + AH_2 - H}{xAB} > 0$

As x > 0, A,B > 1 and H = H_1 + H_2. Which means T_2>T_1. (Why is the forum ending that with a '='?).

$T_1= \frac{H}{A(x+B)}$

$T_2= \frac{H_1}{Ax} + \frac{H_2}{x+B}$

$T_2 - T_1 = \frac{x(H_1 + AH_2 -H) + BH_1}{Ax(x+B)} > 0$

So again T_2 > T_1 for the same reasons.

At one point I figured something like Molten Fury could be considered as a multiplicative cooldown that just isn't popped by the player, which would mean that these results conflict with the earlier. But thinking about that gave me a headache. Have at it.
Wraithlin then summarizes the findings, explaining why we are seeing these results.
Quote:
Originally Posted by Wraithlin
It comes back to something Shalas said earlier.
The length of the <35% buff is dynamic, the more it increases your DPS, the shorter it lasts; trinket effect durations do not have this negative feed-back loop.

Thinking about it a bit harder, we can model the execute dps bonus as a percentage reduction in the bosses health. We cannot model any other dps bonus in this way because they do not scale with the monsters health. This is what makes it behave counter-intuitively when stacked with other cooldowns.
Posted in Shaman , Raiding

 Interesting. Posted 02/10/09 at 11:05 PM by Grenadine Updated 02/10/09 at 11:16 PM by Grenadine
 The 40 second cooldown was for simplification only. Since the no cooldown comparisons are equal, this allows us to see that using any cooldown early, in conjunction with Bloodlust, is better than doing the same thing at the 35% mark. Posted 02/10/09 at 11:17 PM by Binkenstein
 Correct me if I've misunderstood something here, but doesn't all this math assume that the multiplicative cooldowns have an equal relative effect on DPS both above and below 35%? I don't think that's a valid assumption: Warlocks have an execute mechanic in drain soul which drastically changes their rotation and causes their DPS to scale much better with haste. Posted 02/13/09 at 9:06 AM by Zakalwe
 It doesn't. If you re-read it you will see that the gain from the execute abilities/talents is actually less if you use bloodlust at that time. This is the negative feedback loop mentioned. Posted 02/13/09 at 8:43 PM by Binkenstein
 You seem to be talking about the results. I'm saying their method of arriving at those results is wrong - or rather, that their assumptions are wrong. How would the math account for the potential difference in effect on DPS by bloodlust? Ichal calls this effect A and simply multiplies it by uptime and DPS (40 and x/y). Wraithlin calls it B, and, again, simply multiplies it by uptime and DPS (40 and X). To accurately model the situation I'm talking about you would have to multiply with a haste-scaling coefficient as well, with a different value above and below the execute limit. In fact, once you realize that some execute abilities change how DPS scales with haste, it's easy to refute their conclusions: Consider an imaginary class that gets zero benefit from haste, except for an execute ability that allows them to get 100% benefit from haste when the boss is below 35% health. Now imagine a raid stacked fully with this class. For simplicity, let's imagine all of these DPSers are completely without haste gear or haste raid buffs, and that their execute ability doesn't change their DPS under those conditions. With 100k raid DPS and 10M mob health, the fight takes 100 seconds. Add a bloodlust at the start, there is no change and the fight still lasts 100 seconds. Add a bloodlust at 35% (after 65 seconds), and the fight lasts 65 + ( 3.5M / (100k*1.3) ) = 91.9 seconds. Posted 02/14/09 at 2:22 AM by Zakalwe Updated 02/14/09 at 2:44 AM by Zakalwe
 Except that we know that an early bloodlust, with cooldowns, is a faster kill than a late bloodlust. The question is whether the gain for Affliction is greater than the gain from the early bloodlust + cooldowns combination, and for that you'll really need some sort of model/simulation system that can compare things. So unless you have excess Affliction locks, I don't think it's anything to worry about. Posted 02/14/09 at 6:03 PM by Binkenstein
 I really wasn't disputing the conclusion that it may be just as good to pop BL early. And I certainly agree that we can't really answer the question in a generalized way if we include the haste scaling variable I mentioned. I just thought I'd point out that all the math in your blog entry was based on an incorrect assumption. It might not be that unlikely to find a situation where your conclusions are wrong - my guild has had patchwerk kills where affliction locks provided nearly 25% of the DPS, and that's with just 3 locks in the raid. I guess the real answer would be to plug your particular raid makeup into Simulationcraft - you'd just have to wait for it to support all classes first. Posted 02/16/09 at 2:56 AM by Zakalwe
 It seems like the assumtion in the first calculation that certain buffs are additive under heroism is skewing the result. The only buffs additive under heroism are other haste effects. Posted 02/16/09 at 8:53 PM by Jackk
 Zakalwe: Yes, it's an assumption that you get the same relative dps increase above and within the range of execute type effects. And you're right, this isn't true for warlocks. However compared to the original hand waving of "Bloodlust late because it stacks with abilities of three classes", we're down to one class, and a weaker effect at that. And if you do use late Bloodlust, it's virtually impossible to stack cooldowns across a raid. But not stacking cooldowns is a loss of dps. So don't think of the results as "the math says Bloodlust early" but rather: "The math says it's close to a wash, go for practical". Jackk: Your comment is pretty vague, but seems wrong. Is it the A(x+B) term that you think is wrong? If it had been Ax+B then each trinketed spell cast during Bloodlust would deal less damage than without Bloodlust. I don't believe this to be the case. Posted 02/17/09 at 3:45 PM by Ichal Updated 02/17/09 at 3:53 PM by Ichal
 It seems to me that the main arguments for an early bloodlust are essentially: 1) People may die before the bloodlust if we use it late and 2) People may not sync their cooldowns with a late bloodlust. and 3) A late bloodlust may be cut short by the boss dying --- I say this because it seems obvious to me that any one person doing more damage inside of execute range would increase damage over all. Execute range abilities are multiplicative. Mages, for example, specced for Molten fury, do 12% more damage for the entire sub 35%. Of course that means that their normal shots hit 35% harder as well, so it's not important to get any given cooldown in that space, right? Wrong. Let's say a trinket adds +10 damage, flat. Execute improves this to 11.2 damage, a gain of -- surprise surprise -- 12%. Assuming, first of all, that using this cooldown outside of execute range implies it can't be used inside of execute range, the choice of where to use it is simple. And that's where I feel that the math doesn't show "It's close to a wash." Stacking all your cooldowns at once is one of the best things you can do as DPS, along with maximizing uptime of said cooldowns and maximizing your DPS uptime. Therefor, it's obvious that, given a fight with no unknown variables, a late bloodlust will always be better. Wraithlin says you can model execute range as a straight reduction in boss health, and I counter with -- no, it's another cooldown. But all the reasons for an early bloodlust are quite understandable. People might die (which should be avoided to max rDPS but maybe can't be, a la vael), people might not have their cooldowns up for bloodlust (if you're being vocal about when it will happen, this should not happen) and lastly, and really the only two reasons I consider to have a grounding in reality: People might lose out on multiple cooldown uses because they saved theirs to stack with bloodlust which came at a percentage rather than a known time. AND The boss may die before bloodlust finishes, wasting potential uptime. In these last two ways, a late bloodlust can be an rDPS loss, but I put forth that in all other ways it is an rDPS gain. [Edit: Some apologies for the rather jumpy train of thought here.] Posted 02/26/09 at 6:36 PM by Inoko
 Congrats, you just made the same assumption that everyone else had been making before. Stacking all your cooldowns with a static duration is good. Stacking them with a "cooldown" that reduces its duration the higher your dps is while it's up, not so good for stacking. aka: go re-read it again. Posted 02/26/09 at 6:51 PM by Binkenstein
 No. I got that part. The higher you can push DPS inside of the execute duration cooldown, the shorter that cooldown becomes, and the less it gives. What I don't understand, and what hasn't been effectively explained here, is why that means you avoid dealing with it at all. It's still an increase. It becomes less of an increase the more you can stack during it, but it's still an increase. Posted 02/26/09 at 8:54 PM by Inoko
 If you have 16 people doing 5000 dps, and one of these is a mage, you get 15 * 5000 + 5600 sub 35%. Assuming 60 seconds under 35%, that's 4.8 million damage, with the mage doing 33600 damage. With bloodlust, we assume a 30% bonus, so it becomes 15*6500 + 7280 = 104780 dps, taking 40 seconds to do 4.2 million damage, and the remainder takes another 8 seconds. The mage deals 33600 damage here. If there's more reason to use bloodlust sub 35%, we would expect to see the mage dealing more total damage in that situation than otherwise. The reason is that they only have 80% of the 12% bonus that the non-bloodlust situation uses. Posted 02/26/09 at 10:17 PM by Binkenstein
 Okay, I understand. I did the math a few more times for a couple of situations, and while it's true that there's a minor rDPS gain in saving bloodlust, and it would be further magnified by having all cooldowns there. But, it does comes out to be, without any further cooldown stacking, mathematically a wash. So, I conclude that there are too many other risks for it to be worth it, and this has been... quite interesting, and eye opening. Thanks. Posted 02/26/09 at 11:42 PM by Inoko
 Mages aren't the only class who scale better with haste during an execute phase than outside of execute phase. TG warriors gain a much bigger benefit when bloodlust is running <20%. Above 20%, a 30% haste buff only increases our white damage and our heroic strike uptime, and below 20% causes us to execute in free GCDs and maintain our rotation. At a certain gear level (I know I've hit it with 2xBoH, FotFF, 43% crit and ~3.8k AP), when bloodlust is active below 20%, we can spam execute without missing a GCD (or with minimal loss, bloodlust + flurry + speed potion brings white swings close to 1.5secs). The DPS increase is far higher, as it lets us use our most damaging rage-dump (which you can only use below 20%). Bloodlusting early forces us to use Heroic Strike as a dump, which is far less efficient. Posted 03/03/09 at 11:28 AM by ebs2002
 Am I misreading the maths, or does it assume that you kill the boss exactly as Bloodlust fades? Posted 03/04/09 at 8:27 AM by Hythloday