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The Bloodlust Irrelevance Theorem and other results

August 10, 2011

1. The source of the myth and the grain of truth in it

A general rule for damage-increasing trinkets, effects, procs and cooldowns is that they should be used together and at the same time as haste effects to get an effect which is greater than the sum of its parts. The proof is straightforward: suppose that one DPS does D DPS when not using any effect. Suppose that he has a haste trinket (which increases DPS by some fraction H of current DPS) and a damage-increasing cooldown (which increases DPS by some fraction C of current DPS), and both of them have a duration T (suppose a 0-sec GCD for the sake of simplicity). The separate benefit of both of them is DT(H + C) , while the joint effect is DT((1 + H)(1 + C) - 1) = D T ( H + C + H C ) . We can clearly see that the joint strategy outperforms the alternative usage by DTHC .

2. The Bloodlust Irrelevance Theorem

With the same logic it seems optimal to cast Bloodlust when damage-increasing mechanics (Kill, Shot, SW:D, some talents…) go into effect (the ‘Execute range’). However, as savvy people know, that’s wrong. This result is somewhat counterintuitive at first but we can easily prove it: let H be the boss health, D the raid DPS without BL and Executes, B the fraction of current DPS which Bloodlust adds, R the ‘Execute range’, X the fractional increase in DPS added by Execute-ish effects and T the duration of BL/Heroism. Also, let’s add reasonable constraints, such as DT(1+ B)(1 + X) < RH (that is, you can’t kill the boss before BL ends if you pop it once Execute-type effects go into effect) and its symmetric DT(1+ B) < (1 - R)H .

The final DPS of the raid equals  the health of the boss, divided by the kill time. Therefore, we can focus on kill time alone (less time equals more DPS). If we cast BL out of the Execute range, the kill time equals:

K_1 = T + \displaystyle \frac{(1 - R)H - DT(1 + B)}{D} + \frac{RH}{D(1 + X)}

which we can simplify as:

K_1 = T + \displaystyle \frac{(1 - R)H}{D} - T(1+B) + \frac{RH}{D(1 + X)}

while if we cast BL in the Execute range, our kill time is:

K_2 = \displaystyle \frac{(1 - R)H}{D} + T + \frac{RH - (1 + X)(1 + B)DT}{ D(1-X)}

We can simplify this as:

K_2 = \displaystyle \frac{(1 - R)H}{D} + T + \frac{RH}{ D(1-X)} - (1 + B)T

It’s clear that both expressions are the same. The kill time is the same. Why?

The difference between Execute and similar mechanics and other effects is that the former do not have a time limit but a _damage_ limit, while Volcanic Potions and Arcane Power do. Hence, if you use the latter at the same time as BL, you get more buffed casts and therefore more benefit from the Potion, while if you use BL in Molten Fury range you get exactly the same number of Fireballs, you just get them faster. So, under this simple model, it doesn’t matter if you cast BL on sub-35%, sub-30% or sub-20% targets

3. The Bloodlust Relevance Theorem

From a more realistic perspective, it can be even detrimental to cast Bloodlust in the Execute range because some sub-X% spells, such as Kill Shot, have a fixed cooldown and thus do not scale with haste (therefore, they add a fixed amount of DPS, let’s call them W). Under such conditions, BL shortens the sub-X% phase and thus reduces the benefit you’d get from these spells. Let’s prove it. If we cast BL out of the Execute range, the kill time equals:

K_1 = T + \displaystyle \frac{(1 - R)H - DT(1 + B)}{D} + \frac{RH}{D + W}

which we can simplify as:

K_1 = T + \displaystyle \frac{(1 - R)H}{D} - T(1+B) + \frac{RH}{D + W}

while if we cast BL in the Execute range, our kill time is:

K_2 = \displaystyle \frac{(1 - R)H}{D} + T + \frac{RH - ((1 + B)D + W)T}{ D + W}

We can simplify this as:

K_2 = \displaystyle \frac{(1 - R)H}{D} + T + \frac{RH}{D + W} - T - \frac{BDT}{D+W}

Substracting K_1 - K_2 we get:

K_1 - K_2 = - BT + \displaystyle \frac{BDT}{D+W} = \left (\frac{D}{D+W} - 1 \right) BT = \frac{-W}{D+W} BT

which is a negative quantity since W, D, B and T are all positive numbers. Therefore, that means that K_1 - K_2 < 0 or that K_1 < K_2 : that is, the kill time is smaller if you cast BL before sub-X% range as we reasoned earlier. Therefore, from a pure Patchwerk DPS perspective it’s optimal to cast BL _before_ getting to the sub-X% range: probably at the beginning of the fight to get more benefit from early procs and cooldowns.

Of course, in practice what determines the BL moment is the presence of hard enrage or soft enrage phases, phases with improved damage, ‘burn phases’, hard-to-heal phases or standstill phases (most casters* perform better when immobile). But it’s still important to dispel the myth of “BL at Execute range”, because it could mean the difference between victory and defeat on “DPS race” bosses where there is some choice regarding BL.

* That is, all casters save for pre-4.2 Cata moonkins on Maloriak Dark Phase, Halfus Whelps and some other AoE situations.

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From → General raiding

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