Elitist Jerks Execute effects and the use of cooldowns

#1

[top]Introduction

Since the introduction execute-like mechanics such as hammer of wrath and molten fuy there has been a long-standing assumption that the optimal time to use bloodlust and other dps cooldowns is during the period at which dps is at its greatest; namely below 35% health. This article will show that this is incorrect and that, infact, there is no noticeable difference in the total fight length regardless of when bloodlust or trinkets are used. When there is a difference it may favour early or late cooldowns, depending on the circumstances.

The major consideration of this discussion is: When is the best time to use bloodlust? As bloodlust is the only raid-wide fixed duration dps cooldown, it enforces a raid-wide synchronization of dps cooldowns.

Many of the posts linked herein are from the benefactors bar forums, non-members will not be able to see the original posts. These links are prefaced with Benefactors Bar

[top]Summary

The correct time to use dps cooldowns, when there are no other extraneous factors, is at that point when the raid can leverage the most fixed-duration cooldowns simultaneously. Execute is not such a cooldown as its duration is not fixed. This will normally mean using cooldowns early in a fight when it is easier to synchronise across the entire raid.

The exception to the above is when a boss fight is such that a specific phase needs to be completed in the shortest possible time (a la C'thun, M'uru/Entropius, RoS... etc.).

Early use of DPS cooldowns can cause problems for classes with aggro-reduction abilities on a long cooldown. One solution is to ensure at least one of your rogues/hunters keeps their aggro-redirection abilities for during the burn period when their damage, and hence transferred aggro, will be maximized. Alternately, save the cooldowns for 85-80%.

[top]TL;DR

(1) Use cooldowns simultaneously.
(2) Obeying (1), use cooldowns early to maximize the number of people and cooldowns available,
(3) If one particular phase needs to be burnt through in the shortest possible time, use cooldowns during that phase, still obeying (1).

[top]Results from Simcraft

The following results were found from SimCraft (simulationcraft - Google Code), the files are attached the end of this article.

Class/SpecLate BLEarly BLDifference
Druid_T7_00_55_165675566015
Druid_T7_00_55_16_M5616558729
Druid_T7_58_00_13_SF57685774-6
Druid_T7_58_00_13_W546354621
Hunter_T7_06_14_5161116288-177
Hunter_T7_15_51_0555135611-98
Hunter_T7_52_12_0753235425-102
Mage_T7_00_53_1859805987-7
Mage_T7_18_00_535657564116
Mage_T7_20_51_006261623922
Mage_T7_53_18_006010598822
Mage_T7_57_03_116139610138
Priest_T7_14_00_5756135624-11
Priest_T7_38_2828212834-13
Rogue_T7_07_51_135373535419
Rogue_T7_08_20_4363546391-37
Rogue_T7_15_51_055446541630
Rogue_T7_23_05_4364536491-38
Rogue_T7_51_13_075657564116
Shaman_T7_16_55_0057425772-30
Shaman_T7_57_14_0054525466-14
Warlock_T7_00_31_4054525468-16
Warlock_T7_00_41_3058655893-28
Warlock_T7_00_55_16560856071
Warlock_T7_02_13_565720570911
Warlock_T7_55_00_1667256623102
Warlock_T7_55_00_16_CoE6422635567
Warlock_T7_55_00_16_DG7496739898

Av. Raid DPS5775.55778.8-3.3

The Biggest winners/Losers are Warlocks and Hunters, even mages see a net change of less than 50dps between early and late BL. Note, warriors, paladins and deathknights are not included. This Simcraft calculation shows that there is only a 0.056% difference in average raid dps (and therefore fight duration) between using bloodlust early and using it late; this supports the hypothesis that there is negligible difference in the fight length when BL is used below 35%.

[top]The Mathematics of Execute

[top]Assumptions

• Cooldowns provides the same dps boost to the raid as a percentage whenever it is used
Most of the models consider a raid with one bloodlust, and some number of other trinkets/cooldowns.

[top]Proof for multiplicative trinkets

Benefactors Bar
 Originally Posted by Wraithlin $X=DPS$ $B= \text{BL}$ $T= \text{Trinkets}$ $E= \text{Execute Bonus}$ $A= \text{Time Above } 35\%$ $U= \text{Time Under } 35\%$ 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: It does not matter when you pop blood lust and pop trinkets (Above or below 35%), if they are used together.

Benefactors Bar
 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).

[top]Proof that trinkets should be stacked with BL

Benefactors Bar
 Originally Posted by Ichal $T_1= \frac{H}{xAB}$ Unlinked: $T_2= \frac{H_1}{xA} + \frac{H_2}{xB}$ $T_2 - T_1= \frac{BH_1 + AH_2 - H}{xAB} > 0=$ 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 '='?). Additive buffs: $T_1= \frac{H}{A(x+B)}$ $T_2= \frac{H_1}{Ax} + \frac{H_2}{x+B}$ $T_2 - T_1= \frac{x(H_1 + AH_2 -H) + BH_1}{Ax(x+B)} > 0=$ So again T_2 > T_1 for the same reasons.
Because the above is quite general, it applies to all cooldowns, and not just bloodlust/trinket. It is optimal to use all your cooldowns at the same time.

[top]Discussion

Though the results are counter-intuitive there is a (relatively) simple explanation for them. Unlike other dps enhancing effects, execute scales with the mobs HP, and so can be modelled as a flat reduction of the mobs HP; if below 35% every point of damage effectively becomes X points, then the monster's effective hp are scaled by 1/X. If we apply execute as a HP change, not a dps change, it becomes obvious that it does not matter when we BL.

This point was first raised by Shalas:
Benefactors Bar: http://elitistjerks.com/f30/t11107-i...9/#post1089860

[top]Further Examples

Some further examples for those who doubt math:

Benefactors Bar
Originally Posted by Shalas
Vastly simplified example: boss has 800 HP, RDPS is 2, Bloodlust+Trinkets+etc. and low health are both +100% RDPS, low health boost kicks in at 50%.

Bloodlust at beginning:
 Seconds Damage Done 0-39 160 40-159 240 160-260 400
Total fight length of 260 seconds.

Bloodlust at 50%:
 Seconds Damage Done 0-199 400 200-239 320 240-259 80
Total fight length of 260 seconds.
Benefactors Bar
 Originally Posted by Mearis Here are some numerical simulations varying the time when trinkets are used/heroisms are used: For reference: boss_HP=300000; base_boss_HP=boss_HP; base_DPS=1000; execute_DPS=1.2; bloodlust_increase=1.3; bloodlust_duration=30; trinket_increase=100; trinket_duration=15; Trinket Used From: 0 until: 15 Bloodlust Used From: 239 until: 269 Total Fight Length: 280 Time from 1:80: 239 Time from 80:100: 41 Trinket Used From: 241 until: 256 Bloodlust Used From: 241 until: 271 Total Fight Length: 279 Time from 1:80: 241 Time from 80:100: 38 Trinket Used From: 231 until: 246 Bloodlust Used From: 0 until: 30 Total Fight Length: 280 Time from 1:80: 231 Time from 80:100: 49 Trinket Used From: 0 until: 15 Bloodlust Used From: 0 until: 30 Total Fight Length: 279 Time from 1:80: 229 Time from 80:100: 50 Making the numbers slightly 'patchworky': boss_HP=20000000; base_boss_HP=boss_HP; base_DPS=80000; execute_DPS=1.2; bloodlust_increase=1.3; bloodlust_duration=30; trinket_increase=200*18; trinket_duration=15; Trinket Used From: 0 until: 15 Bloodlust Used From: 200 until: 230 Total Fight Length: 232 Time from 1:80: 200 Time from 80:100: 32 Trinket Used From: 201 until: 216 Bloodlust Used From: 201 until: 231 Total Fight Length: 232 Time from 1:80: 201 Time from 80:100: 31 Trinket Used From: 191 until: 206 Bloodlust Used From: 0 until: 30 Total Fight Length: 232 Time from 1:80: 191 Time from 80:100: 41 Trinket Used From: 0 until: 15 Bloodlust Used From: 0 until: 30 Total Fight Length: 232 Time from 1:80: 190 Time from 80:100: 42

[top]Exceptions

This model makes a number of assumptions, the largest of which is that multiplicative effects (such and bloodlust) are symmetric above and blow 35%, when this does not hold then the their use will within execute range can change the total fight length. However, it does not mean that it is always the correct to use them during the last 35%:
 Originally Posted by MatsT Joe the Wizard output 100 dps. When the boss drops below 50%, his damage increases to 200 dps due to cool talents. Greg the Lizard Lord can also output 100 dps, but unfortunately he has no talents that increases his damage at low boss health. When Greg pops his bloodlust, Joe's dps is doubled. Greg does not benefit from bloodlust at all. In conclusion. Above 50%: Joe: 100 dps without bloodlust, 200 dps with. Greg: 100 dps without bloodlust, 100 dps with. Below 50%: Joe: 200 dps without bloodlust, 400 dps with. Greg: 100 dps without bloodlust, 100 dps with. Now consider the two cases of popping bloodlust early and late in the fight. Let's set boss hp to 100k Early bloodlust: 40 seconds of 200+100dps, meaning 12k damage. 190 seconds of 100+100dps, meaning 38k damage. 167 seconds of 200+100dps, meaning 50k damage Total: boss dies after 397 seconds. Late bloodlust: 250 seconds of 100+100dps, meaning 50k damage. 40 seconds of 400+100dps, meaning 20k damage. 100 seconds of 200+100 dps, meaning 30k damage. Total: boss dies after 390 seconds.
 Originally Posted by Wraithlin To reverse your example A = Above X%, Below X%, (with BL) J = 100, 200 (100, 200) G= 100, 100 (200, 200) Joe has an execute talent but does not scale with haste. Greg scales with haste, but has no execute talent. 100k HP, BL early 40*(200+100) = 12k 190*(100+100)= 34k 167*(100+200) = 50K Total time = 397 100k, BL late 250*(100+100) = 50k 40*(200+200) = 16k 113*(200+100) = 34k Total = 403s
What matters is the total percentage change in raid dps when cooldowns are used. It is not always clear which part of the fight is optimum for cooldowns: some classes classes will scale better with haste below 35% than above it (eg mages, warriors, warlocks); some will scale worse below 35% than above it (eg hunters, paladins), and some will not change noticeably (eg rogues, druids, deathknights). Overall the differences are very small, and we can consider them insignificant for all real systems.

 Originally Posted by Nevets_69 I did up an excel spreadsheet based on his scenarios to really take a closer look. Here's the summary:  Boss Health 600000 Early Bloodlust Late Bloodlust Early Late Early DPS time Late DPS time Early DPS time Late DPS time Scenar No BL w/ BL No BL w/ BL BL Time Non BL t BL Time Non BL t Total BL Time Non BL t BL Time Non BL t Total 1 2500 3000 2500 3000 40 72 0 120 232 0 120 40 72 232 2 2500 3250 3750 4875 40 68 0 80 188 0 120 40 28 188 3 2500 3010 2550 3075 40 71.84 0 117.6470588 229.4870588 0 120 40 69.41176471 229.4117647 4 2500 3100 3000 3750 40 70.4 0 100 210.4 0 120 40 50 210 5 2500 3240 3700 4800 40 68.16 0 81.08108108 189.2410811 0 120 40 29.18918919 189.1891892 Here are the differences:  Time Diff % 1 0 0 2 0 0 3 0.075294118 0.008203782 4 0.4 0.047573739 5 0.051891892 0.006856203 So in the most dramatic case, it's still only a 0.05% difference.

[top]No Bloodlust in the raid

There is a small advantage to splitting trinkets when the raid has no bloodlust at all. In this situation, those without an execute multiplier on damage should trinket early, and those with such a bonus should trinket late; again, the difference is marginal. This only applies to a fight length shorter than trinket CD, otherwise one could double trinket early and late.

 Originally Posted by MatsT Super trinket gives 100dps for 10s, Mage does 200sp, rogue does 100dps, MFury is +10% dps, 20k boss health. Rogue trinket early, mage trinket late: 10s @ (100+100) + 200 = 400 = 4k total 51.67s @ 100 + 200 = 300 = 15.5k total 10s @ 100 + (200+100)*1.1 = 430 = 4.3k total 19.4 @ 100 + 200*1.1 = 320 = 6.2k total Total time = 91.07s Both trinket late 65s @ 100 + 200 = 300 = 19.5k total 10s @ (100+100) + (200+100)*1.1 = 530 = 5.3k total 16.25s @ 100 + 200*1.1 = 320 = 5.2k total Total time = 91.25s

[top]Acknowledgements

This discussion was first raised by Binkenstein in the benefactors Bar, who placed it in his blog:
- Benefactors Bar: http://elitistjerks.com/f30/t11107-i...8/#post1088118
- Bloodlust & the "Execute" Talents - Elitist Jerks

The mathematics of the starting models were developed by Wraithlin and Ichal.
- Benefactors Bar: http://elitistjerks.com/f30/t11107-i...0/#post1091180

The assumption of symmetric bloodlust effect was noted by Zakalwe.

Examples were calculated by Shalas, Wraithlin, Songster, Mearis, Neveats, MatsT and others.
Benefactors Bar: http://elitistjerks.com/f30/t11107-i...9/#post1089860
Benefactors Bar: http://elitistjerks.com/f30/t12376-r...8/#post1139701
[Math] Bloodlust/Heroism timing

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Attached Files
 Raid_T7_Late.zip (35.2 KB, 121 views) Raid_T7_Early.zip (35.4 KB, 62 views)

 Contributors: Wraithlin Created by Wraithlin, 03/14/09 at 6:52 AM Last edited by Wraithlin, 05/12/09 at 7:02 AM 2 Comments , 21660 Views
 12/09/09, 10:07 AM #2 Fatoe Glass Joe   Fatoe Blood Elf Paladin   Twisting Nether (EU) Wonderful post, although there are a few things I would like to add. I think that there are two diffrent kind of cooldowns: haste and damage cooldowns.The diffrence between them is that it is useful to stack haste + damage cooldowns at the same time, or damage + damage cooldowns. Although, stacking haste + haste is not worth it, according to my calculations/ Lets take for example a frost mage who is spamming frostbolt. Now assume that he is totally naked and is only wearing [Scale of Fates]. That means that his only haste abilities are Icy Veins and [Scale of Fates]: Normal : 3 sec cast --> 13.33 casts in 40 seconds Trinket : % spell haste = (432) / (32.79) = 13.2 New Casting Time = (3 sec) / (1 + ((13,2%)/(100))) = 2.65 sec cast --> (20 sec)/(2.65 sec) = 7.55 casts in 20 seconds Icy Veins : New Casting Time = (3 sec) / (1 + ((20%)/(100))) = 2.5 sec cast --> (20 sec)/(2.5 sec) = 8 casts in 20 seconds Trinket + Icy Veins : New Casting Time = (3 sec) / (1 + ((13,2% + 20%)/(100))) = 2.25 sec cast --> 8.88 casts in 20 seconds In other a 40 sec test: Normal (40 sec) = 13.33 casts Trinket (20 sec) + Icy Veins (20 sec) = 7.55 casts + 8 casts = 15.55 casts Icy Veins + Trinket (20 sec) + Normal (20 sec) 8.88 casts + ((13.33)/(2)) casts = 15.545 casts Benefit: 0.05 seconds NOTE: benefit will increase by amount of haste you already have (by gear or bufs) (personally I have a ~4% damage boost by not stacking trinket + Icy Veins) Also, I think there is another reason why using bloodlust early is better than using bloostlust when a boss has lower than 35% health: By using it early, it will decrease the amount of time that you can't use any execute effects. In other words: if you use it early, you can start earlier using execute effect against the boss. Which will result into a quicker death (I am sorry but I havent tested this yet, will come soon hopefully.)
 12/09/09, 12:46 PM #3 Erdluf Great Tiger   Erdluf Night Elf Druid   Echo Isles Fatoe, You need to be careful when doing math for stacked Haste effects. Effects that increase HR are additive with each other. Most other affects (and I assume that Icy Veins is in that category) are multiplicative. Changing your 3/(1+.132+.2) = 2.25 sec cast to 3/(1+.132)/(1+.2) = 2.21 sec cast would reverse your conclusion. Spellcasters whose rotation has an instant-cast (or a 1.5s cast) spell, have a soft haste cap at just 50% haste, which is very easy to reach, particularly with Bloodlust. Depending on your gear and complete rotation, you might have a reason to avoid stacking haste cooldowns.

 Elitist Jerks Execute effects and the use of cooldowns