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Heroism timing

Warning: Theorycraft inside.

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Postby Coriel » Wed Mar 04, 2009 10:36 am

MomysLidlMonsta wrote:I think i've found the error in his math:

Potera wrote:Important point is that: Time = Health / (Damage/sec). In other words, Time = Health * Sec/Dmg

which gets used throughout his calculations. But Time = sec and Damage = Health, so he actually says Time = Health/(Health/Time) which simplyfies to Time = Time which doesn't mean anything.


Uhh...units on both sides of the equation always have to be the same. This is the very definition of equality. (This is actually a good check on your math, if the units don't cancel out and become the same on both sides, you've gone wrong somewhere.)

Second, the basic work is not mine, but was done by a few theorycrafters at EJ. I linked to a post showing algebra. I just put in an example with real numbers because sometimes I find using real numbers in a non-intuitive result makes the result more solid for me.

As to why it works, let's take a look at what exactly a percentage multiplier of limited duration (as we assume Bloodlust to be) does.

Let the boss have H health.
Let the base dps be x health/sec.
Let the multiplier be A.
Let the duration of the multiplier be D.
Let the Time to Kill without the multiplier be T1.
Let the Time to Kill with the multiplier be T2.

T1 = H/x
T2 = ((H - DxA) / x) + D
T2 = H/x - DA + D

Now the difference in time is T1 - T2.

T1 - T2 = H/x - (H/x - DA + D)
T1 - T2 = H/x - H/x + DA - D
T1 - T2 = DA - D
T1 - T2 = D(A - 1)

We can see that the change in time does not depend on the boss health or raid dps. It depends solely on the duration and magnitude of the multiplier.

Now does this result make sense? I think it does.

If the multiplier is 0 (no damage can be done), the time increase is D seconds long. Which makes total sense. If you can't do any damage for 30 seconds, obviously it will take you 30 more seconds than normal to kill the boss.

If the multiplier is 1, the time increase is 0. Again, makes sense, as a multiplier of 1 just means you're doing normal damage.

If the multiplier is 2 (double damage), the time is shortened by D seconds. If D is 40 sec, it's exactly like you are doing an extra 40 sec worth of damage.

But again, the key point is that only the magnitude and duration of the multiplier matter.

This leads into why you want to stack fixed duration multipliers. Let's say you have multiplier M1 with duration D and magnitude A1, and multiplier M2 with duration D and magnitude A2. (Same duration to make the math simpler.)

Stacked:

C1 = T1 - T2 = D(A1A2 - 1)

Separate:

C2 = T1 - T2 = D(A1 - 1) + D(A2 - 1)

C1 > C2 when
D(A1A2 - 1) > D(A1 - 1) + D(A2 - 1)
A1A2 - 1 > A1 -1 + A2 - 1
A1A2 > A1 + A2 - 1
A1A2 - A1 - A2 + 1 > 0
(A1 -1)(A2 - 1) > 0

A1 > 1 and A2 > 1
OR
A1 < 1 and A2 < 1

The math is similar, but more complicated for two multipliers with different durations. The conclusion is the same. Beneficial multipliers of fixed duration should be stacked.
Last edited by Coriel on Wed Mar 04, 2009 10:53 am, edited 1 time in total.
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Postby trellian » Wed Mar 04, 2009 10:40 am

Mordinm wrote:Shadow bolts increase in dps just like drain soul. A 30% haste buff will increases shadow bolt dps by the same factor as it will increase drain soul dps. Channeled spells do not gain more from haste then spells with cast times.


You are absolutely right here.

However shadow bolt and immolate are no longer used after 25%, and drain soul is never used before 25%.
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Postby trellian » Wed Mar 04, 2009 10:44 am

Dorvan wrote:The only thing that matter is raid DPS during heroism and raid DPS outside of heroism, whether the raid is 1 person or 25.


no, there are 4 different dps states.

- raid DPS during heroism at 100%
- raid DPS outside of heroism at 100%
- raid DPS during heroism below execute%
- raid DPS outisde heroism below execute%
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Postby theckhd » Wed Mar 04, 2009 10:48 am

cordelia wrote:-PS> For those who need to see the math written out, attempt the whole math proof again assuming that the threshold N denotes a static time into the encounter when the dps increases/decreases by value Y. This will show a notable improvement in time to kill when popping heroism during max dps periods.

Felt like working this out myself. Assuming all of the same variables, and deaths at time T that cause the dps of the raid to drop from X to XY:

Hero late (after T):
Damage before time T: TX
Damage after time T: H-TX

Time before time T... is T
Time after time T: W+(H-TX-XYZW)/XY
Total Time: T + H/XY - T/Y-W(Z-1)

Hero early (before T):
Damage before time T: XZW + (T-W)X
Damage after time T: H-[XZW + (T-W)X]

Time before time T... is still T
Time after time T: (H-[XZW + (T-W)X])/XY
Total time: T + H/XY - ZW/Y -T/Y + W/Y = T + H/XY - T/Y - (Z-1)W/Y

Note the bolded terms are identical. The difference between the two is the last terms,
-(W/Y)(Z-1)(Y-1)
which for Z & Y >1 (i.e. a DPS boost) is negative. Thus Hero at 100% would kill the boss faster by (Z-1)(Y-1)W/Y seconds.

Of course, if it's a DPS nerf (Z<1), it actually takes longer, because you're reducing more people's contributions. And this scales as expected - as more people die, Y gets smaller, increasing the time differential.
Last edited by theckhd on Thu Mar 05, 2009 2:18 pm, edited 1 time in total.
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Postby theckhd » Wed Mar 04, 2009 10:55 am

trellian wrote:no, there are 4 different dps states.

- raid DPS during heroism at 100%
- raid DPS outside of heroism at 100%
- raid DPS during heroism below execute%
- raid DPS outisde heroism below execute%

He was referring to the example he gave in the post which you quoted, which has nothing to do with execute range. He was demonstrating that regardless of whether each player in the raid gets the same benefit from Heroism, it can still be represented as (raid DPS) * (Heroism Multiplier).

i.e.

P1*Z1 + P2*Z2 + P3*Z3 + ... can still be represented as P*Z, where P is P1+P2+P3+.... , and all of the Zi's can be different. Z would be some sort of weighted average of the Zi's in this case.
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Postby trellian » Wed Mar 04, 2009 10:58 am

[quote=theckhd]...[/quote]

You math fails in assuming DPS gains a static increase from heroism. For some classes their DPS will benefit heroism more at <execute% then at >execute% simply because they change their spells and abilities <execute%. Some spells/abilities gain more from haste then others.

See Elsie's comment about the difference between Rets' DPS and warlocks' DPS haste scaling <execute%.


Edit: changed Z in DPS ;)
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Postby theckhd » Wed Mar 04, 2009 11:10 am

trellian wrote:You math fails in assuming Z gains a static increase from heroism. For some classes their Z will benefit heroism more at <execute% then at >execute%.

See Elsie's comment about the difference between Rets' Z and warlocks' Z.

That's not the point, and nowhere in my post did I suggest that what you just said is incorrect.

Dorvan was specifically replying to this:
Elsie wrote:I don't like this assumption. If X is personal DPS, then X*Y is bears no significance for every X unless Y is equally variable. If you think of X as raid DPS, then X is a summation of x1, x2, ..., xn. So you're trying to multiply a summation by a constant Y. This means the affect of bloodlust is evenly distributed among all x when we know this not to be true.

His response and my math both show that this is demonstrably false. Even if each player gets a different benefit from Bloodlust, you can still represent SUM(Xi*Zi) as X*Z, it's just that the actual value of Z is slightly more complicated to calculate.

This does not in any way suggest that the second part of Elsie's post is incorrect. If the Zi's (and thus Z) change above and below a certain health threshold, then of course it's not fair to assume Z is invariant between the two situations.

In short, my math does not "fail," because it's not even attempting to address the question you're asking.
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Postby trellian » Wed Mar 04, 2009 11:12 am

Ah, then I appologize for making a wrong assumption about what you where trying to explain.
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Postby holythief » Wed Mar 04, 2009 11:15 am

the difference between a class Z and another classes Z at execute range does not have any bearing as the calculated value is for the Raid's DPS over all, single players increase does not count

So if at over execute range the raid puts out 1000 (X) DPS
and at under execute range puts out 1200 (X*Y) DPS

in this case X is 1000 and Y is 1.2

So it doesn't matter if you have all DPS being ret pallies or all DPS being warlocks the coefficient by which the DPS of the raid changes as you enter execute range is the same in either scenario before you apply the heroism buff.

I do like the analogy with the video tape, the only place this would matter is if you are trying for the 4 horse man achievement kill when you need them to die in 15 sec of each other so you want the fast forward triggered right at the end

Edit: Fixed confusion between Z and Y
Last edited by holythief on Wed Mar 04, 2009 1:55 pm, edited 1 time in total.
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Postby theckhd » Wed Mar 04, 2009 11:36 am

holythief wrote:the difference between a class Z and another classes Z at execute range does not have any bearing as the calculated value is for the Raid's DPS over all, single players increase does not count

So if at over execute range the raid puts out 1000 (X) DPS
and at under execute range puts out 1200 (X*Z) DPS

in this case X is 1000 and Z is 1.2

So it doesn't matter if you have all DPS being ret pallies or all DPS being warlocks the coefficient by which the DPS of the raid changes as you enter execute range is the same in either scenario before you apply the heroism buff.

First of all, Z was being used for Heroism's bonus. Y was the boost due to execute range. So in your example, all the Z's should be Y's, and Y is 1.2. And you would be correct if you said Y is averaged over the entire group, and thus any particular class's Yi doesn't matter.

However, Bloodlust is different. Because Heroism benefits the classes differently, Z can have different values above and below execute range. All of the calculations, however, have assumed that Z is invariant. Since this is probably not the case, it means they're not a complete (nor accurate) model of what's happening.
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Postby holythief » Wed Mar 04, 2009 2:06 pm

Corrected the post above with the right letters

So what you're saying is that using the math in the OP post

Time to kill boss when Heroism was used outside of the execute range is
= H/X - HN/X + W + HN/XY - Z1W

Time to kill boss when Heroism was used inside of the execute range is
= H/X - HN/X + W + HN/XY - Z2W

Z1 is the overall Damage improvement the raid gains when heroism is used outside execute range
Z2 is the overall Damage improvement the raid gains when heroism is used inside the execute range

Assuming that Z2>Z1

In which case for raid groups for which Z1/Z2 approaches 1 using heroism inside the execute range has only marginal benefit
But as Z1/Z2 becomes smaller heroism should be triggered in the execute range.

Which all adds up the fact that heroism should be triggered in the execute range for every group.
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Postby theckhd » Wed Mar 04, 2009 2:27 pm

Yep, I believe that's what Elsie was getting at.

If Z2>Z1, then using it during execute range is preferable.
If Z2<Z1, then it's better to use it early.
If they're approximately equal, it doesn't matter at all, which is the case the math in the OP is modeling.

And of course, all of this assumes that there's not a more important segment of the fight to use it on (Maexx enrage, pre-Vesperon OS3D, etc).

Patchwerk would be a good example, since the fight doesn't change significantly at any point (5% enrage can basically be ignored, since even without heroism finishing him off is faster than the duration of Bubblewall).

Unfortunately, I'm not sure what the most realistic estimate of Z1/Z2 would be. I would suspect that Z2 is at least slightly larger for most classes, but I could imagine a scenario where it's smaller (you swap a different spell into your usual rotation that does more DPS, but doesn't scale much with haste). I'd guess that Ret paladins are fairly close to the Z1=Z2 range, since my understanding is that they're mostly cooldown/gcd-limited and HoWrath doesn't scale with haste at all, as far as I remember.
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Postby Dorvan » Wed Mar 04, 2009 2:31 pm

holythief wrote:Which all adds up the fact that heroism should be triggered in the execute range for every group.


*Unless you have DPS dying early in a way that not pegged to a percentage of the bosses health. Also, if there are particularly dangerous phases, you should pop heroism during them. Whether it's worth waiting until execute range depends on the tradeoff between (Z2-Z1) and these other factors. My guess is that for most progression kills you're better off popping it early, as either:

-- you'll lose some DPS early, so you want to pop Heroism early to get a greater benefit from it
or
-- you've gotten to a point where you're reliably solid on the fight anyway, so it really doesn't matter.

The only times it really seems like it'd be worthwhile to save it for execute is if:

-- you've completely mastered the fight mechanics but are having problems with an enrage timer.
or
-- you're going for a DPS record


Of course, a rough idea of the ballpark values of Z2 and Z1 would be very helpful for the conversation. It seems like that there are DPS models for all of the classes, so that calculation shouldn't be too bad (if a little time consuming to download/set up all the existing spreadsheets) to rough in for a given raid set up.
Last edited by Dorvan on Wed Mar 04, 2009 2:35 pm, edited 1 time in total.
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Postby Markoh » Wed Mar 04, 2009 2:32 pm

holythief wrote:Which all adds up the fact that heroism should be triggered in the execute range for every group.


THere have been a couple posts in different places on this recently. Basically the problem with lusting at the end of a fight is that it is almost impossible for everyone to have cooldowns ready to use without wasting them (aka sitting on cooldowns b/c your saving for lust). I don't have a link off hand but basically what the math needs to come down to is the dps gain by lusting in execute range vs lusting 10-15 seconds in the fight with everyone blowing full cooldowns.

Bloodlusting at the end of a fight is only beneficial to certain classes/specs and getting a lust with the ability to blow full cooldowns is beneficial to every spec/class.

There were a few caveats, which were basically assuming that there was not a particular phase that needs a burn or the fight was shorter than 5 min, where anything longer would defiantly benefit from an early lust due to the ability to work in two lusts.
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Postby holythief » Wed Mar 04, 2009 4:14 pm

Lets assume a straight fight with no phases that need to be burned through and a enrage cap of 5 minutes.

So adding into consideration cool-down and trinkets while we're at it and assuming everyone saves cooldowns/trinkets for the particular phase they are popping heroism we'd be looking at this:
for outside execute range
Time to boss death = H/XCT - HN/XCT + W + HN/XY - Z1W
and similar for execute range
Time to boss death = H/X - HN/X + W + HN/XCTY - Z2W

Where C is the damage increase coefficient for popping Cool Downs
and T is the damage coefficient for popping Trinkets.

So going by the conclusion in the OP's quote that both these times should be equal those two equations should solve to Z1=Z2 i don't have time to try do it right now, also someone might want to double check the math on it.
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