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#1 Tifi

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Posted 05 April 2009 - 01:15 PM

How much damage/DPS am I going to lose if I keep MB off cooldown for x seconds/for y% of the fight?
How much DPS am I going to gain from adding DP to a rotation?
If two priests A and B have VT uptime percentages tA and tB, how will their DPS differ because of that?
If I have to choose between two nukes, how much DPS will I gain from each choice?

An easy way to answer them all is with the concept of effective damage per cooldown, or eDPCD. This concept is not limited to Priests, I assume it can be applied to all casters (classes with filler+nuke mechanics). I haven't seen this way of looking at things anywhere, hence this post.

Before we get started I'd like to give a few definitions.
Acronym|Meaning|Dimension|Definition
DPE|Damage per execute|damage|Average damage dealt when using the spell one time.
ET|Execute time|time|Maximum of cast time and the global cooldown after haste.
CD|Cooldown|time|Smallest possible time from one execution to another. A DoT's "cooldown" would be its duration, MB's "cooldown" is ET(MB)+5.5s.
DPET|Damage per execute time|damage/time|DPE/ET.
DPCD|Damage per cooldown|damage/time|DPE/CD

By filler I mean the spell with the highest DPET, but no cooldown. A nuke is a spell with a cooldown, and a higher DPET than the filler.

Now exactly how much DPS do we gain from using a nuke? There are two scenarios we have to look at: One where we only cast the filler, and another where we use the nuke on cooldown, and the filler while the nuke is cooling down. While casting the filler we deal DPET(filler).
In the second scenario we use the nuke on cooldown. Here's one way to look at it: While casting the nuke, we deal DPET(nuke), and in the remaining time we deal DPET(filler), so our average DPS is

$\frac{DPET(nuke)\cdot ET(nuke) + DPET(filler)\cdot (CD(nuke)-ET(nuke))}{CD(nuke)}.$

But I'd like to look at it in another way. By casting the nuke we made a choice. We also could have cast the filler, but by replacing it with the nuke, we deal more damage. We gain damage from using the nuke over not using the nuke, but we can only do it once for ET(nuke) every CD(nuke). Our average DPS is

$DPET(filler) + \frac{DPET(nuke)\cdot ET(nuke) - DPET(filler)\cdot ET(nuke)}{CD(nuke)} = DPET(filler) + \frac{DPE(nuke) - DPET(filler) \cdot ET(nuke)}{CD(nuke)}$.

This motivates the definition of the effective damage per cooldown of our nuke,

$eDPCD(nuke) := \frac{DPE(nuke) - DPET(filler)\cdot ET(nuke)}{CD(nuke)}.$

Here's a DPS diagram for MB+MF to illustrate both points of view. Yellow means nuke, grey means filler. Note that the yellow shapes have the same area.

Now we can say: By keeping the nuke perfectly on cooldown, we gained eDPCD(nuke) over not using it at all. And we can easily answer the previous questions.

How much damage am I going to lose if I keep MB off cooldown for x seconds? Answer: x*eDPCD(MB).
How much DPS am I going to lose if I keep MB off cooldown for y% of the fight? Answer: y/100*eDPCD(MB).
How much DPS am I going to gain from adding DP to a rotation? Answer: At most eDPCD(DP).
If two priests A and B have VT uptime percentages tA and tB, how will their DPS differ because of that? Answer: (tA-tB)/100*eDPCD(VT).
If I have to choose between two nukes X and Y, how much DPS will I gain from each choice? Answer: eDPCD(X) and eDPCD(Y).

But enough with the algebra: What's the eDPCD for Shadow Priest nukes? The exact values of eDPCD of course depend on a lot of factors like gear, buffs, talents, trinket procs, partial resists, etc, so there's not one answer. What I can show you is the eDPCD for my own values, assuming zero latency. I happen to have very good gear as of 3.09, and after all modifiers I have on average 3072 spell power, 36.97% crit, and 28.77% haste.

My DPET(MF) is 3,927.38, and DPS(SW:P) is 605.70, so the base DPS is 4,533.07.
[TABLE]Nuke|eDPCD|eDPCD/base
DP|508.73|11.22%
MB|273.31| 5.18%
VT|667.27|14.72%
SW:D| 14.47| 0.32%
SW:D (4xT7)| 45.21 | 1.00%
SW:D (Glyph)| 54.04 | 1.19%
SW:D (both)| 87.86| 1.94%[/TABLE]
While the absolute values are highly dependent on the input, the relative values are probably pretty much the same for everyone.

What I found both interesting and disturbing is how small these values actually are in comparison to the base DPS. By using MB for example, the maximum gain is 5.2% of the base DPS if we somehow manage to keep it on cooldown all the time. If we manage to raise our MB uptime from 80% to 90%, all we get is is a measly 0.5%.
We can also use it to determine the expected value of the theoretical max. DPS by simply adding up all the eDPCD values. For me this means that even if I somehow manage to have perfect uptime on all the nukes (not possible in practice), the best I can possibly do is 5,958.16 DPS on average.

#2 Elimbras

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Posted 05 April 2009 - 01:48 PM

I agree on the analysis, which leads to good estimation of gain.

In practice, you may want to add the fact that spells have a discrete cast time.
Therefore, if you have a cd of 5s on a spell, and all you spell have 6s cast time, you want be able to have the perfect rotation leading to your expected gain eDPCD.

This also means that using a nuke with a positive little eDPCD, you may screw your rotation, and therefore in fact loose dps. In short, you modelize here perfect rotation, and managing perfect rotation is not always possible, meaning that you have to consider the whole rotation for exact results.

#3 Tifi

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Posted 05 April 2009 - 03:14 PM

Indeed, care has to be taken due to the discrete nature of spells. We can't cast MF2.4, it's either MF2 or MF3.

Another effect is what I call cooldown collisions, i.e. when two or more than nukes are off cooldown at the same time. The most interesting example is when deciding between MB and VT at 0% haste. When looking at the eDPCD alone, one might think that it's better to use VT>MB, but simcraft disagrees. The priority list VT>MB>MF induces the following rotation:
VT, MB,MF,MF, MB,MF,MF
At this point of time, the situation is equal to the start: Both MB and VT are off cooldown again (colliding), and we have to choose again. This means that every 15s, we have to keep MB off cooldown for 1.5s. We lose 1.5/15*eDPCD(MB).

If, however, we use MB>VT>MF/SWD, we get a slightly longer rotation between MB/VT collisions:
MB,VT,SW:D,MF, MB,MF,MF,
MB,SW:D,VT,MF, MB,MF,MF,
MB,MF,VT,SW:D, MB,MF,MF,
MB,MF,SW:D,VT, MB,MF,MF,
MB,MF,MF
Due to the collision we lose 1.5s*eDPCD(VT), which is more than what we lost before. But we only lose it once every ~60s, and 1.5/60*eDPCD(VT) is less than 1.5/15*eDPCD(MB). This is the main reason why simcraft bots like it better to use priorities with MB>VT.

Now this still doesn't mean that MB>VT should always be used. It should only be used if we're certain that the fight is going to last longer than 60s. Also, it's generally better to cast VT,DP,MB if that's possible. The overall gain is small though. Using the above values, 1.5/15*eDPCD(MB) = 27 DPS, and 1.5/60*eDPCD(VT) = 19 DPS, so we gain at most 8 DPS by using MB>VT.

#4 Ledha

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Posted 05 April 2009 - 06:02 PM

What if we cast SWD?
Shall it be casted instead of MF or not?
Especially in heroism when MF becomes faster and faster
What do you think about it?
In tbc we had to cast it but now...it does not provide a great dps improvement when you have a quite good spell haste on MF, isn't it?

#5 tedv

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Posted 06 April 2009 - 01:39 PM

As a side note, the Latex formatting options will make your equations much more readable.

To briefly summarize for the TL;DR crowd, this is a table of how much additional damage over mind flay each cast provides:

[TABLE]Nuke|eDPCD|eDPCD/base
DP|592.60|11.22%
MB|273.31| 5.18%
VT|777.28|14.72%
SW:D| 16.85| 0.32%
SW:D (4xT7)| 45.21 | 1.00%
SW:D (Glyph)| 54.04 | 1.19%
SW:D (both)| 87.86| 1.94%[/TABLE]

In other words, spending a GCD on Devouring Plague and then 22.5 seconds on Mind Flay is 11% more damage than casting Mind Flay for 24 seconds. Casting Mind Blast for 1.5 seconds and Mind Flay for 5.5 seconds is 5.18% more damage than casting Mind Flay for 7 seconds. I haven't double-checked these numbers but some of them look suspiciously low, especially Mind Blast.

What this table is most useful for is telling us what spells are actually worth casting. By looking at the ratio between the additional mana cost incurred and the additional damage provided, you can figure out what spells to cut in low mana situations. In particular, Shadow Word: Death is a waste of time. It's technically an incremental upgrade at a huge mana cost, and I believe the argument about accidentally messing up your cycle holds weight here. It's still worth casting Death when you have a cooldown up in 1 to 1.5 seconds and don't want to spend a full 2 seconds on a Mind Flay, however, so don't take it off your bar.

This table does NOT define the casting priority queue, however. To look at that, you want an estimate of the deferment cost for each ability. When you could cast both Devouring Plague and Mind Blast, one of them is going to be delayed by 1.5 seconds. In the case of Mind Blast, you lose 1.5/7 = 21% of a Mind Blast as all mind blasts will be cast 1.5 seconds later. You lose 1.5 / 24 = 6% of a Devouring Plague. It's easy to see that 21% of average Mind Blast damage is higher than 6% of Devouring Plague damage, so Mind Blast should be cast first. In general, the shorter the cooldown and the higher the total spell damage, the sooner a spell should be cast. Vampiric Touch is the only spell that beats Mind Blast, despite having a longer cooldown, because of its absolutely enormous damage coefficient.

#6 Elimbras

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Posted 06 April 2009 - 02:16 PM

r.
This table does NOT define the casting priority queue, however. To look at that, you want an estimate of the deferment cost for each ability. When you could cast both Devouring Plague and Mind Blast, one of them is going to be delayed by 1.5 seconds. In the case of Mind Blast, you lose 1.5/7 = 21% of a Mind Blast as all mind blasts will be cast 1.5 seconds later. You lose 1.5 / 24 = 6% of a Devouring Plague. It's easy to see that 21% of average Mind Blast damage is higher than 6% of Devouring Plague damage, so Mind Blast should be cast first. In general, the shorter the cooldown and the higher the total spell damage, the sooner a spell should be cast. Vampiric Touch is the only spell that beats Mind Blast, despite having a longer cooldown, because of its absolutely enormous damage coefficient.

Actually, you also want to include the "phase-lock" probability when doing such a choice.
I mean here that the question is also when your next "synchronisation" problem will occur. This time will be obviously different for the two choices you have, and it will be non-negligible in the final dps, because it can divide the number of "problems" by a significant positive number.

#7 tedv

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Posted 06 April 2009 - 02:32 PM

Actually, you also want to include the "phase-lock" probability when doing such a choice.
I mean here that the question is also when your next "synchronisation" problem will occur. This time will be obviously different for the two choices you have, and it will be non-negligible in the final dps, because it can divide the number of "problems" by a significant positive number.

Honestly I haven't seen a lot of phase problems. Vampiric Touch lasts 15 seconds, Mind Blast lasts 7, and Plague lasts 24, so you don't get a continual deferment problem if you choose to cast one before the other just based on durations. Including Shadow Word: Death is a problem because it conflicts with every other Mind Blast, but Death isn't that great anyway. (Still technically a DPS increase though.) In my experience, if you have a conflict in one frame, you won't have a conflict between that pair of spells for at least half a minute.

Of course, the amount of haste you have for Mind Blast does affect this a little. Phase issues really come up more with bosses that have sporadic delays in them, like Thaddius where you lose 2 seconds traveling between piles. But you can't really theorycraft the cast rotation given a 50% chance of needing to move in a few seconds.

#8 Falim

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Posted 06 April 2009 - 03:22 PM

SampleOutput - simulationcraft - Google Code

They have a damage per execute time there.

#9 Elimbras

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Posted 06 April 2009 - 03:28 PM

In fact, the best case is when all spells have a common denominator. Then they want collide anymore during the fight, except at the beginning, when they all are free to cast. That's a real phase lock, but it's good ;-)

In practice, the collision can happens, due to the discrete problems (for computer scientist, this is called the knapsack problem, and it's NP-Hard). Tifi gave an example for MB and VT, when no haste is used.

#10 tedv

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Posted 06 April 2009 - 03:42 PM

In fact, the best case is when all spells have a common denominator. Then they want collide anymore during the fight, except at the beginning, when they all are free to cast. That's a real phase lock, but it's good ;-)

In practice, the collision can happens, due to the discrete problems (for computer scientist, this is called the knapsack problem, and it's NP-Hard). Tifi gave an example for MB and VT, when no haste is used.

My point was more that this is a problem worth ignoring because even if you think hard about it, it's not clear you'll actually produce a better result.

#11 eap

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Posted 07 April 2009 - 09:54 PM

I understand the original intent of Cooldown damage, but it seems dedicated to fights like patchwerk. In a fight where movment is more often (maybe Grobbulus, Thaddius, or Sapphiron) DOT timing and uptime become more important then what DD spell to cast during a CD.

I have no experience on the PTR, but I'm hoping for a nice variety of fights in Uldaar. So the question to myself is, do I change rotations for each boss to completely maximize damage per boss, or find a nice baseline "rotation" for maximizing damage for the duration of any boss.

#12 jdgaynor

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Posted 08 April 2009 - 05:47 AM

Phase issues really come up more with bosses that have sporadic delays in them, like Thaddius where you lose 2 seconds traveling between piles. But you can't really theorycraft the cast rotation given a 50% chance of needing to move in a few seconds.

Actually, you could Theorycraft fights like these by using expected value calculations. In the Thaddius example, you have a 50% chance of getting to stand still for two seconds and mind flay (because casting anything else would be ill-advised), or a 50% chance of having to SWD and maybe refresh VE when you have to switch sides. Take the expected damage of either case, and multiply by the probability and number of expected side switches; viola, instant theorycraft.

I don't think it is particularly elegant to model fights based on the stand and nuke paradigm, but that's what the whole theorycrafting community has accepted.

#13 Elimbras

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Posted 08 April 2009 - 08:13 AM

One of the nice point of this analysis is that it can be adapted to both cases.
For still fights, you just use priorities as given on the table.
For fights where mechanism may prevent you from casting spells in the future, you just use the highest damage per cast time priority.

I do agree that maximizing dps on Patchwerk is not equivalent to real dps for most situation.
The trouble is that theory needs to be tractable. For any boss with movement, you need to have a model of this movement (or capacities).
These models will be as particular for the considered boss as Patchwerk is. And Patchwerk is at least easier to analyze ;-)
That's the same problem for healing theorycraft, even worse. Healing is not about hps, it's about keeping people alive. But the trouble of the second metric is that it's lot dependant on the fight, especially on damage patterns. That's why people talk about max hps, which is at least "nearly" independant of damage and fights. In practice, all healers know that the theoretical best rotation is not to be used in fights ;-)

#14 Tifi

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Posted 08 April 2009 - 08:53 AM

This table does NOT define the casting priority queue, however. To look at that, you want an estimate of the deferment cost for each ability. When you could cast both Devouring Plague and Mind Blast, one of them is going to be delayed by 1.5 seconds. In the case of Mind Blast, you lose 1.5/7 = 21% of a Mind Blast as all mind blasts will be cast 1.5 seconds later. You lose 1.5 / 24 = 6% of a Devouring Plague. It's easy to see that 21% of average Mind Blast damage is higher than 6% of Devouring Plague damage, so Mind Blast should be cast first.

What you're saying is basically: The number we should look at is ET*DPCD, not ET*eDPCD. But what you're not taking into account is that while casting a nuke we don't cast the filler.

To clarify let me construct an example. We'll look at the Bard class, which works similar to a Fury Warrior: Autohit+two nukes, but no filler. Let's call the autohit MF', and the nukes MB' and DP'. Both nukes have cast times ET(DP'):=ET(MB'):=ET(DP), and cooldowns CD(DP'):=CD(DP) and CD(MB'):=CD(MB).
We set the autohit DPS to the DPS of MF, DPET(MF'):=DPET(MF).
We set DPE(MB') := DPE(MB) - ET(MB')*DPET(MF'), and DPE(DP') := DPE(DP) - ET(DP')*DPET(MF'). In other words we subtract the damage done by the autohit. Here's a picture of what we did here:

Let's say the Priest uses MB at exactly the same times as the Bard uses MB', and DP whenever the Bard uses DP'. It should be clear that they deal exactly the same DPS. Let's review what you said before: We should look at ET*DPCD. That is obviously the case for the Bard. Now you may have noted that DPCD(MB') = eDPCD(MB), and DPCD(DP') = eDPCD(DP), which means: If the Bard should look at DPCD, the Priest should look at eDPCD.

In other words: For a class with nuke+filler mechanics, eDPCD has the same meaning as DPCD has for a class without a filler.

#15 Elimbras

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Posted 08 April 2009 - 10:43 AM

I guess you didn't understand Tedv's point, even if you're right.

Here is what Tedv is saying.
Your eDPCD analysis is fine : it just tells you how much each spells gains you, if you maximizes it's usage, compared to using a filler in replacement.
But if you have the choice between two nukes, you have to take into account their cd also.

More preciseley : you have DP and MB free at the same time. You can either cast DP first, and loose 1.5s of MB. Or cast MB first, and loose 1.5s of DP.
If you can use only one spell, then DP is the spell to go, because it has higher eDPCD.

But if you know that you'll cast both spells, you need to consider the loss of each scenario.
If you delay DP, you loose 1.5s of DP, that is to say 1.5s/(CD(DP) * eDPCD(DP) ~ 0.06 * eDPCD(DP).
If you delay MB, you loose 1.5s of MB, that is to say 1.5s/CD(MB) * eDPCD(MB) ~ 0.2 * eDPCD(MB).
Basically, delaying a long timer spell is less problematic, because you loose only a small portion of it's uptime. For short cd spells, you loose a high portion of it.

It's not contradictory of your analysis. Using DP is more important that MB : for 1cd every 24s only, we get a significant increase of DPS. MB is lower gain despite it's high dps, especially because it needs to be cast often, and therefore use more MF time.
But delaying DP of 1 gcd is not problematic either. We loose only 6% of uptime, whereas delaying MB is more costly : you loose a significant part of the MB potential.
That sound reasonable.

EDIT :
But I think that Tedv's point is not valid.
What you want to consider is not your current dps, but your total dps (or total damage) for the fight.
And whilst your current dps loss is lower when using MB first, you have it for a full 24s cycle duration, whereas your MB loss, whilst higher, is for a shorter cycle duration. In other word, you need to multiply by the cycle duration, and your are back to 100% of eDPCB.

But in fact, there is no reason to think about cycles:
what you loose is 1.5s of the "dps" gain from the nuke you gain.
The total damage you loose is then 1.5 * (dps(nuke_delayed) - dps(filler)), independantly of your cd.
Whereas you need to cast this spell every 8s, 15s or 24s is not relevant. That's 1.5s where this nuke is not effective, period
To realize this, best way is probably to imagine nukes doing a filler equivalent direct damage, then additionnal damage being dots, for the duration of their cd (and ajusting their strenght so that the total damage is the same). Then what you want to maximise is their uptime, with prioritie to higher ticking dots.

#16 Tifi

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Posted 08 April 2009 - 11:51 AM

The difference is what exactly "loss" means. Tedv said that we lose DPCD(MB)*ET(DP) if we cast DP first.

I was trying to show that we can't look at two nukes in isolation, because casting a nuke always means that we can't cast the filler. The decision between two nukes is not "Should I cast DP or MB?" but "Should I replace MF by DP or by MB?" And the reason is indeed that we have to look at the entire fight. We can't only look at the 1.5s while we cast one of those nukes.

#17 tedv

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Posted 09 April 2009 - 06:34 PM

I think it's pretty much accepted that all nukes deal more damage than Mind Flay, even Shadow Word: Death, and that's what your analysis is (correctly) saying. Death is certainly the lowest damage increase over Mind Flay, meaning that it's often not worth using because of the extra mana burden and/or cooldown intersections. But every other shadow spell should be used in place of Mind Flay at all times. No one is debating otherwise.

The point I'm making is that once you know you that Mind Flay is your lowest priority spell, how do you determine the priorities of other spells? And for this you need to analyze the fraction of the spell lost through deferment when there's a cooldown intersection between two spells. Note that all the numbers I posted assumed an infinitely long fight (or at least 24 seconds long, as that's our longest cooldown). Clearly it's not worth casting Devouring Plague or Vampiric Touch when there's 6 seconds left in the fight. That's a situation where Mind Flay is in fact more damage. But theorycrafting the tail end of the fight isn't that important in the grand scheme of things.

#18 Tifi

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Posted 10 April 2009 - 08:55 AM

The point I'm making is that once you know you that Mind Flay is your lowest priority spell, how do you determine the priorities of other spells? And for this you need to analyze the fraction of the spell lost through deferment when there's a cooldown intersection between two spells.

Yes, I agree. What we disagree on is what this deferment cost is. You simply hold on to your claim that it's DPCD(deferred nuke)*ET(cast nuke), but without giving any reasons. In an earlier post I replied that this is only correct for classes with autohit+nuke mechanics. What I did is basically convert the Priest into an equivalent class with autohit+nuke mechanics (the "Bard" is really just a Priest in disguise) in order to prove that for a class with filler+nuke mechanics, the deferment cost is eDPCD(deferred nuke)*ET(cast nuke).

BTW the source of those numerical values is this google spreadsheet. I just updated it for 3.1 (added Imp. DP, critting DoT ticks, and lowered Imp. Scorch to 5%).

DPET(MF): 3,779.41
DPS(SW:P): 625.23
base DPS: 4,404.64
[TABLE]Nuke|eDPCD|eDPCD/base
DP|730.63|16.95%
DP (T8 2pc)|836.10|19.70%
MB|225.78| 5.13%
VT|710.11|16.12%
SW:D| 13.46| 0.31%
SW:D (Glyph)| 51.49 | 1.17%[/TABLE]
Theoretical max DPS: 6,138.48.

I should also mention that the value for the glyphed SW:D simply adds the 10% bonus, i.e. it's only valid if the target is below 35% health. It doesn't mean that the Glyph grants 51-13=38 DPS.
I assume that the T8 set bonus stacks additively with

#19 tedv

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Posted 10 April 2009 - 02:13 PM

Yes, I agree. What we disagree on is what this deferment cost is. You simply hold on to your claim that it's DPCD(deferred nuke)*ET(cast nuke), but without giving any reasons.

No, that's not what I'm claiming. I will try to explain it in your notation. But first...

For those who are confused by the random string of letters, let me give some simple definitions that might help you follow what Tifi and I are discussing.

DPCD: Damage per cooldown. This is the total damage a spell does.
eDPCD: Incremental damage. This is the amount of additional damage a spell does relative to alternate option of casting the filler spell (Mind Flay).
ET: The cast time of the spell. Generally 1.5 seconds.
CD: The cooldown of the spell. For DoTs, this is their duration. Note that for Mind Blast, this is 5.5 + GCD = 7 seconds. For spammable spells, this is equal to the cast time.

I claim when you want to cast two spells at the same time, A and B, and choose to defer B, the deferment cost is:

$TotalDamage_B \cdot \frac{CastTime_A}{Cooldown_B}$

In your acronym notation, this would be:

$DPCD( \cdot \frac{ET(A)}{CD(}$

I did give a simple explanation, however. When you spend 1.5 seconds casting spell A, then all future cooldowns of B occur 1.5 seconds later. That means you lose a fraction of B's damage equal to 1.5 / Cooldown. It's intuitively obvious enough that I didn't think a larger explanation was warranted.

In an earlier post I replied that this is only correct for classes with autohit+nuke mechanics.

For the record, that equation in that earlier quote (the one you thought I suggested but did not) is never correct, not even for autohit+nuke mechanics.

#20 Tifi

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Posted 10 April 2009 - 03:44 PM

Woops, apparently I failed to define what *I* meant by DPCD. You just used DPCD where I would use DPE.

I use it like in the DrDamage addon:
$DPCD(x) := DPE(x)/CD(x)$,
so you take the damage done by one execution of the spell, and divide by its cooldown. So it's damage per time, and the default unit is damage per second.

Similarly, the dimension(?) of eDPCD is damage per time. Plugging in we get
$eDPCD(x) = \frac{DPE(nuke) - DPET(filler)\cdot ET(nuke)}{CD(nuke)} = DPCD(nuke) - \frac{DPET(filler)\cdot ET(nuke)}{CD(nuke)},$
so eDPCD is like damage per cooldown, but we factor out the DPS lost by not casting the filler while casting the nuke.

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