While thinking about Shadow Priest DPS, I tried to find answers to the following questions.
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



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

.

This motivates the definition of the effective damage per cooldown of our 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.

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%

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.

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.

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.

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?

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:

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.

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.

SampleOutput - simulationcraft - Google Code

They have a damage per execute time there.

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.

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.

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.

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.

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.

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.

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.

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

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%

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

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:
,
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

so eDPCD is like damage per cooldown, but we factor out the DPS lost by not casting the filler while casting the nuke.

All of that is well and good, but I still contend that when you delay casting Mind Blast by 1.5 seconds, you've given up the potential of 1.5/7 * Mind Blast Damage. In other words, damage times ratio of cooldown time lost is the correct measure for deferment penalty.

Using info from Tifi's analysis, would it be correct for me to say that I should (almost) ALWAYS start a Mind Flay cast as long as Mind Blast is on cooldown, rather than wait for Mind Blast to be ready? Considering that keeping Mind Blast perfectly on cooldown is only a 5% increase over the base DPS of SWP + Mind Flay, it would seem that even waiting a fraction of a second for Mind Blast to be ready would be a loss of DPS compared to starting a MF2 cast.

Guess that the best theoretical answer is to cast MF, and interrupt it after the "right" tick (ie. the first one following the event "MB is free").

The original poster's analysis only relates to mana efficiency, not cast priority. The cast priority is still:

Shadow Word Pain (only with 5x weaving stack and crit buffs)
Vampiric Touch
Mind Blast
Devouring Plague
Shadow Word: Death (only when there is between 1 and 1.5 seconds before the next non-mind flay should be fast)
Mind Flay

On the subject of delays, it depends heavily on how much time is left until Blast is up. If there's more than 1.5 seconds, it's best to Mind Flay and interrupt at the 2 second mark. If there's less than 1.5, you want to use Shadow Word: Death as filler. If it's less than .5 seconds, you should just wait for Mind Blast and cast nothing.

Tedv is correct, and the OP's method does not correctly prioritize spells, however I think there is a better way of showing it. Lets say we had a nuke that did 1k more damage than a MB, with a 2 minute cooldown. The OP's method would have this be the first thing cast would be this spell, however its long cooldown makes it unimportant to maximize (the actual DPS of the spell is low).

Additionally, all that matters with Mind Flay is that cooldown spells (nukes are the same as dots here) have a higher Damage_per_Cast_Time. The "filler" spell should be the lowest damage per cast time spell you cast. If any of the nukes are worse in this regard, they should never be cast.

Once we have that out of the way, it does not matter what the damage per cast time of our spells are!!

The value of interest here is the DPS per cast time, where DPS = Damage/Cooldown.

We can call this DPSPC, where DPSPC = (Damage/Cooldown)/Casting_Time.

This will lead to the same outcome as the "deferment" calculations, but allows each spell to get a "damage" like number for easily comparing the relative importance of each spell. The SPriest community has gotten away with confusing this for a long time due to the fact that all of our spells have the same cast time. Therefore the order does not change, and we just cast the higher Damage/Cooldown spell first.

As Tedv said, think about the damage lost by postponing the spell. Assuming each spell is constantly on cooldown, it has a certain DPS. If we have two spells come off CD at the same time, we need to postpone one of them. If they both have the same (Damage/Cooldown), we choose the quicker one, so as to get the second spell cast more quickly. Similarly, if they both have the same damage and cast times, we cast the one with the shorter cooldown.

There was a demonstration of this on shadowpriest.com that I'll try to find.