On the Value of Haste
 

The value of haste in regards to healing seems to come up in arguments every week or two on these boards. Often these arguments consist of uninformed opinions relying on anecdotal evidence, but every once in a while we get a pretty good discussion going. However, due to the complex nature of this topic it is almost impossible for any discussion inside one of the other healing threads to truly encompass all aspects of this debate. While some good points are made each time this topic is discussed, it is rare that all aspects of it are expressed in a comprehensive way and a good discussion can occur without getting derailed. This post and thread are an attempt to do that. I look forward to everyone's comments in this debate, and welcome all criticisms of this post, which I will admit contains some of my opinions to go along with the math.

Before we go any further, I think it is important to note how haste calculations are conducted. To find the cast time of a spell with X% haste, simply divide the original cast time of the spell by 1.X, as in the Flash Heal example above. If there are multiple haste effects then the calculation becomes more complicated. Generally, haste effects apply multiplicatively, so if you have 6% haste from enlightenment and 20% haste from gear, then the cast time of your Flash Heal will be 1.5/(1.2*1.06) = 1.18. This means that the 6% haste from Enlightenment and 20% haste from gear are collectively worth 1.2*1.06 = 1.272, or 27.2% haste, not 20% + 6% = 26%. However, not all haste effects are multiplicative, so one must be careful when calculating cast times with multiple haste effects. Eventually it would be nice to compile a list here of all possible haste effects and if they are multiplicative or additive, but I have no such list as of yet.

Contents

1. Effect on throughput
   1. Holy
      
   2. Disc
      1. Raid healing
         
      2. Tank healing
   3. Relative to Spell Power and Crit
      
   4. Spells with cooldowns
2. Effect on mana consumption
3. Effect on reaction speed

In progress or need help with:
-There is more to say about haste and Holy priests, but Holy is not my expertise so I need some help there

---

I. The effect of haste on throughput.
The classic way to measure this is by the effect of haste on a certain spells HPS (health per second).

Example
Let’s say your Flash Heal heals for 5000 health on average.
An unhasted Flash Heal has a 1.5 second cast time, giving an HPS of 5000/1.5 = 3333 HPS.
If you have 30% haste, the cast time becomes 1.5/(1.3) = 1.15 seconds.
The HPS of your Flash Heal is now 5000/1.15 = 4348 HPS.
Now, for a general rule we can say that 1% haste equals 1% higher HPS: 4348 is 30% larger than 3333. There are some exceptions to this rule, but in most cases this is sound, and will be referred to in the rest of this post as the “1%:1% rule”.

Clarification: when I say that 1% haste increases HPS by 1%, I mean that it increases your HPS by 1% of your unhasted HPS. What this means is that going from 30% haste to 31% haste yields an increase equal to 1% of your unhasted HPS, not 1% of your 30% hasted HPS. So when you go from 30% haste to 31% haste, what you will actually see in game is really a 0.01/1.30 = 0.77% increase in HPS. Therefore one could say that haste does not scale with itself, and that the value of adding haste decreases with the more haste that you already have.

Quote:

Originally Posted by Elimbras (Post 1548357) This "diminushing relative value" means that stacking only one stat is usually counter-productive.

The 1%:1% rule can be extended from individual casts to sequences of casts, assuming that as the amount of haste in question changes it does not also change the choice of spells used or their order:
See Elimbra's post for the proof.

It should be noted that once the cast time of a spell has reached 1 second, any further haste will not increase the HPS of that spell in the strict sense of over a series of casts, because the next spell cannot be cast until 1 second after the previous spell began casting (once the Global Cool Down, or GCD, has finished). However, there is still some benefit to haste past that 1 second mark. For instance, consider two heals, spells “A” and “B”, which are identical in every way except that spell “A” has a 1 second cast time and spell “B” is instant cast. In the long run, chain casts of either spell will have the same HPS, but we can all agree that an instant cast is much more valuable than a spell with a cast time. In that same way, a 0.9 second Flash Heal is slightly more valuable than a 1.0 second Flash Heal, even though your HPS over the course of the encounter will not differ.

Another way to think of how haste increases throughput is that it allows one to cast more spells in a given length of time. For instance, with zero haste you can cast 10 Flash Heals in 15 seconds. If you have 50% haste you can cast 15 Flash Heals in those 15 seconds. This is essentially the same argument as the HPS example shown above, just stated differently.



A. Holy Priests
One of the in vogue styles of play for Holy Priests at the moment is as a renew-centric raid healer, with spell casts consisting primarily of Renew, Prayer of Mending (PoM), Circle of Healing (CoH), Flash Heal (FH) and Prayer of Healing (PoH). Renew, PoM, CoH and FH fall under the 1%:1% rule up to a haste value of 50%. PoH falls under the 1%:1% rule up to a haste value of 300% for non-Serendipity hasted casts, or 221% for fully Serendipity hasted casts. PoM and CoH are also special cases due to their cooldowns, which will be addressed in section I.D of this post.

B. Discipline Priests
Discipline priests are often asked to wear two different hats depending on the encounter, raid makeup and personal preference. One must keep in mind the innate 6% haste given by 3/3 Enlightenment, which all PvE Discipline Priests should have. The haste granted by Borrowed Time following a PWS cast is important for this discussion. Assuming 5/5 in the talent, which all Discipline priests should have, the first spell cast within 5 seconds following at PWS gets an additional 25% haste. Note that instant cast spells (and Penance) do not consume the BT buff, although most appear to benefit from it in regards to reducing the GCD they generate. This means a priest can cast a PWS, then a Penance, then a Greater Heal and have both Penance and GH benefit from the 25% haste. I believe all instant casts benefit from the haste, but there have been some reports that maybe PoM does not.

The haste from Borrowed Time is what gives rise to the idea of a haste “soft cap” for Discipline priests: the amount of haste one needs from gear to reach a 1 second GCD with BT haste up. This soft cap varies depending on raid buffs. Assuming all applicable raid buffs (BT haste – 25%, 3/3 Enlightenment – 6%, Wrath of Air totem – 5%, Swift Retribution or Improved Moonkin – 3%, but not Power Infusion or Heroism/Bloodlust), this soft cap can be derived as follows:

1.5/(1.25*1.06*1.05*1.03*X) = 1
X = 1.0468 = 4.68% haste
1% haste = 32.79 haste rating
4.68% haste = 154 haste rating

1. Tank Healing
There are two common approaches to tank healing which differ by choice of “filler” spell: FH or Greater Heal (GH). Like a holy priest, Flash Heal falls under the 1%:1% rule up to a value of 50% haste. Greater Heal falls under the 1%:1% rule up to a value of 300% haste (or 250% haste if 5/5 Divine Fury). Penance falls under the 1%:1% rule up to a value of 100% haste.

2. Raid healing
This role typically means heavy use of PWS, which in turn means high uptime of BT haste. If two or more PWS are cast in a row (and less than 5 seconds apart), then any haste rating you have beyond 154 is wasted on each PWS after the first. This also applies if you weave in a PoM in place of a PWS. However, even while raid healing BT haste will probably not apply to every single spell cast, nor can we assume that every spell cast will have a 1.5 second base cast time. A raid healing Discipline Priest still uses Penance (2 second base cast) and PoH (3 second base cast), which will benefit from additional haste past 154 rating even when BT is up. Flash Heal will still benefit from additional haste rating past 154 in that the heal will land sooner, but the GCD it triggers will still be capped at 1 second.

C. Haste Relative to Spell Power and Critical Strike Rating
In order to appreciate the value of haste, it is necessary to consider it in the context of its, for lack of a better word, competitors. Also, since most of the arguments on this board regarding haste tend to take the form of “should I stack haste or crit or spell power?” this post would be remiss not to address it.

In this section we will only consider the increase to throughput provided by the three stats. We’ve already seen that 1% haste provides a 1% increase in HPS. However, 1% critical strike provides a 0.5% increase in average HPS if you are Holy, and 0.95% increase in average HPS if you are Discipline (due to Divine Aegis).

Example
Your Flash Heal heals for an average of 5000.
If you gain 1% crit your Flash Heal now heals for an average of (5000*.99 + 5000*1.5*.01) = 5025.
And 5025/5000 = 1.005, or a 0.5% increase.
For Discipline the calculation becomes (5000*.99 + 5000*1.95*.01) = 5047.5, and 5047.5/5000 = 1.0095, or a 0.95% increase.
One can argue that the DA shield is less likely to overheal than the crit heal itself and therefore the value of crit to a Discipline priest is actually higher than 0.95%, but that argument is beyond the scope of this post.

It should also be noted that the value of Critical Strike Rating decreases when taking into account that Renew cannot crit (outside of Empowered Renew), and PWS cannot crit (outside of the glyph). And of course arguments can be made against the value of Critical Strike Rating due to the unreliable nature of critical heals and their higher chance to overheal.

Spell power is a little trickier because different spells have different coefficients and increased spell power will provide no benefit to heal that already partly or completely overheals. A full listing of Priest spell coefficients can be found at Spell power coefficient - WoWWiki - Your guide to the World of Warcraft.

Spell power is widely regarded as the largest throughput increasing stat, but the exact value of a certain amount of spell power in terms of HPS improvement is hard to quantify. The strength of spell power is also noticeable in that every spell you cast (ignoring dispels and the like) will take advantage of spell power, however not every spell will take advantage of haste or crit (as in the PWS and renew examples cited earlier).

We cannot neglect the itemization cost of the three stats. Haste rating and Critical Strike rating have equal 1.00 StatMods when it comes to item budgets. However, the conversion between rating and percentage are different between the two. Specifically, 32.79 haste rating equals 1% haste, while 45.91 crit rating equals 1% crit. This means that crit rating is 40% (45.91/32.79) more expensive than haste rating. So when considering gems/gear, note that the choice is not between 1% haste or 1% crit, but rather between 1.4% haste and 1.0% crit.

Spell power has a StatMod rating of 0.86, meaning that for every 1 haste or crit rating a piece of gear has, it could have 1.16 spell power (1/0.86).

The interdependency of the three stats should also be noted.

D. Spells with Cool Downs
The value of haste also depends on if the spell you are using has a cool down. For instance, adding haste will make Penance cast faster, but no amount of haste will allow you to cast more Penances in a given amount of time. That extra time gained from a faster Penance can be put toward filler spells, but not towards more Penances.

Adding haste means casting more spells in a given amount of time. Because the number of Cool Down spells is capped for that length of time, the extra spells must be "filler spells" like Renew or Flash Heal. This means that the ratio of CD spells to filler spells will decrease with increasing haste.

We can all agree that our CD spells (PoM, CoH, Penance) are superior to our "filler spells," so as the ratio changes to favor more filler spells it makes sense that the HPS gain will not be as large as anticipated.

Example (courtesy of RootBreaker)
Your renew heals for 18k over its duration and your circle of healing heals for 26k (both pre-overheal). If you have no haste, you can fit 3 renews between each circle of healing, doing 80K healing every 6 seconds. If you have 25% haste, your global is 1.2 seconds and you can fit 4 renews between each circle of healing, doing 98k healing every 6 seconds. In this example, 25% haste gives you a 25% increase in the number of spells you can cast, but only a 22.5% increase in healing.

The exact effect of CD spells on devaluing haste depends on the relative strengths of the CD spell and the filler spells. We can assume our CD spells are stronger (otherwise we wouldn't be casting them), so generally speaking the value of haste to us will be somewhat less than 1%:1%, but the exact difference will vary.

---

II. Effect on Mana Consumption
Although mana consumption is of little concern in the current tier of content, it has been an important factor in the past and will supposedly be important again in Cataclysm, therefore I think it prudent to include this section on how haste affects mana consumption.

A. More casts One argument that is occasionally made, and that we must pay heed to, is that adding haste can have a negative effect on longevity. If haste allows one to cast more spells in a given length of time, those additional spells cost mana that would not have been spent if the player did not have that haste. The haste, therefore, essentially had a negative mp5 effect.

There are some counterarguments to this negative mp5 theory. For a given fight there will only be a certain amount of damage to be healed. If we assume that all of that damage is healed (everyone survives the fight and is at full HP at the end), then additional spells allowed by more haste are actually not needed, and therefore do not need to be cast and would not result in extra mana spent. Of course, that makes certain assumptions that are perhaps not very accurate, namely that the healers play as efficiently as possible.

B. Reduced Overhealing Theoretically, haste should also reduce overhealing and therefore increase mana efficiency. Think of it like this, if every heal that everyone in the raid has is instant cast, then there is no chance of a different persons heal landing on your target while you are still casting, so overhealing would be minimized. However, I don't think we play with enough haste for anything approaching that kind of effect to occur.

Also, just because your heal lands before a heal from another healer does not really mean reduced overhealing, just a shifting of the overheal from you to the other healer, so no mana is saved.

C. Holy Concentration Uptime In a more specific case for Holy Priests, increasing one's haste allows more casts in a given length of time. More casts means more opportunities to crit, which should mean a higher uptime on Holy Concentration.

The size of this effect is of questionable value, as pointed out by Elimbras here. Essentially, higher HC uptime due to increased haste should be seen as simply a minor fringe benefit, with 1% haste giving only a few additional mp5. Gearing for crit rating and adjusting spell usage to more non-SoL FH/GH/Renew/BH would provide much larger increases in HC uptime.

---

III. Effect on Reaction Speed
One of the most common arguments seen on these boards is that haste saves lives. This is commonly stated as something like,

That increased haste should result in more saved lives seems obvious and needs no proof, but to say that adding haste will save more lives than adding spell power does requires some supporting evidence. Unfortunately, this kind of claim is something that would require the intricate analysis of large amounts of data, which we have neither the ability to do nor access to.

It seems to me that this argument is perhaps due, at least in part, to our own bias. What I mean is that, at least for me, when I have a raid member die just before my heal lands it is easy to notice, extremely frustrating and I tend to remember it well. It is much harder to notice if your raid suffers deaths because your heals are not large enough. A deficiency in heal size would more likely manifest itself as slowly decreasing HP levels throughout your raid – if each heal you cast is a couple hundred HP smaller than it needs to be you probably won’t notice any problems at first, but after a minute or two those couple hundred HP deficits begin to add up and it appears that the raid damage is just overwhelming. Or if you are tank healing, it is easy to say that a tank death was due to a bad spike of damage, but it’s not always easy to notice that if your heal 5 seconds before the tank died had actually been slightly larger it would have saved him from that spike.

Another argument that is often seen is, Which is of questionable validity. Hard mode fights are more difficult all around and should theoretically require more of everything – spell power, haste, crit, longevity, reaction speed, better decision making, etc. There is nothing inherently unique about haste in regards to hard modes.

The 1%:1% rule doesn't really follow for prayer of mending and circle of healing. Let's assume you're doing blood queen or twin valkyrs with exclusively with renew, and circle of healing. Obviously you'd always use prayer of mending too, but that complicates the math a bit. Your renew heals for 18k over its duration and your circle of healing heals for 26k (both pre-overheal). If you have no haste, you can fit 3 renews between each circle of healing, doing 80K healing every 6 seconds. If you have 25% haste, your global is 1.2 seconds and you can fit 4 renews between each circle of healing, doing 98k healing every 6 seconds. In this example, 25% haste gives you a 25% increase in the number of spells you can cast, but only a 22.5% increase in healing.

Edit: I left my previous comment below but it is incorrect. I assumed you were including the 2nd cast of circle of healing but you were not.

You aren't considering the benefit of a reduced global cooldown. Your sample sets cost 6.5s and 6.2s respectively and if you calculate the HPS based on those values you will see a larger increase in healing.

This is a good point, and illustrates the different effect that haste has on spells with cooldowns.

Your example correctly shows that the casting sequence does not follow the 1%:1% rule. However, the individual spells do.

CoH did 26k healing in 1.5 seconds with 0% haste, 26000/1.5 = 17.3k HPS
CoH did 26k healing in 1.2 seconds with 25% haste, 26000/1.2 = 21.7k HPS
21.7/17.3 = 1.25, or 25%

Renews did 18k*3 = 54k healing in 4.5 seconds with 0% haste, 54000/4.5 = 12k HPS
Renews did 18k*4 = 72k healing in 4.5 seconds with 25% haste, 72000/4.5 = 15k HPS
15/12 = 1.25, or 25%

---

Adding haste means casting more spells in a given amount of time. The number of Cool Down spells is capped for that length of time (can only cast X number of CoH no matter how much haste), so the extra spells must be "filler spells" like Renew or Flash Heal. This means that the ratio of CD spells to filler spells will decrease with increasing haste.

We can all agree that our CD spells (PoM, CoH, Penance) are superior to our "filler spells," so as the ratio changes to favor more filler spells it makes sense that the HPS gain will not be as large as anticipated.

A thought experiment: what if CoH was actually a terrible spell? Let's say it heals zero health, but we still want to cast it. Then your example becomes:

A. 1 CoH + 3 Renews in 6 seconds = (0 + 18k*3)/6 = 9k HPS

B. 1 CoH + 4 Renews in 6 seconds = (0 + 18k*4)/6 = 12k HPS

12/9 = 1.33, or 33%.

So in this case the 25% haste actually results in a 33% increase in HPS.

So we can say that the exact effect of CD spells on devaluing haste depends on the relative strengths of the CD spell and the filler spells. We can assume our CD spells are stronger (otherwise we wouldn't be casting them), so generally speaking the value of haste to us will be somewhat less than 1%:1%, but the exact difference will vary.

This is exactly what I needed to flesh out the CD spells part of my original post, thank you.

I'll have some more substantive comment later, but my biggest takeaway is that actual choice among gear is limited. We can visualize caster gear (minus the hit stuff) as falling into 5 categories:

1. Haste/Crit
2. Haste/Spi
3. Haste/MP5
4. Crit/Spi
5. Crit/MP5

We can make tradeoffs less granular through gems and enchants (though enchant options are pretty limited), but that's the set of options available to most priests (and casters generally). Where our theorycrafting suggests some interaction between haste and need for MP5 (in that we burn through mana faster), 2/5 possible options are optimal and 3/5 are not. While that doesn't speak to the value of quantifying haste, it does limit the usefulness of theorycrafting for gear selection--simple heuristics are likely to suffice.

As a totally extraneous comment, I like the approach of comparing marginal benefits from haste for various specs and spells. We do this quite naturally for DPS and often shy away from it in healing discussions. One caution is that the benchmark need not be 1:1--it is only sufficient that haste be the biggest marginal gain at the locus of decision. In other words, in the spec, gear and playstyle I am at right now, do I get the highest marginal benefit from haste, crit or sp (or regen, if you prefer)? In some cases we may not see a marginal benefit equal to one for any stat.

The 1%:1% rules can be easily extended to sequence, as follows:

The proof is quite easy : consider a sequence of time T, composed of a sequence S1 of spells with cd, that takes (as a whole subsequence) a time T1 to cast and heal for H1 hps, and a sequence S2 of spells without cd that takes time T2 to cast and heal for H2 hps.

Add now x% of haste. The time needed to cast the same cd spells sequence is now T1/(1+x). Same goes for the spammable sequence. This leads you to the same healing total that you had without haste. In time interval T, the time leftover is T-(T1+T2)/(1+x) = T - T/(1+x) = xT/(1+x).
During that time, you can spam your sequence S2, which has now (with the additionnal haste effect), an HPS equal to H2*(1+x). The additionnal healing of the extra-sequence is then H2*x*T.

In long term, in the time interval T, you had an increased healing of H2*x*T, which means that your hps was increased by x*H2.



Second point is different : you don't consider that any of the stats don't scale with ifself (except for haste from different effects, but most of the concern is haste from gear with is additive). When going from 0% to 1% haste, you indeed increase your throughput of 1%. But if you go from 50% to 51% haste (from gear), then your throughput increased only from 1% of your "unhasted" throughput, that is the throughput you have without any haste from gear. Your going from 1.5*BaseHPS to 1.51*BaseHPS, which is only 0.66% relative increase.
The same is of course valid for crit and spellpower (and applies to haste for sequence that I stated in the first part of the post).
This "diminushing relative value" means that stacking only one stats is usually counter-productive, except if the stats has from the beginning a very clear lead in value (as hit for dps, or spellpower for PW:S spam).

That's true, with one minor caveat: as haste changes, the relative value of the cd limited spell and the cd-free spell changes, and presumably our hypothetical healer could allow for the selection of spells as a choice variable. That's partly what makes for the wonky sim results vis a vis haste/sp for shadowpriests--spells are being added and subtracted to the "rotation" as haste impacts their value.

Going too far down the rabbit hole is academic for healers, but a rough real world example can be seen in discipline priest timing BT, penance and flash heal. As their haste goes up, the need to conserve BT procs in order to both speed up penance and FH drops (it doesn't go away, obviously). At some lvl of haste on gear, the cast order becomes somewhat irrelevant. Clearly at that lvl of haste you would be substituting GH for FH, but I hope the point is made.

If I may be so bold as to make this thread about the relative value of haste, spellpower, and crit (which it really already was despite the title), I'd like to elaborate on the following point (emphasis added).

It is not true that the DA proc is "less likely" to overheal. Rather, the DA proc cannot overheal. The question is really whether the shield will be consumed before it expires. As the original post points out, there are two principal roles for the discipline priest in the raid: tank healing with a PW:S/PoM/Pen/FH rotation or shield-spamming the raid during heavy, predictable raid damage (e.g., Prof. Putricide phase 3, Blood-Queen). I will consider the first case since neither crit nor haste really helps for shield-spam encounters.

The argument for haste on holy priests and the one I would like to make for crit on disc priests is that players don't frequently die from heals being too small, but rather from one not landing between big hits. Consider this parse as a test case. The tank put on an avoidance set (60% dodge + parry if memory serves) and tanked Razuvias. Note how DA shoots to the top of the effective healing. This is because in a fight with high, spiky, and unpredictable tank damage the shields will almost always be consumed. Additionally, when we consider that it is the spikes that generally kill tanks rather than sustained damage, these empirical considerations become more important relative to throughput calculations. The hit at 23:47:52.541 (only time Ardent Defender procs) which hit the tank for 37k after 32k was absorbed serves as an example. Without a cooldown, no amount of throughput would save the tank from such a hit.

As for attempting to quantify the throughput effect of crit for the disc priest, this can be done simply by considering the percent of overhealing a and assuming a constant rate of overheals:

Holy: (1-a)(1+.05)
Disc: (1-a)(1+0.5) + 0.45

This can be normalized for the standard rate of overhealing as:
((1-a)1.5 + .45)/(1-a)

For a value of a=0.42, an admittedly anecdotal value from one of my recent ICC runs, this give a bonus of 127% rather than the 95% above. This goes to 134% with the [Revitalizing Skyflare Diamond], which becomes valuable when we consider that these haste vs. crit vs. spellpower discussions are typically conducted with the understanding that mana regeneration is already irrelevant, thereby making regen metas ([Insightful Earthsiege Diamond] and [Beaming Earthsiege Diamond]) likewise irrelevant.

These are both good points and I hope you don't mind if I edit your work into my original post.

My apologies on my original wording, when I said that DA shields were less likely to "overheal" I was of course referring to them expiring without being consumed. I agree that when it comes to tank healing, it is safe to assume almost 100% usage of DA shields because it is rare that after a critical heal your tank will take no damage for 12 seconds AND you will not crit another heal on him/her to refresh DA.

On a side note, I couldn't find the line in your World of Logs parse, but 32k absorbed on a single hit? DA caps at 10k, plus another 8000ish from PWS still leaves you 14k short. Even if someone had the proc from the legendary it caps at 10k, so still short of 32k. How was that 32k absorb possible?

Regarding the stat weighting of crit for a disc priest - let's not ignore the difference in overheal percentages between crit heals and non-crit heals. If you can calculate those overheal rates seperately and then adjust your calculation accordingly, I would be interested to see how much HPS 1% crit gives, closer to 1.27% or 0.95%?

@Squeakster : Please proceed. Had I wanted to keep it secret, I wouldn't have posted it here (and I think I already posted these points in other threads of the EJ priest forum).

I'm note sure I understand clearly your point, most likely because of my poor english understanding.
I don't care about the relative value of cd spells and non-cd spells. I just care about how much hps you can get when you don't cast any cd spells : that's the spammable throughput I defined. And in fact, haste will just allow you to spam these spells, so that's the throughput you'll gain.
Now, you can say that you have haste plateau, as shadowpriest haves. As holy, it is good to be able to fit an exact number of GCD in your "7s cycle (POM - COH - N X renew for example). This effect will happen, and is not really negligible. Trouble is that in order to quantify it, you need to consider the precise sequence of spells and their cd, and you won't get any general result. However, in long term (considering high changes in haste value), you can ignore them, and just look at the mean increase per haste rating. And that one will be the result I gave --- that's why I began with the "Ouside granularity questions". In fact, in more practical terms : you'll have plateaus where the local derivative haste value is near zero, and you'll have places where it will be your whole cycle hps. The proportion of those periods depends precisely on the relative hps of cd spells and cd-free spells, as well as on their cd.

In fact, maths would be similar to the one on the shadow thread "Damage per Execution time". Just consider the "extra healing" from the cd-spells as a instant gcd-free hot that you can apply every T seconds for the whole T seconds (you can think of holy fire, if the direct damage had exactly the dps of smite), and replace the cd-spells with the cd-free spells (or fillers). Now, the question of granularity is : assuming that you can't cast an exact integer number of fillers in your T second cycle, do you prefer to cast one more, and loose the hot for the extra-time --- that's the case where haste is really interesting, because it would reduce the whole cycle time ---, or do you prefer to wait till you can recast the hot (and lose some casting time) --- that's the case where haste has local zero derivative value.


Detailed (and long and painful formulas) math :

Now, you consider you are in a perfect case, and increase gradually haste. Call HPS_Hot and HPS_Filler the respective unhasted hps, and consider that the filler has a an unhasted cast time CT (and the hot a duration and cd T). Your local haste rating is x, and you can cast exactly N fillers in Ts (ie, N * CT = T * (1+x) ). You increase your haste of dx.
If you choose to wait, your hps is precisely Hps1 = (1+x) * HPS_Filler + HPS_Hot.
If you choose to delay your cycle, you can get a cycle of length (N+1) * CT / (1+x+dx). During that cycle, your mean hps is
Hps2 = (1+x+dx) HPS_Filler + T /[(N+1) CT / (1+x+dx)] HPS_Hot
= (1+x+dx) HPS_Filler + T * (1+x+dx)/[(1+x)*T+CT] HPS_Hot
= (1+x+dx) HPS_Filler + [[ 1 - (CT - T*dx)/[(1+x)*T+CT] ]] HPS_Hot
= (1+x+dx) HPS_Filler + [[ 1 - (CT - T*dx)/[(N+1)CT] ]] HPS_Hot

For simplification, call RHPS the ratio HPS_Filler / Hps_Hot.
Haste value is zero iff
Hps1 >Hps2 <=>
dx * HPS_Filler - (CT - T*dx) /((N+1)CT) * HPS_Hot < 0 <=>
dx * [[ (HPS_Filler) + T / [(N+1)CT] *HPS_Hot ]]< [CT / (N+1)CT] * Hps Hot <=>
dx * RHPS * (1+N) * CT + T< CT
dx < CT / (T + CT * (N+1) * RHPS).

I had to do some major digging with the log query tool to find it, and unfortunately the AJAX on WoL works prevents linking to it directly. The lines are copied below:

On further inspection it's clear that we can't credit all of this to DA, so perhaps linking to the parse as an example of my point was not the best way of articulating it. The point was that the goal on almost every encounter is not to maximize HPS but to keep people alive, and DA procs on an overheal have a benefit disproportional to the amount healed by reducing the spikiness of incoming damage.

I completely agree with your assessment of the different proportion of overheal for crits and non-crits. Using a_r for overheal fraction of non-crits, a_c for overheal fraction of crits, and m as a binary variable indicating the presence of the 3% crit meta the new 1%:x becomes:


Unfortunately, I can't get empirical values for a_c or a_r from WoL, and my hard drive is too slow to log during a 25man raid. If someone can send me a raw combat log, I can extract those with a perl script pretty easily. However, the reduction of spike damage by increasing effective health on an overheal that I mentioned above, adds an additional benefit that is extremely hard to quantify.

Regarding BT impact on the GCD triggered by PoM, I think it makes sense to resolve that issue in this thread now.

In the healing compendium thread I recently stated that the PoM's global cooldown probably isn't affected by BT. This was supported by preliminary testing (excerpt):

1/31 23:16:33.803 SPELL_CAST_SUCCESS,"Milstaff Sturmauge",0x10a18,10900,"Machtwort: Schild",0x2
1/31 23:16:34.861 SPELL_AURA_APPLIED,"Hegen",0x511,48111,"Gebet der Besserung",0x2,BUFF
1/31 23:16:36.046 SPELL_CAST_SUCCESS,"Hegen",0x511,48066,"Machtwort: Schild",0x2

Final closing words were:

In a number of test runs (a dozen or so), I always see a bit more than 1s between Pw:S and PoM, and around 1.2s (or a bit more) between PoM and the following cast. While this may still be within the bounds of statistical bad luck, it sure looks like the GcD triggered by PoM doesn't profit from BT.


Today I did some more testing. I removed all haste from gear (but not the 6% from talents) in order to get clearer results. This revealed that the testing above isn't nearly reliable enough - at least not with the current state of servers. Even though not wearing any haste, I frequently get Pw:S sequences a tiny bit less than 1s apart (which is too low without buffs and haste on gear). I also get sequences where the glyph heal and shield are more than 0.2s apart, and others where both hit at exactly the same timestamp.

So - at least for me - this testing methodology isn't worth a thing because the variance in results is close to the effect we look at.

An alternative approach might be to test throughput by using a continuous sequence of Pw:S with PoM on cooldown. Due to weakened soul, this requires more than 10 people. If executed for several minutes and with enough haste to bring the GCD exactly down to 1s with BT, we should see very close to one cast per second if PoM's GcD is affected by BT.

Any other - better - ideas regarding testing this? Perhaps using a custom addon?

Just goes to show my lack of knowledge about the mechanics of other classes, but it turns out an Ardent Defender proc actually shows up as an absorb equal to the amount that would have overkilled the paladin, so that is how the 32k absorb exists.

I agree about the disproportional benefit of DA in regards to smoothing out damage spikes, and also that it would be extremely hard to quantify. I guess the best we can do is a "trust me on this."

I wish I had a combat log to supply you with, but I'm afraid I don't. Anyone brave enough to bare their data for the good of science?

I'll try and log this coming week's raids and shoot it over to you. I think that you have shown a good upper bound for absorbed damage from DA. It remains to be seen if we can estimate how much absorbs in the logs can be linked to DA vs. PWS or Sacred Shield or AD (etc. etc. etc.).