Kavan, this is getting closer to addressing the practical problem we as raid mages must solve. Let's say that under common 25-man raid conditions (replenishment, JoW, Divine Spirit), we need to solve a problem for depleting our mana pool slowly over the course of a typical Naxx fight (5 minutes).

A typical mana pool is about 30,000 mana for an arcane mage (20,000 plus 2 x 4500 for two 2T7 mana gems used in five minutes). If we want to end up at ~0 mana after 300 seconds (5 mins) then we have only 100 mps to burn. This means we need to make up the difference in spirit, evocate, innervate, etc. According to your calculations, if we are aggressive (and, to be competitive with the new TtW Fireball spec we have to be so) we will aim for AB3->AM->ABar. Your figures give us a mps of 229. So, we need 229-100 ~= 130 mps (or 650 Mp5) to keep pace. This is untenable using Molten Armor and 3/3 Arcane Meditation where typical spirit amounts will hover around 400 at most (until and unless we gear for spirit). It forces (glyphed) Mage Armor to be used plus the aforementioned raid buffs to bring us within range.

Of course, this is all napkin math, but what it tells us is the value of spirit (and int) is much higher than it otherwise would be for the "aggressive" track of AB3->AM->ABar. More specifically, it looks like we are going to need to aim for about 600 spirit unbuffed, include mage armor for 80% mana return and get full raid buffs to maintain. Otherwise, AB2->AM->ABar (165 mps) may be our most practical solution until gear allows otherwise. That would require only an additional 65 mps (325 Mp5) that we could get "naturally" from the typical gear that many raid mages already wear.

It may be worth considering: haste is very easy to aquire, and benefits (point for point) the arcane spec more than crit does. If the mana consumption can be accomodated for in an arcane spec, you end up with the interesting meta-game of stacking spellpower and haste, not really worrying about crit (the same way other specs don't worry about haste) and balancing your haste with spirit and int in order to feed your mana-habit.

Kavan, I'm curious. What are you doing to calculate DPM? Your numbers are out of scale with what I normally encounter when I see DPM expressed as Damage Per Mana calculated by Damage Done / (Mana Spent - Mana Regened through Procs or Regen)

Physicist, you have nailed the challenge (and the fun) of this spec: the meta-game, as you termed it. Haste, spirit and int work in concert to create a "steady state" environment for sustainable DPS. Clearly, the values of which are malleable...unfortunately for us, we don't have infinitely adjustable gear (but we might just be able to stack properly based on the total cast time and mana pool available for each fight...adjusting gear possibly as needed on a per-boss basis). I think we should solve this steady-state problem set (I feel like I'm getting my graduate degree all over again) with some vanilla (but plausible) gearing, revealing where our spirit, haste or int shortfalls may lie.

Since Int is the easiest stat to obtain in our gear naturally, we can probably fix int to a typical value (like 1000) and solve for the other two variables. The beauty of Kavan's calculations are that they calculate dpm without mana refunds or natural regen, allowing us a "pure" solution. If we back off the haste value to something more like 20% (closer to real-world talent and gearing), we can recalculate the dpm values (they are obviously a little more than 10% more favorable under those conditions) and then solve for SPI.

As an aside, you can solve for haste, given the int and spirit values we may be relegated to. In any case where we are "haste capped" for our current int/spirit, we can compensate with comparable crit items (although clearly not ideal).

For anyone that is familiar with Rawr, the numbers that I'm reporting are what you would see in the spell info section, so the mps figures already include regeneration. The dpm is the dpm tradeofff as used first in the analysis of arcane two-cycle systems of 2.4. I compute it as (dps(cycle) - dps(maxdpscycle))/(mps(cycle) - dps(maxdpscycle)), it is used to express the actual role of mana to damage conversion as you're balancing the two cycles (meaning lower numbers are better).

That effectively nullifies half of my previous argument. Oh well. Now, we have to back out the calculations to *not* include the dpm tradeoff. We need that number to be around 100 dpm. We're not even close, apparently. As I am not 100% comfortable with Rawr yet, I will have to see if I can remove all mana regen conditions to get a better read on the kind of dpm number I was referring to above.

is anywhere the tradeoff between mage armor and molten armor made?
Or exists a calculation how much mana a mage armor would give in a given time?

Lovely crunchy math bits. I'll need some time to digest this (my Markov/Linear Algebra has been a while).

On the one hand, though, I'll note that a Glyph of Arcane Blast is now in the Patch Notes (under Inscription). It doesn't say what it does, however. It is presumably the 5% increased buff glyph, but no telling whether or not it stacks.

Anyone know if this is in the PTR for testing?

5% static increase regardless of debuff is logical and fits the numbers. 5% per debuff would be grossly outdamaging any other mage talent build, as well as other classes which does not fit blizzards design philosophy.

Glyphed = 20/35/50

Having read your conclusion I believe your arcane rotation is not the best DPS. Your rotation is likely the best DPM while sustaining a decent DPS, but its likely not the best DPS.

From my calculations you want to continue casting Arcane Blast until Missle Barrage procs. The only time to use Arcane Barrage is immediatly after a Missle Barrage, or to dump Arcane Blast.

5 minute fight is really pushing it I think. Most fights I have to save Combustion for Heroism because I don't have it back in time. The fights I can think of that last longer than 4 minutes are Saph, Raz, Loatheb, Thadius. DPS is only getting better each week, so fights are getting shorter. Arcane rotation is about 66% effeciency compared to a fire rotation. I've never had to Evocate on any boss encounter, I've never even come close (I don't count KT though because he destroys mana). I don't think mana will be an issue with Arcane.

Just some helpful information I'd like to pass along:

Haste Cap is 797, anything over 797 will not benefit you during a heroism + icy veins + trinket proc or haste potion.

In a 300 second fight, 3 arcane powers, 2 icy veins, 3 mana gems, 2 evocates. If the fight goes longer than 5 minutes, your evocation and icy veins are available at the 304 and 308 second mark respectively. There are 3 intervals in which you aren't going to mana dump: 77.83 seconds (1st interval), 24.95 seconds (2nd interval), and 53.32 seconds (3rd interval). I would suggest the dpm rotation for the 77.83 and 53.32 seconds, and a dps rotation going for the 3rd debuff of blast in the 24.95 second interval. Haste for this test was 797, fyi.

I was initially thinking of a 55/3/11 build. But that 6% crit to arcane blast might not be worth it, it would probably be best to go 53/0/18. Only getting 1 magic attunement and 4/5 arcane mind. The only thing you sacrifice is 3% int and 6% AB crit for 10% reduced spell costs. I like it! However I haven't tested it in the raid environment so I'm not sure if mana will be an issue.

With precision, focus and channeling thats 84.68% mana cost.

Now that we know that there is an AB glyph and how it works I've redone the test. Before I get to the numbers I'll spend some time on the reasoning behind the selection of cycles based on dpm tradeoff.

Everyone has probably heard about the two cycle theory. This is a result that follows from linear programming. In general we would formulate the system by specifying all possible cycles and using a LP solver to determine which pair gives the optimum solution. Since we have a lot of possible cycles we would like to use some preprocessing to limit the number of cycles that we have to consider. The process I'll outline does not guarantee that we'll get all cycles that are relevant, but in practice it works quite well.

If we have two cycles A and B, one of them being sustainable and one not sustainable, then we can formulate constraints in a simplified system as follows (where M is total available mana and T is total time):

M = t * mps(A) + (T-t) * mps(B)
damage(A,B) = t * dps(A) + (T-t) * dps(B)

Solving for t we get:

M = t * (mps(A) - mps(B)) + T * mps(B)
t = (M - T * mps(B)) / (mps(A) - mps(B))
damage(A,B) = (M - T * mps(B)) * (dps(A) - dps(B)) / (mps(A) - mps(B)) + T * dps(B)

By introducing the dpm tradeoff K(A,B) = (dps(A) - dps(B)) / (mps(A) - mps(B)) we can express total damage as

damage(A,B) = (M - T * mps(A)) *K(A,B) + T * dps(A) = (M - T * mps(B)) *K(A,B) + T * dps(B)

Lets say that A is the sustainable cycle and B is the unsustainable. If we fix A then we see that total damage will increase if tradeoff coefficient increases, so the tradeoff coefficient of the high dps cycle must be as high as possible given a fixed sustainable cycle. If we fix B instead then total damage increases as the coefficient decreases so the dpm tradeoff of the sustainable cycle must be as low as possible.

Knowing this we can look for efficient pairs of cycles in the following way. First find the highest dps cycle, call it B. Next select a mps that corresponds to sustainability limit. Look for a sustainable cycle with the lowest dpm tradeoff against max dps cycle, call this cycle A. Then look for cycle with higher dps then A that maximizes dpm tradeoff against A, this is your new cycle B. Repeat until there are no more changes. Note that we ignore cycles with negative dpm tradeoff (they are dominated, higher dps at lower mps). Another interesting note is that the common dpm that we usually see is basically comparing efficiency of the cycle against not doing anything, not necessarily a good measure of efficieny.

Lets move on to the numbers with AB glyph. Highest dps cycle is AB spam with MBAM-ABar on proc at 3 stack with 4924 dps, 412 mps. Using a sustainability level of 300 mps we find the best sustainable cycle to be ABx3-AM-ABar regardless of proc with 4791 dps, 229 mps with the highest dps cycle being the best pair. If we lower the sustainability we find that the best pair is ABx2-AM-ABar regardless of proc (4472 dps, 165 mps) and ABx3-AM-ABar, but doing AM-ABar at 2-stack if we notice the proc then (4725 dps, 211 mps).

Using the cycles on a typical 5 minute raid setting we find that the optimum is AB3-AB-ABar and AB3+MBAM-ABar pair, this is with molten armor and using 2 evocates.

The problem with the two-cycle theorem is that cycles are not infinitely divisible (you can't cast half a fireball), which makes you wonder if you should be using an integer program instead of a linear program. If so, the theorem goes out the window: integer solutions to linear programs do not, in general, have to even remotely resemble their linear counterparts. They're also significantly more unstable, as the solution will generally depend on various divisibility criteria. Nonetheless, given that the fight length is generally unknowable with sufficient precision, a linear program is a decent approximation. I object to using the word 'theorem' since we're now dealing with acceptableness of approximations, but that's personal preference. Since this is the hardcore-math thread, I just want to make sure we're all aware of the various caveats we use, such as the inapplicability of linear programing to cycle modeling.

You definitely have a point about. The problems of divisibility become especially apparent when trying to apply the linear programming model to cooldown management since individual cooldowns have much shorter durations. It is still a decent approximation, but translation to real play becomes a lot more questionable.

You pretty much pointed at the major design issue of simulating fights.

You can do everything in integer steps. But that gives you very exact but generally useless information.
Small values of haste is usually useless, only haste thresholds where you get one more cast/GCD/something in matters.
Mana stats don't matter unless you can shave off a whole mana pot or a whole Evocation tick or a whole cast more of your burn cycle.
Your simulation results are completely irrelevant for practical purposes since 299s, 299.5s, 300s, 300.5s and 301s fights are totally different from each other.

Each fight can probably have a different spec and set of gear for maximum damage.


So, allowing continuous cycle splitting with a fixed time frame works for simulating, estimating and averaging fights with a small variance in fight length.
I'm fully aware that were doing two completely different things here.
We want to know the average result of a discrete solution when we have a slightly variable fight length.
Instead we compute the continuous solution of a fight with a fixed length.


Is that good enough as result? I don't actually know mathematically, but it's what I'd want to have as result judging from gut feeling.
Gut feeling and math don't usually work together (just check the official forums ), but in this case gut feeling is needed to determine what we actually want to find out.

Personally, I don't even want a discrete solutions because haste and mana only have threshold values and the optimised cycle is very volatile.
It would always end with Fire Blast, perhaps a Scorch even and cause lots of headache with lost Ignites from crits in the last 4 seconds.
There are probably people who want that. Even I might want that for a fight with an explicit berserk timer like Archimonde, where everyone gets oneshot when the time is up. Where damage after 600.00s does not matter at all.

But for the majority of the fights, and for choosing gear, spec and tactics in general, that model is not what I want at all.

Regarding guessing fight length.

Now this may not be completely on topic and I just might be drooling too much for my own good(but hey, I'm trying).

As I understand it damage done in t, time units is generally calculated by guessing the length of the fight(which you actually don't know). But what if you instead of finding damage done/dps based on guessing the time would find the time based on damage done. This you actually do know: (boss health) - (damage done) = 0

This would of course mean you're not modelling individual dps but raid dps, which would of course be a lot bigger of a task than the previousl but my questions are simply:
1) would it be solvable?
2) advantages/disadvantages of doing it that way instead?

After some consideration (and softening my tone in a rewrite) I have to say that if you understand we are doing two completely different things, then what is the issue? The idea behind most of the posts in this particular thread to me are to get a feel for the DPS to DPM variances we have to deal with in this spec. It is quite a different DPS model from what we have with fireball/ffb. In those cases, the DPM is effectively static. You follow the cycle until either the boss is dead or you are out of mana--"partial cycles" be damned. As a matter of fact, the calculations and postulates posted above are *not* discrete solutions.

They are more like what your college physics professor gives you in a problem set--"clean" solutions that are supposed to reveal larger truths--in this case the "most reliable" cast sequence to fall back upon when you have little else to do other than stand-and-cast for a finite period of time. What the numbers say is that your "gut" *can and should* work with some informed analysis. As an example, informed analysis on this forum revealed that AB3->AM->ABar is better then AB3->ABar at three cycles. Your *gut* wouldn't tell you that (without seeing some data to inform it).

I think what you wrote was frankly a cop-out and not an attempt to understand the analysis and consideration of the dynamics of this spec. Hey, maybe we're all wrong, but it won't be because "that model is not what I want at all."

Basically your dps improvements changing the fight length is a second order correction, and most second order corrections in WoW are neglicible. If your entire raid does 1% more dps, then the boss will die 1% faster (actually I again neglected the second order correction of 1/99 %). That means your 6 min fight will turn into a 5:56.4 min fight. Doesn't really make any difference of real importance on your actual game plan (gear choices, cooldown useages, etc), and on top of that this kind of fight duration difference is actually quite smaller than the deviation in fight durations you will get by repeating the same fight with the same raid, not to mention repeating with a different raid.


Regarding modeling fractured spells/abilities/etc VS averaging over various fight durations, it should give extremely similar results as long as you're not fracturing anything that has a cooldown that is not << fight duration. Say, if you know the fight is 5 mins give or take 30 seconds, dont count 5/3 or 8/3 or whatever, instead count 2 combustions, since 3 mins is not << 5 mins, however you can count your 45s internal cooldown trinket to proc 5.45 times if it procs every 55 seconds on average, since 55 seconds is quite close to being << 5 minutes, plus with the randomness of the fight duration you'd get fractured usage anyway when averaging over different fight durations.

In other words, if averaging over various realistic fight durations would give you fractured useages of something, then it's OK to fracture it. If all realistic fight durations give the same number of uses, you should not fracture it.

Of course, if we really want to be totally accurate we should fracture nothing and always average over various possible fight durations and weight each duration with its probability, however that will take a lot of calculations and in most cases is completely unnecessary as will give practically the same result.

Keep in mind that you don't really care whether or not 1 spell power is worth .0781 crit or 0.782 crit, as not only both values will pick the same items in 99.9% of the cases, but the error caused by the assumptions you made for the calculations is probably much bigger than this kind of error anyway. It's like using a 10X zoom sniper scope for a shotgun - you can aim exactly where you want to hit 100s of meters away but you're still going to spread all over the place.

I did another interesting experiment today. Instead of looking at a model without latency combos and one where they're always executed perfectly I instead looked at a model where there is a 50% chance that the combo will be successful. I also updated the AB base damage as is current on PTR, so overall numbers will be a bit higher.

The highest dps cycle becomes AB spam with MBAM-ABar combo on proc at 3 stack, if ABar procs again immediately follow with just MBAM. This gives 5090.468 dps, 431.850 mps. Looking at the best pair of cycles according to dpm tradeoffs we find the best pair (at 300 mps barrier) to be ABx3-ABar with MBAM-ABar at 3 stack (4852.069 dps, 250.270 mps) and AB spam with MBAM-ABar on proc at 3 stack (5089.973 dps, 411.778 mps). This pair has a dpm tradeoff of 1.473 dpm. For comparison ABx3-AM-ABar regardless of proc is 4812.87 dps, 228.12 mps, 1.50 dpm.

It is interesting that while ABx3-ABar with MBAM-ABar only on proc has a better dpm tradeoff, when we use the cycles in a full linear program with cooldowns, the ABx3-AM-ABar still wins. Only when reliability of executing the combo drops to 35% does ABx3-ABar actually win over.

Fascinating...from my tests (an actual caster with some fallability), I was able to execute ~96% of ABar after unhasted AM (in the AB3->AM->ABar cycle) and ~71% after MBarAM->ABar. Would you mind terribly running an experiment for something ~80% combo proficiency? It would seem that AB3->AM->ABar still works fine and AB spam in short-burst situations still holds, but I would be interested in seeing the dpm and mps numbers on those.