I'm going to keep this simple, and a bit abstract. This thread exists as an explication of a simple concept that I often see grossly misunderstood. The application is most simple when applied to a Mage casting one spell (to some extent, this thread is a reaction to the Mage +hit one elsewhere on this forum), but the concept is extensible to any class. At the very least, this is the sort of analysis most applicable to comparing different stat choices.

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The following are fixed parameters:
A: Base damage of a shot, with 0 +dmg (substitute AP if you wish, everything which follows is identical).
B: Damage coefficient.
C: Critical bonus.

The following are variables:
d: +dmg
h: chance to hit, i.e. 1-(chance to miss)
c: chance to crit
s: haste

Expectation damage per unit time is:
E = (A+Bd)*(h+Cc)*(1+s)

Taking partial derivatives:
dE/dd = B*(h+Cc)*(1+s)
dE/dh = (A+Bd)*(1+s)
dE/dc = (A+Bd)*C*(1+s)
dE/ds = (A+Bd)*(h+Cc)

These derivatives, whatever form they take for your class, are all you need to know to evaluate gear options. Further analysis (such as using a spreadsheet to compute expected absolute DPS taking a variety of different factors into account) serves only to muddy the results.

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A fully-worked example:

In my 10/48/3 build, let's say I have +1000 to Fire, 0 haste, 130 HR (10.3%), and 300Int/240CR (16.9%).

Evaluating parameters (note that all multiplicative bonuses, such as Fire Power and Imp. Scorch, are irrelevant):
A = 719
B = 1.05
C = 1.1 (Ignite)

Variables:
d = 1000
h = 0.963 (Elem. Precision)
c = 0.259 (Critical Mass, Pyromaniac)
s = 0

Results:
dE/dd = 1.05*(0.963+1.1*0.259) = 1.310
dE/dh = (719+1.05*1000) = 1769
dE/dc = (719+1.05*1000)*1.1 = 1946
dE/ds = (719+1.05*1000)*(0.963+1.1*0.259) = 2208


Dividing out relative itemization costs, I find the value of each stat:
+all damage: 1.31/0.855 = 1.53
+Fire damage: 1.31/0.7 = 1.87
Hit rating: 1769/(12.62*100) = 1.40 (factor of 100 because these things are expressed as a % in-game).
Crit rating: 1946/(22.08*100) = 0.88
Haste rating 2208/(21.0*100) = 1.05

Which is all I need to keep in mind when evaluating or comparing items.

Doing the same for a Shadow Priest spell priority (can't even use the word "rotation") is considerably harder.

Certainly not impossible....... but perhaps just a little more complicated.

Personally, I prefer to write simulators..... but I can understand why folks would want to "plug-n-chug" with a pre-made spreadsheet.

How can you tell? The subject thread doesn't indicate class applicability nor does the OP have his/her profile properly completed. I'd summarily ignore this thread.

Uh, what? The OP has an armory link in his profile, and the post is quite clear about how to apply it -- you're looking at the effect on single spells as per the parameters. Obviously you can't summarize a SPriest rotation with this system, but for any class that primarily uses a single spell to DPS, it's a good guide.

On the contrary, I'd rather recommend not ignoring Arawethion's posts/threads.

FYI, I reported this post. Maybe Praetorian is correct, but if so, the subject line and/or the begnning statement should be completely reworked.

This does not make sense for melee attacks where weapon speed, weapon skill, set bonuses, and procs are very important as well.

Please make very obvious that this is not supposed to be a generalized formula in any sense, and take out statements such as "This is all you need to know to evaluate gear options."

I hope other people get a lot more value from your post than I did, and if you rework the verbiage to reflect what I wrote above, perhaps other people won't immediately say "this is a complete waste of a post" like I did before reading Praetorian's defense of it.

You're right (above), I worded it too generally. Will change after raid. It's obviously not worth trashing the whole post over.

No, this is part of the whole point. Though the computation may be much more difficult for some classes than it is for Mages, all of the things you mentioned can be incorporated somehow. I'd imagine it's not extremely difficult for, say, Rogues; you'd just have two terms in the basic equation except one. Especially in the cases where everything is linear (nearly everything in this game), the only thing standing in the way is a little algebra.

I think it's really useful for elemental shamans - we really do use only one spell for dps. I always thought about change in dps / change in crit, holding AP constant, or vice-versa. The same obviously applies with spell damage.

Question is, when selecting gear - how do you weight int, stam, mp5, and other non-damage stats?

Correct me if I'm wrong, Arawethion, but shouldn't your damage coefficient (B) for the 10/48/3 build be 1.035 [((3.5/3.5)+.15)*.9] instead of 1.05 [(3.5/3.5)-.10+.15]? Most of the theorycrafting I've seen has treated Improved Fireball's coefficient penalty and Empowered Fireball's bonus as multiplicative instead of additive.

I thought I remembered this board deciding a long time ago that they were additive. Not at all certain, though.

I am not sure if this has been mentioned, but a very knowledgeable guy has made a quite impressive wow raid simulator, check it out here:


SimulationCraft - Shadowpriest.com Wiki

you can deal with 3 second lock outs of shared wf procs using this? I believe you if you say you can, but I'd have to see it to understand it.

The average proc rate per minute can still be calculated, right (I'm not saying that I've done it)?

Or for the Shadow Priest example, assume that your DoT's are always up, and use an empirical approximation for how many Mind Flays you cast per minute, and throw it all in.

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DPS spreadsheets exist for basically all classes at this point. In order for them to exist at all, the computations for the more complex abilities in the games have been performed to at least some degree of accuracy. When people compare gear options, they usually look at a DPS spreadsheet for a computation of their theoretical absolute DPS output in a variety of setups, and compare the gross values. This has a number of problems:
1) The underlying reasons why one item is better than another are not self-evident. Thus people choose items without a real understanding of how each stat contributes to their total output.
2) Many factors that might be entirely unnecessary to answer the actual question at hand may be taken into account. This not only might throw an arbitrary bias in favor of certain stats or items, but introduces computational or other propagating errors due to the frequent approximations in the process of overall DPS estimation.
3) Ultimately, it doesn't teach people to take a logical approach to item selection. There are tons of people who can't correctly and coherently explain to you whether 1 crit or 1 hit would be better for them at the moment. Much more frequently, in my experience, you get an answer which is more along the lines of, "well I have Ruin, so I just want to stack crit out the wazoo" or "spell haste is great! It makes all my spells faster!"

Instead, if the tools to exists to at all model the DPS of your class, the capacity will exist to throw some simple calculus at them and have a set of coherent results that not only further your understanding of your class, but allow you to objectively evaluate gear options in a direct and accurate way.

I totally agree with your last statement.
I think its far better to pick every item on its individual merits using a simple stat weighting system than to have some 'stat' that you decide is the good one, and try to pick a set of gear with that in mind.

I hear so many melee DPS talk about how good 'AP' or 'Hit' or 'Crit' is for their class/talent and then use a one eye'd aproach to gearing up that stat.

Casters may find the following tables useful:

SimulationCraft/Scaling - Shadowpriest.com Wiki

Hi, long time lurker, first time poster. I may be wrong, but here's my take on the current Windfury.

Since Blizzard linked the MH/OH Windfury procs, I've been modeling it as a single Poisson distribution. To calculate the average time between procs, you merely need the MH weapon speed, OH weapon speed, and a rough estimate of the percentage of time you spend in Flurry. From these, you can calculate the swings-per-second you generate with each weapon and from there, the mean time between WF procs:

MH = Main Hand Speed
OH = Off-Hand Speed
F = Time Spent in Flurry

MHS = MH swings/sec = F*1.3/MH+(1-F)/MH
OHS = OH swings/sec = F*1.3/OH+(1-F)/OH
Mean time between WF procs = 1/(MHS+OHS)/0.2

Since a Poisson distribution is memoryless, we can factor in the 3-second internal cooldown by just adding 3 to the mean time:

Actual mean time between WF procs = 1/(MHS+OHS)/0.2+3

For MH=OH=2.8 and F=0.6 (corresponding to about a 30% crit rate - there's a formula for F if you're interested), the proc rate is about once per 8.93 seconds.

Okay, back on topic: I've been using the same concept of differential analysis for my mage spreadsheet, and am very glad to see that my weighted values correspond closely to Arawethion's. SPriest and Warlock calculations will be more complicated, but not grossly so: after figuring out the weights of the stats for each spell, they merely need to weigh them further by the percentage of time spent casting spell X, spell Y, etc.

This tool was very valuable in the days of DM blues vs. Arcanist, however Blizzard nowadays is much more adroit with its stat allocation. Unless the item has an overabundance of Int/Stam or if there's Spirit (hello mage sets!), you generally can't go wrong with grabbing the highest ilvl item available.

While it's difficult to use this information to determine the optimal DPS cycle for a shadow priest, I'd assume most priests have a good feel for that already: keep DoTs up, then use cooldown damage spells, then mind flay.

Even though this just gives relative stat values, you should be able to convert all stats into "X points of stat = +1 DPS" for every spell you use in a rotation. Then using a mod like recap you can determine what percent of combat time you spend casting each spell. Just compute the average of each stat for each spell, weighted by the percent of time that spell is cast. That should give a single composite stat rating for an spell cycle you actually use.

I actually use (partial) differentials to calculate DPS increases from gear too. Heck I even did it before even knowing what partial differential is (very long time ago), but knowing how to do it makes things even easier - especially when everything is linear when you keep the other variables constant.

Another thing to point out is that the other variables aren't really constant, however when you upgrade your gear the change in your DPS is pretty much always << your current DPS, therefore if E = (A+Bd)*(h+Cc)*(1+s), then you can say that if you change s enough to give X DPS increase or change d to give X DPS increase - while changing s enough to give X/2 dps increase while also changing b enough to give X/2 DPS increase, the DPS increase may be more than X, but in reality it'll be almost X, as if the change in dps is << the actual dps then the difference between the real dps increase and X is of the same scale (X/DPS)^2 which since X<<DPS means (X/DPS)^2<<X so X+(X/DPS)^2~=X

While my mathematical terminology may be a bit off (english isn't my first languege), that assumption is very useful for comparing items. Of course when you make that assumption you have ot remember it only holds true when the change to DPS is << the actual DPS, so when your DPS is changed significantly, you have to recalculate the derivatives. Example for such a situation is when you add up flask+oil+food+totem for spell damage.

This can also be used for any spell/attack rotation. Any DPS can be worked into a linear function (or at least a linear approximation even though I can't think of any case where you can't make a true (when holding all but 1 variables constant of course) linear function such as the one showed in the OP), and then you can calculate the partial derivatives of that function by the each variable.

Having a nice linear function for your dps analysis just means that the calculated slope (your partial differential) is valid over a larger region.

Given current WoW mechaincs, it is probably safe to say that we can always determine a slope for one dimension of the dps surface at any point on that surface.

It wouldn't be easy. I can only give a basic sketch, since I don't know Shaman DPS in detail.

Start with three terms of the basic form given in the OP: one for white hits, one for Stormstrike, and one for WF procs.

To model Flurry, I imagine you'd make an estimate of Flurry uptime that looks something like (1-(1-c)^3). (Note that the first-order Taylor expansion of this is simply 3c, which might be a perfectly acceptable approximation for taking derivatives).

The estimation of Flurry uptime allows you to head over to Desiderata's post. You can compute the mean time between white swings. If you know the total number of white swings and Stormstrike hits per minute, you can make the Poisson model for the expected Windfury procs per minute.

Unleashed Rage uptime can be computed in a similar way. Make a Poisson estimation that your crits are randomly spaced in time, and compute the chance that you haven't crit in the previous 10 seconds.


All in all, you'll wind up with a bunch of factors tacked on to basic formula, all of which are themselves functions of c, h, and s. Take the derivatives (using Taylor if necessary to drop useless high-order-terms) and go.

Hrm yes, there is a mistake. I forgot that WFs proc off of hits and not swings--dividing my final result by your hit rate (0.76 by default, more with talents/items) should bring you closer to reality.

Stormstrike poses an interesting problem: you can mash it as soon as the cooldown is up, or you can try to time it to go off as you exit the 3s WF lockout. The former is easy to model--just add an extra 0.1 swing/sec to the term (MHS+OHS)**. Doing the latter would refute my original assumption that the system is memoryless, and you would be better off writing a simulation to obtain the procs per minute.

You're quite right, melee dps models have greater complexity than caster models. Enhancement Shamans and Survival Hunters both have nonlinear systems with feedback. It is possible to create fairly accurate models for these on a spreadsheet, but considerably harder to distill the interactions of the mechanics into a single, elegant formula.
Another excellent point--differential analysis doesn't factor in weapon damage and weapon speed, even though they are likely the biggest factors in determining Enhancement dps. Again, the method works better for casters, whose attack method is independent of gear selection.

**Note: here I'm assuming that Stormstrike is counted as a single attack and has a 20% chance to proc WF. But since it displays two numbers, the proc rate may instead be 1-0.8^2 or 36%. Not sure if there's been any work done on this, but I'm too lazy to look it up.

Alright, the easier to answer ones first... :-)

Not only that, but when you're dual wielding, WF proc chance per hit when you're outside of the 3 second cooldown is actually 36%, as opposed to the 20% you'd have with only one weapon imbued with windfury.

You are very quickly coming to the conclusion that we shaman made, and why the vast majority of shaman dps modeling happens through simulators instead of a formula (or set thereof) :-)

I'd throw warriors in there with the feedback that rage adds to the computations.

Stormstrike is intersting. If the mainhand lands (assuming dual wield), then they both can proc windfury. If the first stormstrike results in a dodge/parry, the second one never activates at all. A further wrinkle.

Further complicating matters is that you are in flurry a lot more with slow weapons than fast weapons. Very counterintuitive to the Blizzard forum trollers, but true.

Take a crazy example - with a single 10 speed weapon (yes, a 10 second swing timer to make this very easy to understand), and a 30% crit chance, in any 10 second interval I will have:

1 white attack
1 stormstrike attacks
on avg 1 windfury attack.

That's 3 attacks. Your long term flurry uptime will be close to 100%, since the yellow attacks dont eat flurry charges, they only produce them. Only 1 attack per 10 seconds could consume the flurry charges, with 3 attacks possibly procing them. That means once flurry procs the first time, you have 30 seconds worth of time ( = 8 attacks) to get another crit.

Take the same crit rate, but with a 0.1 speed weapon. Your long term flurry uptime will be much closer to the advertised (1 - (1-c)^3) = 65% advertised, since a huge percentage of your attacks are now both consuming and refreshing flurry.

Same crit rate, different speed weapons = a 30% flurry uptime differential. The example is absurd in absolute values, but rock solid in the theory behind it and the point it makes.

For more fun, see the thread linked below about how flurry doesn't even work like it's supposed to in game. Calculations don't really reflect how the game actually treats flurry.
[Melee Combat] How does Flurry Work?

Ok, on to the harder to answer bit!

see above.

I agree with this. My issue is in the formula whereby adding crit, you not only add UR procs, but you add extra attacks per minute via flurry, which then in turn are also benefitted by having UR up. I dont see an easy way to discern the value of this through the formulas given.

No. You *have* to include weapon speed. You also *have* to include which weapon is which speed. Having a 1.8 speed mainhand and 2.6 speed offhand will make for hugely different results than a 2.6 speed mainhand and a 1.8 speed offhand. You also *have* to have some way to account for the fact that different attacks use different factors -

Using the example above of Yellow attacks not glancing, not missing, with white attacks doing both, and the +hit derivative is meaningless if you choose just one.

If you choose to have 2 separate differentials added together, you must then weight them.

Please understand that as confrontational as I may appear about this, I'm trying to find every hole in the logic possible. It's what I expect of anyone looking over my own shaman simulator and other people's shaman dps calculations and simulators. If they can't answer every concern, they can't be correct. So, it's not personal, it's just that shaman are in a desert looking for the oasis of truth, to be melodramatic, and we've seen enough mirages already that we'd rather not pin hopes on something that can't be either
a) useful due to theoretical limitations (not having a sound theory behind the calcs)
or
b) useful due to practical limitations (having to come up with hugely complicated formulas, or thousands of person-hours of testing)