[Math] Paladin threat generation

Most of the analysis in here, and then some, is summarized in the spreadsheet at
http://www.29tharmor.com/WoW/Tankadata.xls

Still an ugly work in progress, but I've crunched my way to some numbers that I don't remember seeing anyplace else - so I'm now inviting comments and critiques.

Black calculations are for a paladin with a Continuum Blade and +40 SD enchant for a grand total of +161 spell damage. Red calculations are for a paladin with no spell damage whatsoever.

Weapon
Assume Continuum Blade, and we get 41.4DPS + 37 more from attack power, so 78 DPS from weapon. Since this is a pretty minor contribution to our total damage I'll just pull 50% out of the air for the mitigation number; furthermore 15% of our attacks fail due to miss/dodge/parry leaving us with 33 DPS and 33 TPS from white damage.

Seal of Righteousness
Seal of Righteousness on a one-handed weapon. 22 + (2 * (WeaponSpeed*10) - 30) + (WeaponSpeed*0.092*SpellDamage) = Damage per Second

Note: Threw in a factor of 0.85 because about 15% of our attacks fail to connect.

0.85 x (22 + (2x18-30) + 1.8x0.092x161)=0.85 x (22 + 6 + 27) = 47 DPS = 89 TPS.

0.85 x (22 + 6) = 24 DPS = 45 TPS

Holy Shield
155 per block + 0.05xSD

Pessimistic: 1 block per 10 seconds = 155 + 0.05x161 = 163 damage per block = 16 DPS = 39 TPS

Conservative(Realistic?): 2 blocks per 10 seconds = 32 DPS = 78 TPS

155 per block = 16 DPS = 38 TPS


Consecration
64 per second + 0.96x161/8 = 83 DPS = 158 TPS

64 per second = 64 DPS = 122 TPS


Judgement of Righteousness
Average damage 218 + 0.5x161 = 299 damage, every 8 seconds means 38 DPS = 72 TPS but 5%-ish are resisted, so we get 36 DPS = 68 TPS.

218 per 8 seconds = 27 DPS = 52 TPS

Total so far (not updated yet):
DPS = 33 + 47 + 32 + 83 + 36 = 231 damage per second
TPS = 33 + 89 + 78 + 158 + 68 = 426 threat per second


DPS = 33 + 24 + 32 + 64 + 27 = 180
TPS = 33 + 45 + 76 + 122 + 52 = 328


So a paladin with zero spell damage spamming all abilities at max rank with 3/3 Improved Righteous Fury should average about 328 threat per second.

Adding 161 spell damage increases that number to 426 threat per second, which indicates that 1 spell damage converts to 0.63 threat per second in the limiting spam-max-rank case.

Cleaned up: Paladin generates 252 TPS + 38 per HS block per cycle + (0.63 x Spell Damage) + (82 if Exorcism applies). Remember to multiply by 1.05 if you have 1-H Specialization.
These values assume the target has 50% physical mitigation, 5% chance to be missed, 5% chance to dodge and 5% chance to parry, and resists direct-damage effects 5% of the time. Also assumes that on every application of Holy Shield the tankadin blocks twice.

Here are some hopefully-useful data I use when calculating threat stuff; they haven't yet been updated for the mana cost changes in 2.3.

Mana Consumption
A tankadin running flat-out will be cycling some or all of the following abilities. (Mana costs assume the tank has Benediction.)
Seal of Righteousness - 221 mana every 8 seconds = 27.6mps
Seal of Vengeance - 213 mana every 8 seconds = 26.6mps
Judgement - 124 mana every 8 seconds = 15.5mps
Holy Shield Rank 1 - 135 mana every 10 seconds = 13.5mps
Holy Shield Rank 2 - 175 mana every 10 seconds = 17.5mps
Holy Shield Rank 3 - 215 mana every 10 seconds = 21.5mps
Holy Shield Rank 4 - 280 mana every 10 seconds = 28.0mps
Consecration Rank 1 - 120 mana every 8 seconds = 15mps
Consecration Rank 2 - 205 mana every 8 seconds = 25.6mps
Consecration Rank 3 - 290 mana every 8 seconds = 36.3mps
Consecration Rank 4 - 390 mana every 8 seconds = 48.8mps
Consecration Rank 5 - 505 mana every 8 seconds = 63.1mps
Consecration Rank 6 - 660 mana every 8 seconds = 82.5mps
Exorcism Rank 1 - 85 mana every 15 seconds = 5.7mps
Exorcism Rank 2 - 135 mana every 15 seconds = 9mps
Exorcism Rank 3 - 180 mana every 15 seconds = 12mps
Exorcism Rank 4 - 235 mana every 15 seconds = 15.7mps
Exorcism Rank 5 - 285 mana every 15 seconds = 19mps
Exorcism Rank 6 - 345 mana every 15 seconds = 23mps
Exorcism Rank 7 - 400 mana every 15 seconds = 26.7mps

A maximum-threat rotation with perfect cooldown management and no Exorcism will cost 154 mana per second, sustainable if you're taking 1540 DPS.

Damage Output (DPS) and Effect of Spell Damage D
These figures assume 2/2 Improved Judgement and 2/2 Improved Holy Shield.
Seal of Righteousness - 27.8 + 0.09D
Seal of Vengeance - 44 + 0.05D
Judgement of Righteousness - 27.3 + 0.09D
Judgement of Vengeance - 66 + 0.054D
Holy Shield Rank 1 - 19.8 + 0.017D
Holy Shield Rank 2 - 28.9 + 0.017D
Holy Shield Rank 3 - 39.3 + 0.017D
Holy Shield Rank 4 - 52.1 + 0.017D
Consecration Rank 1 - 8 + 0.06D
Consecration Rank 2 - 15 + 0.075D
Consecration Rank 3 - 24 + 0.095D
Consecration Rank 4 - 35 + 0.115D
Consecration Rank 5 - 48 + 0.12D
Consecration Rank 6 - 64 + 0.12D
Exorcism Rank 1 - 6
Exorcism Rank 2 - 10.8
Exorcism Rank 3 - 15.4
Exorcism Rank 4 - 21.5
Exorcism Rank 5 - 27.7
Exorcism Rank 6 - 35.6
Exorcism Rank 7 - 43.7 + 0.029D

Threat Output (TPS) and Effect of Spell Damage D
These figures assume 2/2 Improved Judgement and 2/2 Improved Holy Shield as above, and 3/3 Improved Righteous Fury. I'm also sticking with an overall threat coefficient of 2.25 for Holy Shield until proof of the multiplicative 2.565 figure is shoved in my face.
Seal of Righteousness - 52.8 + 0.171D
Seal of Vengeance - 83.8 + 0.095D
Judgement of Righteousness - 51.9 + 0.171D
Judgement of Vengeance - 125.4 + 0.103D
Holy Shield Rank 1 - 44.6 + 0.038D
Holy Shield Rank 2 - 65.0 + 0.038D
Holy Shield Rank 3 - 88.4 + 0.038D
Holy Shield Rank 4 - 117.2 + 0.038D
Consecration Rank 1 - 15.2 + 0.114D
Consecration Rank 2 - 28.5 + 0.143D
Consecration Rank 3 - 45.6 + 0.181D
Consecration Rank 4 - 66.5 + 0.219D
Consecration Rank 5 - 91.2 + 0.228D
Consecration Rank 6 - 121.6 + 0.228D
Exorcism Rank 1 - 11.4
Exorcism Rank 2 - 20.5
Exorcism Rank 3 - 29.3
Exorcism Rank 4 - 40.9
Exorcism Rank 5 - 52.6
Exorcism Rank 6 - 67.6
Exorcism Rank 7 - 83.0 + 0.055D

Efficiency
Finally, the threat outputs divided by their respective mana costs.

Seal of Righteousness - 1.91 + 0.0062D
Seal of Vengeance - 3.15 + 0.0036D
Judgement of Righteousness - 3.35 + 0.011D
Judgement of Vengeance - 8.09 + 0.0066D
Holy Shield Rank 1 - 3.30 + 0.0028D
Holy Shield Rank 2 - 3.71 + 0.0022D
Holy Shield Rank 3 - 4.11 + 0.0018D
Holy Shield Rank 4 - 4.19 + 0.0014D
Consecration Rank 1 - 1.01 + 0.0076D
Consecration Rank 2 - 1.11 + 0.0056D
Consecration Rank 3 - 1.26 + 0.0050D
Consecration Rank 4 - 1.36 + 0.0045D
Consecration Rank 5 - 1.45 + 0.0036D
Consecration Rank 6 - 1.47 + 0.0036D
Exorcism Rank 1 - 2.00
Exorcism Rank 2 - 2.28
Exorcism Rank 3 - 2.44
Exorcism Rank 4 - 2.61
Exorcism Rank 5 - 2.77
Exorcism Rank 6 - 2.94
Exorcism Rank 7 - 3.11 + 0.0021D
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In my day, we didn't have gear. Or tanking talents. Or - actually, you don't want to hear this story.

Oooh. would be nice to have this as a simulator where you'd input weapons, spell damages, talents.... and possibly buffs!
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"Anatheme c'est la reroll de Benediction."
- Yshuta

My armory

So far this theorycraft is in pretty good agreement with what I'm getting from my threat meter; it normally puts me around 400-450 TPS, and if I'm getting Reckoning procs and the enemy's eating most of my shield charges it can go much higher.

Last night in Heroic Ramparts, against either a Ravager or a Destroyer, I broke 750 TPS because he was hitting me so fast. With Improved Holy Shield I would probably have hit 800.
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In my day, we didn't have gear. Or tanking talents. Or - actually, you don't want to hear this story.

The thing is that we have SPELL damage.
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I'd really love some simple formulas to help me evaluate the value of my gear.

(White DPS) *Reckoning coverage*Armor Reduction*Target Avoidance* Crit boost

+

(Base SoR DPS+ Coeff*spell)*Imp SoR * Reckoning Coverage* Target Avoidance * RFfactor

+

(Base JoR Damage + Coeff*Spell)*Imp SoR * Target resist * RFfactor / sec between use

+

(Base Consecrate DPS + Coeff*Spell) * RFfactor

+

(Holy Shield base + Coeff*Spell) * RFfactor * Charges/sec


Then we could factor down to some simple formulas based on our talents:

Low Mana Seal and Shield alone - X + Y% spell + Z% white DPS
Extra From frequent Judgements - add X plus Y% spell
Extra from using Consecrate - add X plus Y% spell

In low mana situations (hello to YOU AD nerf!) spelldamage weapons may no longer be king.

SHield slam: scales with block value

Block value is multiplied by 1.3 due to their shield mastery
Threat is multiplied by 1.45 due to Defiance and Defense stance
Crit multipliers are assumed to average out miss/dodge/parry
Their 10% 1h spec increase is balanced by defensive stance's 10% loss
Shield slam has a 6 second cooldown so divide by 6 for TPS

So every point of shield block value provides more than 0.3 TPS for a Shield Slamming warrior (reduced by armor of the target)

But wait, there's MORE! Block value is ALSO cheaper than spell damage in regards to itemization (as cheap as stamina):

http://www.wowwiki.com/Formulas:Item_Values

.855/.65 = 1.3 scaling amount compared to a point of +damage

So on a squishy target, if an identical item has +spell for us and +block value for them, they get the mitigation of block, as well as shield block's threat:

(.3*1.3):0.63

Shield Block Value provides Shield Slamming warriors with 64.8% of the scaling threat that Spell Damage provides for us.

Gonna have to post this to the Blizz forums and ask them to PLEASE give us some stats that do double duty, as agi and Block Value do for them.

They did it in deep holy, but there are already plenty of folks specced deep Holy, and we can't get that and push crushings off at all.

It would be interesting to have these numbers to post for all those people who only use Sun Eaters or King Defenders and think they do just as well.

I'm pretty sure that in a standard single target tank and spank situation I'm putting out between 600 and 800 TPS, big range I know.

Now I readily see posts from Warriors who state they are in the 600-700 range and a few in the 700-800 range while I've also seen some state they can hit between 800-1000. I've seen warriors in my own raids put out 600-700 and even spike up to close to 900 and those are my OTs, for whatever reason our 2 MTs don't run KTM.
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I can put out about 700ish threat on most encounters.

Edit: This is based off of last night's Kara I led.
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I am fucking awesome.

These were Exorcisable targets and I use Seal of Crusader, I am not sure if you considered that.

My mainhand is a Stormcaller with +40 spell damage.

My Armory Profile
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I am fucking awesome.

So my total spell damage would be 272 in my Tank Gear... so 190 from Judgment of the Crusader, makes 462 in my tank gear; plus Exorcism.

The formula does not account for Reckoning Procs?
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I am fucking awesome.

I could throw it in, but it's a pig determining how much time an ability like that is up since you can get a proc while it's already up. I could get a number using Excel and Monte Carlo techniques but it sounds like too much work and would require numbers for the bad guy's attack speed and the tank's avoidance (the more you avoid, the less Reckoning you get).

Autoattacking with SoR (Continuum Blade) gives 33 TPS from white damage and 45+0.27D TPS from the seal.

You generate that much additional threat while Reckoning is up.

So if Joe Tankadin does 328 + 0.63D threat per second, he'll do 396 + 0.90D TPS during Reckoning.

If Reckoning is up, oh, I don't know, a quarter of the time, your average-over-time threat would be

328 + 0.63D + 1/4(33) + 1/4(45+0.27D)

328 + 0.63D + 8 + 11 + 0.07D

347 + 0.70D

(I don't like the gear-specific terms in this, and since it's actually attracting some notice I'm going to come up with a general version so you can plug in the numbers from your own weapon. Not today, though, 'cause it's kind of busy.)
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In my day, we didn't have gear. Or tanking talents. Or - actually, you don't want to hear this story.

You also forgot crits on melee and Judgements.