Theck's MATLAB TPS analysis - WotLK/3.x

[Thalastor82]

JotJ adds 464 damage (SoV proc) every 9 seconds, for around 51 DPS. That'll go up a bit with new gear, since we'll have more STR. I'd guess about 55-60 DPS with 264 gear.

Triple that for SoCommand, of course.

As for the SoCom as a tanking seal question, I just ran about 4-5 heroics with SoCommand, and was pulling 4k+ dps on most trash packs. I managed to break 5k dps on a few of the bigger pulls. That's with 0/5 Reckoning as well - I'm guessing with 3/5 Reckoning instead of Divine Guardian, I could break 6k.

I just tried the SoComm in instances and I must say that it is awesome on packs, this forum proved again to be really useful.

But at the end of the instances my dps-meter (Skada) recorded a global 3k DPS (0/53/18, 0/5 Reckoning) while in blocking stuff (~3k blocking value, 2000force plus 300 of the Greatness proc), which is far below the reported 4,5,6k DPS ...
In any case, I never had any problem with aggro, event with my guilde-mate fury warrior which was used sometimes to pull mobs from me

I'm looking forward for the new calculations, thanks again!

[Candiru]

Hmm, with ShoR procing seals, we now have a seal procs 7 times in a 18s rotation window. IE one every 2.6s as well as the ones from our natural swings. This is going to make slower weapons even more attractive...

[theckhd]

But at the end of the instances my dps-meter (Skada) recorded a global 3k DPS (0/53/18, 0/5 Reckoning) while in blocking stuff (~3k blocking value, 2000force plus 300 of the Greatness proc), which is far below the reported 4,5,6k DPS ...

The 5k+ DPS figures are for trash packs, not overall instance data. Any time you hit a boss that has to be single-targeted, your dps will drop back down to reasonable values.

...Unless you chain pull the bosses along with everything else, I guess...

[theckhd]

3.3 Paladin Ability Damage/DPS/Threat/TPS Breakdown

I've decided to do something a little different with the ability damage calculations this time. Usually, we look at each ability in isolation, after the effects of boss avoidance and crit chance. While I'll still include those, there are some other, perhaps more relevant ways to present the data.

Note: For this and subsequent 3.3 analyses, I'll be using a boss swing timer of 1.0, or 1.2 after JotJ. This doubles Holy Shield DPS, but doesn't change any of the conclusions in the text. The plots are all updated to reflect the new values.

Setup
Rotation: 969
Seals: SoV
Glyphs: V only, VJ and VA for glyphed Judgement and AS calculations
Talents: 1V+3C build, on a "no-DUH" mob type
Gear: T10 Progression Set
File: calc_ability_dmg.m

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Ability  Raw Dmg.  Net Dmg.   Glyphed   Net Threat   Glyphed
ShoR       3608      3833         0       9866          0   
HotR       4294      4561         0      11740          0   
Cons       4755      4197         0      10803          0   
JoV        2628      3023      3326       7781       8561   
JoC        1300      1495         0       3848          0   
JoR        2516      2895         0       7452          0   
AS         2391      2750      5501       7079      14160   
HoW        3250      3739         0       9624          0   
SoV         502       590         0       1519          0   
SoC         547       582         0       1498          0   
SoR         506       480         0       1236          0   
HV         6239      5914         0      15223          0   
HS         1281      1131         0       2911          0   
Exor       3103      3071         0       7905          0   
Melee      1346       953         0       1363          0   

The first new feature is the "Raw Damage" column in the table above. Raw damage is the average damage the ability does on a successful non-crit attack. There's a few reasons for including this, one of which is to see how boss avoidance and player crits change the expected outcome. More importantly, I wanted these numbers tabulated and put someplace that I can reference them later on, and it's closer to what you'll actually see in your recount/WoL parses in-game.

"Net Damage" is the average damage once crits, boss resistance, and misses/dodges/parrys are incorporated. As you can see, abilities that can't crit (Cons, Holy Vengeance, SoR, HS) tend to do less net damage than raw damage, and abilities that can crit but can't miss (SoV procs mostly) do notably more net damage than raw. Abilities that can be crit and can miss tend to go up overall.


Threat graph
With the diminishing returns looming on ShoR and higher DPS weapons from ICC, HotR has convincingly overtaken ShoR for our highest damage ability (short of Glyphed AS). Interestingly they've both outpaced Hammer of Wrath, which puts HoW in an odd place as far as substitutions. While it's a 6-second cooldown, we don't want to use it in place of our 6's. That means that we either substitute it for Consecration (if we're mana-starved) at a slight TPS loss, or substitute for Judgement or Holy Shield for a slight DPS gain if we want to go all-out DPS on a boss below 20%.


Next, let's look at DPS and TPS.

DPS/TPS
To plot each ability's DPS and TPS contributions, we divide by the cooldown for a realistic 969 scenario. For ShoR, HotR, HoW, Consecration, and Judgement this is easy: just divide by 6 or 9 depending on what slot it fills in the rotation. For Avenger's Shield, we divide by 36, since the most likely way to weave it in is to substitute for a 9-second ability every 4th rotation.

Melee swings get divided by the player's raid-buffed swing speed (which for a 1.7 speed weapon comes out to 1.334 seconds after parry haste (20.3% parry) but in the absence of Reckoning). The SoV proc gets divided by the proc rate, which is calculated to include procs from melee attacks, HotR, ShoR and Judgement. This works out to
(1/1.334 + 2/6 + 1/9)^(-1)= 0.8375 seconds
The first term is the contribution from melee swings, the second from HotR & ShoR, the last from Judgement.

Holy Shield gets multiplied by its average procs per second, which is a slightly more complicated function:

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avg_HS_pps=min([8/9.*ones(size(boss_swing_speed)); (max([(player_block./100) 0])./boss_net_swing_speed)]);

This works out to whichever's smaller:

• 8/9 (if all charges are used up, 8 procs every 9 seconds) or
• (player_block./100)./boss_net_swing_speed (chance to block divided by time between incoming attacks, assuming block-capped).

In practice, this works out to 0.1848 pps in this gear set (41.33% block) for a boss swing speed of 2.4 (2.0 base), or an average of 1.66 procs every 9 seconds (or a proc every 5.4 seconds).

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Ability   DPS   Glyphed    TPS   Glyphed
ShoR      639       0     1644       0 
HotR      760       0     1957       0 
Cons      466       0     1200       0 
JoV       336     370      865     951 
JoC       166       0      428       0 
JoR       322       0      828       0 
AS         76     153      197     393 
HoW       623       0     1604       0 
SoV       704       0     1813       0 
SoC       695       0     1789       0 
SoR       626       0     1612       0 
HV        394       0     1015       0 
HS        209       0      538       0 
Exor      205       0      527       0 
Melee     714       0     1021       0 

Not much has changed since last time. HotR and ShoR are still on top, and now that I'm using 6 seconds for HoW's cooldown, it's in a close third place as far as activated abilities go. However, as we've seen before, it still gets relegated to being subbed in place of Judgement or Holy Shield. SoV procs still form a large part of our DPS and our second largest TPS source.

Finally, there's one more way we can look at the data. Up until now, we've treated the SoV procs as a separate entity. However, we know they're linked to our melee attacks and abilities. Skipping a Judgement doesn't just cost you Judgement damage, it also costs you an SoV proc. What happens if we divide the SoV procs up amongst the abilities that cause them?

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Calculation with SoV procs included in ability damage
Ability  Damage    Glyphed   TPS    Glyphed 
ShoR      4423        0     1897        0   
HotR      5151        0     2210        0   
Cons      4197        0     1200        0   
JoV       3613     3916     1033     1120   
AS        2750     5501      197      393   
HoW       3739        0     1604        0   
HV        5914        0     1015        0   
HS        1131        0     1039        0   
Melee     1543        0     2223        0       


I'm only gong to show two plots here, but they're two very relevant ones - Damage and TPS:


On the first graph, we see that the seal damage boosts Judgement up to be nearly as good as HoW for net damage per cast - with a little more expertise on the gear set they would be tied. This means that as our gear progresses a little more, we will likely reach the point where HoW is only a valid substitution for Holy Shield, at least until cataclysm gives us new ranks to play with.

On the second graph, I'd like to draw your attention to the Melee bar. Our melee attacks are our third highest TPS source after seal procs are included, higher than consecration or Judgement. That means it's incredibly important to keep ourselves in melee range if possible, even while moving or repositioning a boss. Ducking out of melee range costs us about 1600 TPS.

TLDR

• Gear scaling has lead us to the point where Hammer of Wrath should really only be substituted for Holy Shield, if ever
• Remaining in melee range is vital for threat, e.g. while repositioning bosses.

Hopefully I'll have time to get another post up around lunchtime, the rest will probably have to wait for tomorrow or Thursday.

[Iselian]

Now that makes things interesting. By the time we're killing Arthas, I think HoW will be far out of any use.. hell, I'll probably take it off of my keybindings tonight and quit worrying about it.

Curious though, do we know if HoW can be parried or dodged? I think I've seen it dodged before, but I can't recall it ever parried. I'm assuming the threat plots already take all that into account though.

[Argali]

Don't take it of your bars. It's still an instant cast ranged attack.

[Iselian]

Aye, it'll be there for snap at ranged times (ie, running around chasing Marrowgar) but it won't be nearly where it was (closer to "=" as opposed to "6" or "7")

[theckhd]

3.3 Talent/Spec Analysis - Single Target

First, we'll look at the breakdown of TPS per talent point spent.

We simulate the DPS and TPS for each of these talent configs, and calculate the value of each talent point through simple subtraction and division:

1. 5/5 Divine Strength and 3/3 Redoubt, but no other threat talents
2. #1 + 3/3 One-Handed Weapon Specialization (OHWS)
3. #1 + OHWS + 3/3 Touched By The Light (TBTL)
4. #3 + 3/3 Crusade
5. #3 + 5/5 Conviction
6. #3 + 5/5 Seals of the Pure (SotP)
7. #3 + 5/5 Reckoning
8. #3 - 5/5 Divine Strength
9. #3 - 3/3 Redoubt

For example, to evaluate Crusade, we subtract spec 3 from spec 4, and divide the result by 3 (the number of points in Crusade). Since OHWS and TBTL are so far ahead, we use those as the base spec with which to compare the other talents.

Setup:
Rotation: 969
Seals: SoV
Glyphs: V only
Talents: Various, in combinations to isolate the benefit of each talent individually.
Gear: T10 Progression Set
File: calc_talent_analysis.m

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Talent     DPS      TPS   (both per point)
Reck       51.0     94.4                 
Conv       25.5     58.2                 
Crus       42.2    100.6                 
SotP       40.6    104.4                 
Red        36.3     93.4                 
DivStr     71.4    171.5                 
TbtL       88.4    227.5                 
C(6%)      85.6    204.2                 
1HWS      119.7    284.1

Some interesting changes this patch. Everything has gone up across the board thanks to the STR boost in the gear set. SotP gained a little ground on Crusade thanks to the change to ShoR (it now procs seals), but Crusade is still solidly ahead for DUH mobs. Interestingly, we've finally gained enough STR from gear for TbtL to pass Crusade(6%) hands-down. And Redoubt has dropped behind both Crusade and SotP.

However, perhaps the most startling result is from Reckoning. I've changed the boss base attack speed from 2 to 1 in the code to reflect a more realistic Icecrown boss, which helped increase Reckoning's effectiveness. It now out-threats Conviction point for point, making Conviction our weakest threat talent.

The Redoubt question hasn't changed at all since last time, but I'll repost it here since this is the relevant analysis:

.... the value of each point in Redoubt will decrease exactly the same way block value does in the region of diminishing returns, and will drop to 0 once you're above the ShoR block value cap. Keep in mind that dropping the talent might bring you back into the DR region again though.

The BV break points at which the points in Redoubt actually become useless for threat should be easily calculated. The cap is 3161 BV, so if you let B be the base block value break point when 1.2*B=3161, then the break point with 3/3 Redoubt is 1.3*B=1.3(3161/1.2). Thus:
If you have over 3434 block value with 3/3 Redoubt, you'll see no difference in threat between 2/3 and 3/3 Redoubt
If you have over 3448 block vlaue with 2/3 Redoubt (3736 with 3/3), you'll see no difference in threat between 1/3 and 2/3 Redoubt
If you have over 3477 block vlaue with 1/3 Redoubt (4109 with 3/3), you'll see no difference in threat between 0/3 and 1/3 Redoubt

Next we'll run through the talent spec comparison again.

Setup:
Rotation: 969
Seals: SoV
Glyphs: V only
Talents: Various combinations of SotP (S), Reckoning (R), Conviction (V), and Crusade (C)
Gear: url=http://maintankadin.failsafedesign.com/forum/index.php?p=518670&rb_v=viewtopic#p518670]T10 Progression Set[/url]
File: calc_talent_analysis.m

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Spec         DPS      TPS      TPS(e)   Util
3S           4357     10405     10405     12
4S           4397     10507     10507     11
5S           4436     10609     10609     10
4S+3R        4560     10814     10814      8
5S+3R        4602     10921     10921      7
5S+5R        4702     11110     11110      5
5S+3R+1V     4630     10985     10985      7
5S+3R+3V     4686     11113     11113      5
5S+1V        4463     10670     10670      9
5S+3V        4517     10793     10793      7
5V           4366     10391     10391     10
1V+2C        4349     10360     10567     12
1V+3C        4392     10462     10776     11
3R+1V+3C     4555     10763     11086      8
3V+3C        4444     10582     10900      9
5V+3C        4497     10702     11024      7
3R+5V+3C     4666     11014     11345      4
5V+2C        4453     10599     10811      8
5V+3C-1T     4409     10476     10790      8

This first plot is sorted by "no-DUH" TPS. The next two are sorted by tree (Ret-heavy specs at the top and Holy at the bottom) and sorted by utility points, respectively:





Not much has changed here. SotP has pulled a little further ahead, but Crusade still dominates. The standard "4S vs. 1V+3C" comparison (comparing a 4/53/14 utility spec with a 0/53/18 that takes exactly the same utility talents in prot and ret) shows that 4S is about 50 TPS ahead for no-DUH mobs, but 200+ behind for DUH mobs. So Crusade is still king for Icecrown.

If you're not going deep enough into Ret to pick up Vindication or PoJ, then you can do pretty well with Seals of the Pure and Reckoning. 5S+5R actually beats out 1V+3C even on DUH mobs, and frees up some points for other Prot or Holy talents.

TLDR Summary:

• 0/53/18 provides the highest threat potential on DUH mobs, with or without Reckoning.
• For no-DUH mobs, specs with 5/5 SotP and 5/5 Reckoning are very strong, and will still be decent for DUH mobs (though about 200 TPS behind).
• As a general rule of thumb for maximizing threat, fill points in the following order:
  
  • OHWS, Divine Strength, and TbtL before anything else.
  • For DUH mobs, Crusade > SotP > Reckoning > Conviction
  • For non-DUH mobs, SotP > Crusade > Reckoning > Conviction.

[Awyndel]

Sky remains blue. Interesting read, thnx again Theck.

[theckhd]

3.3 Stat Threat Analysis - Single Target

Note: I've arbitrarily set the expertise and hit contributions from gear to 0 for this, so we get a clear picture of the entire range of rating.

The major change this patch has been ShoR moving to the melee table. Now that it can be dodged or parried, we expect that expertise should become a more important stat for us. This has some advantages - warriors and DKs like expertise, so it makes gear itemized with them in mind a little better for us than it was before.

For those interested in the "under-the-hood" mechanics of how this simulation works, see the 3.2.2 Analysis. Nothing in the code has changed since last time.


Setup:
Rotation: 969
Seals: SoV
Glyphs: V only
Talents: 1V+3C
Gear: T10 Progression set
File: calc_stat_tps.m

As usual, we'll look at STR first. For a point of reference, the unbuffed character-sheet STR of the default gear set here is 2070.

DPS plot

The amazing news here is that STR has finally been un-seated from the top spot. Above 1942 unbuffed character sheet STR, both Expertise (up to the soft-cap at 26) and Hit (up to the a few %, see hit graph) give better TPS per point of rating.

We had already expected this would occur for hit, but the real news is expertise. The ShoR change has drastically changed our reliance on expertise. Soft-capping expertise at 26 becomes highly recommended.

Also note that since this gear set is below the region of diminishing returns on ShoR, we don't see the DR on STR. We'll get to see this later in the BV plot though.

Hit:

DPS plot

No surprises here. Hit drops off when you reach the melee and spell hit caps. Everything else is pretty much the same, with slight scaling of damage-increasing stats until you reach the appropriate caps.

The interesting thing to note is that after about 50 hit rating, STR increases enough to equalize the two stats. Note that this is with 0 expertise rating from gear; at the expertise soft cap this number will go up. What this means is that if you're really optimizing for threat on a limited budget, you'd want to balance hit, expertise, and STR. On the other hand, it's hard to come up with a rule of thumb for this, since it will depend on your current STR, hit, and expertise values. In practice it will be more common to just gem STR if you want sustained TPS, and hit if you want snap threat. The difference in sustained will be small enough (~2-3 TPS per 10 points of rating or stat) that hit is probably the better choice.

Expertise:

DPS plot

The dark horse of this patch, Expertise has risen to top dog status with the change to ShoR. As always, expertise loses effectiveness when you hit the soft and hard caps (82 and 328 rating or 26 and 56 skill after talents, respectively), and damage-increasing stats scale differently above and below those caps (see the different slopes on the STR and BV lines in the three regions).

Past the soft cap, expertise drops back down to be roughly equivalent to AP for threat purposes. That would put it at a "don't gem" level as far as threat is concerned, though there are also avoidance benefits to consider.

Block Value:

DPS plot

Here we see the effects of the ShoR diminishing returns formula. We're actually closer to the edge of this, since the default set is sitting at 2216 BV.

The other thing to note is that BV has dropped a little bit across the board this patch, which is directly due to ShoR dodges and parries. STR suffers from this as well, but not as directly - it spreads your threat across many sources, so the ShoR change is a smaller relative effect. However, with BV you put all your eggs in one basket (ShoR), so the change is more noticeable. As a result, BV is generally not going to be our best option for threat (compared to STR) since most people won't be hard-capped on expertise or hit-capped.

The discontinuity at 2400 is due to the piecewise nature of the function and the fact that I'm stepping by ~3 BV at a time in this simulation. The TPS value of each point of BV decreases linearly as we get more BV, which is exactly what we'd expect since the diminishing returns formula is modeled best by a quadratic. By 3161 BV, we find that BV no longer contributes any threat.


For the bar plots, I've decided to just generate one plot for 2300 BV. In the past, there were four threat stats that could all be the best in one configuration or another. Now, BV is basically out of the running, and the stepwise nature of hit and expertise makes it easy to determine whether either is significantly weaker than STR.





And finally, here's the single-stat values pulled off of the first graph (STR scaling) at 2000 STR (with 0 expertise and hit):

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         TPS per 10
Stat   ipoint    stat
Exp     37.86   37.86
Hit     31.80   31.80
STR     30.80   30.80
BV      21.88    7.11
AP      18.21    9.10
Crit    11.62   11.62
AGI     11.27   11.27
Haste    6.98    6.98
SP       4.93    4.22
ArPen    1.97    1.97

TLDR Summary:

• The ShoR change, while seemingly minor, made a huge difference in stat weighting.
• Expertise is now our best threat stat below the soft-cap of 26 (above that it's fairly weak).
• Hit is better than STR, point for point, after you reach around 2000 unbuffed character-sheet strength. But it will remain close, and they synergize such that adding one makes the other better.
• STR is our third best threat stat, but it becomes our best after we soft-cap expertise. Adding more expertise or hit will make STR even better.
• BV, while nice, isn't really something we should stack for threat. STR is better point per point, and doesn't push you into the DR range of ShoR as quickly. Obviously take what you can get as far as gear goes, but if you get to choose between 10 STR or 31 BV, always take the 10 STR.
• General gearing strategy for threat:
  • Stack Expertise to 26 (soft-cap)
  • Stack hit rating to 8%
  • Stack STR

[theckhd]

I'll probably be busy for the better part of the day today, so chances are the rest will have to wait until tomorrow.

[Saiph]

Past the soft cap, expertise drops back down to be roughly equivalent to AP for threat purposes. That would put it at a "don't gem" level as far as threat is concerned, though there are also avoidance benefits to consider.

Just so I'm completely clear on this point, does that mean that below the soft cap it may be acceptable to gem expertise, or that at all levels you shouldn't gem for it? With the amount of defense on gear we are seeing now, would it be an option to gem expertise/stam in a yellow slot instead of defense/stam if you weren't ignoring the socket bonus on a particular piece for whatever reason (and below the soft cap)?

[Grehn|Skipjack]

Well, for one, expertise/stam gems are purple, and have nothing to do with yellow sockets.

[theckhd]

Just so I'm completely clear on this point, does that mean that below the soft cap it may be acceptable to gem expertise

Yes.

or that at all levels you shouldn't gem for it?

No.

• General gearing strategy for threat:
  • Stack Expertise to 26 (soft-cap)
  • Stack hit rating to 8%
  • Stack STR

[beornus]

<------ Has been twatted by this information

/bow you bow to theckd

thanks again for all the great work you do!