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KeyboardNinja
10.28.2012 , 05:08 PM | #1
Update: Corrected some of the probability munging.

Time for more theory-crafting with Tam! Today on the menu we have the question of "effective HP": what is it, and do we really care? For those of you who already know what it is, skip to the "do we care" section.

Definition

Effective HP is basically a measurement of how much "worst case buffer" a tank has. Some of a tank's defenses are RNG-based, such as defense and shield. Also, certain attacks will bypass some or all of a tank's defenses. Thus, there are scenarios in which a tank is literally just a meat shield, absorbing damage with a (slightly) deeper health pool than the rest of the raid. Effective HP is a measurement of just how good of a meat shield a particular tank is.

The obvious starting point when measuring something like this is the amount of hit points a tank has. Let's take 26k as a pretty average number. This is an immediate buffer that is easy to reason about, but it's not the whole story. Even when RNG defenses fail to proc, or when attacks are bypassing said defenses, tanks still have forms of mitigation which will reduce the incoming damage by a certain percentage. We need to consider those as well.

Taking my shadow tank main as an example, we have two forms of static mitigation to consider: armor and resistance. Armor is the "Damage Reduction" percentage in your character sheet, and it is a static reduction applied post-shield to incoming kinetic/energy damage. It does not apply to internal/elemental damage (e.g. Scream in TFB HM, or Stormcaller in EC HM). Resistance is really the internal/elemental analog of armor. This is usually much lower than the damage reduction, but that's ok, since very little endgame damage is internal/elemental!

My shadow has the following numbers (with stim):
  • HP: 26588
  • Armor: 40.48%
  • Resist: 23%

If we consider TFB HM, EC HM and Nightmare Pilgrim, we can determine the ratio of damage that is kinetic/energy vs internal/elemental. This will be important in determining exactly how heavily to weight armor vs resistance in our calculations:
  • Kinetic/Energy: 71.80 + 18.78 = 90.58%
  • Internal/Elemental: 9.41%

Thus, our net static damage reduction can be computed as follows:

0.9058 * armor + 0.0941 * resist

In the case of my shadow, this works out to:

0.36666784 + 0.021643 = 0.38831084

That is to say, 38.83%. As a shadow, this is by far the lowest static, non-RNG mitigation you will see. (which is part of why I'm using my shadow as an example)

Now, we need to somehow calculate the total, effective health pool for my shadow, assuming that 38.83% of incoming damage is negated, and thus not subtracted from our hit points. This can be done with the following bit of algebra:

(1 - 0.3883) * x = 26588

Another way of putting this would be to ask "what is the smallest hit required to completely deplete my HP?" The answer to this question is:

43466 HP

So that's a lot of effective HP. Note that this is actually not the whole story, since shadows have a self-heal. As long as we aren't completely stunned while the boss beats on us, we will be healing back damage at a rate of (roughly) 160 HPS. If we wanted to really get the full picture, we would take this into account. For simplicity's sake, we will skip the self-heal for now and simply consider the naive effective HP calculation.

Do we care?

Effective HP (eHP) is a way of looking at tank performance under an absolutely worst case scenario. Basically, the scenario in question is where all RNG-based defenses are failing to proc…consistently. How well will the tank do under those circumstances?

It's a really good idea to be thinking about this sort of situation, since tanks don't die under "normal" raid conditions. They die under worst-case conditions. So, optimizing for effective HP is really an attempt to stave off death just a little longer, at least in the worst-case scenario.

Here's the problem: optimizing for eHP means deemphasizing RNG-based defenses, like shield and absorb. Since there is no way to stack armor rating in TOR, the only real way to optimize for eHP is to stack Endurance, and a lot of it. This results in a lower shield chance, a lower absorb percentage, and a much lower defense chance. This in turn means that the average mitigation of the tank in question drops precipitously, meaning that they require a much higher HPS (from the healer) to keep them alive.

So that's the tradeoff. Here's the question: is the tradeoff worth it? The answer is: it depends. Really, it depends on two things: how common is the worst case scenario, and just how bad is it when it happens? Fortunately, given boss DPS numbers and swing-timers, we are actually able to answer that question with comparative ease.

Considering just EC HM, TFB HM and Nightmare Pilgrim, here are the average (pre-mitigation) DPS and swing timers:
  • Average DPS: 2408
  • Average swings/sec: 0.0703631
  • Max swings/sec: 0.86533

The reason for the excessive long-tail in the set of boss swing timers stems from the fact that most bosses have a lot of very small, very infrequent attacks. Middle-of-the-road attacks tend to be very frequent, with extremely large attacks (e.g. TFB's Scream) being extremely infrequent. Going with a somewhat sane estimate within the bounds of these metrics, we can say that the really interesting attacks are going to be somewhere in the realm of 0.5 swings/second. This indicates that the average hit is 4816, which seems within the realm of plausibility.

Ok, so here's the golden question: how many hits is it going to take at 4816 per hit to bring down our tank, assuming all RNG-based mitigation is failing? The answer to this question is very closely related to time-to-live (TTL), but actually more useful for the purposes of probability munging:

43466 / 4816 = 9.025 ~ 10

Since we can't have an "0.025th of a hit", we round up to 10. With a swing timer of 0.5/sec, that gives us a TTL of 20 seconds. Now, right away, we're running into interesting results, because 20 seconds is a very long time. A very, very, very long time. Doesn't seem too concerning at face value, but let's press on.

Here's the money question: what is the probability of the "worst case scenario"? This turns out to be a fairly easy question to answer. My shadow has a shield chance of 65.28% (with Kinetic Ward) and a defense chance of 28.77%. These are my RNG-based mitigation mechanisms. We also need to consider the fact that only 71.80% of damage in EC/TFB/NMP is subject to defense/shield. Given all of that, the probability of both shield and defense failing on a single attack is as follows:

0.7180(1 - 0.2877)(1 - 0.6528) + (1 - 0.7180) = 0.4595

So, 45.95%. Those are comparatively high odds, but remember, that's only the probability of a single attack passing through all of my RNG-based defense. What is the probability of the full-on, worst case scenario from above? In other words, what is the probability that 10 consecutive attacks somehow pass through my shield and defense? Well, this is also fairly easy to calculate. Shadows have no abilities that affect shield or defense chance based on whether or not an attack has been shielded/defended (aside from the Kinetic Ward rate limit, which is a non-issue due to swing timer). In fact, none of the tanks have such a mechanism. Thus, each shield/defend event is independent, and the probability of 10 such events failing consecutively is simply the probability of one event failing raised to the 10th power:

0.4595 ^ 10 = 0.0004196

This is to say, my shadow has a 0.042% chance of hitting a worst case scenario. A worst case scenario which is, btw, 20 seconds in duration.

(for those interested, here is a graph which indicates the curve along which the probability falls of consecutive failed procs over a span of time approaching 10 seconds)

The average boss fight in TOR is about 390 seconds (6.5 minutes). That's 185 overlapping 20 second periods in which a worst-case scenario may be taking place. The algebraic leg-work required here is somewhat...involved, but the answer works out to 0.091%. With five bosses per instance, we're up to 0.45% odds that you will see a worst case scenario in any given raid.

Let's extend this a bit further. Let's imagine that your group clears 10 bosses a week (TFB HM, EC HM and Nightmare Pilgrim). That's 0.913% odds per week of a worst-case scenario. This means that from a purely statistical standpoint, you would need to raid in TOR for 2 years, 1 month and 1 week to see a worst case scenario…once.

A worst case scenario that requires a full 20 seconds to bring down your tank.

Granted, you might not one-shot every boss. Maybe you require an average of 2.5 attempts per boss. that's still about 10 months of raiding before you see your first "worst case" scenario.

Conclusion

From a purely statistical standpoint, we have answered two questions here.
  • Q: How bad is the worst-case scenario? A: Not very. 20 seconds is enough time for a single healer to put out 40k worth of healing. That's almost the full effective HP of the tank in question. Oh, and that's just one healer. 8 man raids have two; 16 man raids have four. As long as they're not dead, mind trapped, AFK or similarly incapacitated, they probably won't even notice the "worst case" scenario.
  • Q: How common is the worst-case scenario? A: Much less than you would think. 0.45% odds per raid. If you keep raiding long enough, you'll see it eventually. Maybe. If you're paying attention. More than likely you'll just heal through it, down the boss and collect your loot.

Now, the counter-argument does exist that these sorts of statistical simplifications are less relevant in a progression setting. All of my numbers above are making the implicit assumption that you have these bosses on farm (e.g. not taking unnecessary damage, not requiring a dozen attempts per night to down the boss, etc). This is true. However, those assumptions really only have a significant effect on the probability of the worst-case scenario happening in any given raid. According to my combat logs, the healing I require in a progression setting does not differ significantly from the healing I require once the boss is on farm. Thus, I think we can assume that the damage numbers and swing timers aren't being significantly affected by our fight proficiency. This in turn means that the severity of the "worst case scenario" is not any more dire in a progression environment. Twenty seconds! That's almost the half the cooldown on Resilience!

Even assuming the tank archetype with the least effective HP (shadow/assassin), and even discounting important non-RNG survivability mechanics like the self-heal, I still have a hard time seeing how a "worst case scenario" poses a significant threat. It almost never happens, and even when it does, it really doesn't pose a problem. The reason for this stems from the fact that bosses in TOR simply don't hit very hard.

In conclusion, I would say that while effective HP may be an important metric in games like WoW or RIFT where tanks take a serious beating in every fight, I don't think it really merits concern in TOR. Bosses don't hit very hard, and RNG-based mitigation procs at a comparatively high probability.

As a tank, you should be itemizing for mitigation. I don't think there is a plainer way to put it. Big, juicy health pools are fun, and they certainly make for a great vanity stat for group finder randoms, but in the end you are doing your raid group (and especially your healers) a disservice if you prioritize effective HP over mean mitigation.

(note: data for this post was drawn from this spreadsheet, which was assembled from my own combat logs as a progression tank)
Computer Programmer. Theory Crafter. Dilettante on The Ebon Hawk.
Tam (shadow tank) Tov-ren (commando healer) Aveo (retired sentinel) Nimri (ruffian scoundrel)
Averith (marksman sniper) Alish (lightning sorcerer) Aresham (vengeance jugg) Effek (pyro pt)