Friday, December 17, 2010

Train of Thought

Train of Thought is a very interesting talent, and fairly hard to evaluate. It's a talent that saves you mana, but only if you spend mana on an expensive spell. The question of how much mana it actually saves you is a difficult one to answer.

If you don't cast a good number of Greater Heals, Train of Thought is a very bad talent. Suppose you mostly use Heal as your single target heal, and use Greater Heal only when Inner Focus is making it free. Because Inner Focus must be available at the start of the cast, but it not used - triggering the cooldown - until the end of the cast, the real time between Greater Heals would be 45 seconds plus the casting time of Greater Heal. We'll use 2.28 seconds for the casting time of greater heal1.

So in this case Train of Thought would reduce the time between free Greater Heals from 47.28 seconds to 42.28 seconds. These free heals also benefit from +25% crit, so they are more powerful than typical Greater Heals. In order to figure out how much mana this saves us, we can't look at the cost of Greater Heal, because we wouldn't have been casting Greater Heal if it weren't free. Instead, we should look at how much mana we would have spent healing the same damage with our spammable spell of choice, Heal. In my current gear, a free Greater Heal is worth around 3.21 Heals of healing. Since heal costs 2118, that means we save 6799 mana per Inner Focus boosted Greater Heal. This is actually more than the cost of Greater Heal, thanks to the 25% crit bonus. If we save 6799 mana every 47.28 seconds, that gives us 143.8 mana per second. If instead we save 6799 mana every 42.28 second, that gives us 160.8 mana per second, only 17 mana per second for two talent points. Definitely very bad points.

The question is, what if we cast Greater Heal instead of Heal to heal the damage between free casts, reducing the time between free heals more. Things get a little more complicated in this case. We can figure out how many of our Greater Heals will actually be free to cast - and +25% critical strike chance - using the following:

In order to calculate the time between casts, we would need to know how much our average heal heals for, and how much damage we are trying to heal per second:

To get the average heal, we'd need to know the ratio of free heals, as well as the average amount a normal Greater Heal heals for (G) and the average amount an Inner Focus Greater Heal heals for (G'):

Fortunately, three equations is exactly the right number for three unknowns. The simple formula for the ratio of free Greater Heals is:

So, given a damage amount we can get the average free Greater Heal ratio and the time between casts, which allows use to easily calculate the amount of mana spent per second. The question is, is it cheaper to cast Greater Heals than Heals? If it is not, then Train of Thought saves us mana only when the damage exceeds the amount we can heal with Heal, and we should cast Heal as often as possible. If it is cheaper to heal the same damage with Greater Heal, then we should ignore Heal, even if the incoming damage is low.

There are two results from this simple analysis. The first is that it is cheaper to cast Heal, the second is that the amount of incoming damage you can actually heal with Heal is so low that Heal is basically never a viable option so it is a bit of a moot point.

Of course these results come because the model is too simple. In reality we have three spells - Power Word: Shield, Penance and Borrowed Time hasted Renew - that are all strictly better than Heal but that have real or effective cooldowns. We would never cast Heal if casting one of these were an option instead. In additional, while Power Word: Shield isn't strictly better than Greater Heal, it would have to be a very bizarre situation for us not to use it, and both Penance and the hasted Renew are better than Greater Heal in both efficiency and throughput.

So, what happens to the calculations when we cast Penance, Power Word: Shield, and hasted Renew on the "cooldown"?2 Fortunately we can use the same formula above for the ratio of free Greater Heals. What changes is that the damage we are trying to heal with them is much lower, and our mana expenditure depends not only on how many Heals or Greater Heals we cast, but also has a baseline set by the cost of our superior spells. When we factor this both of our conclusions from the simple case are reversed. While Heal becomes a viable option for some reasonable levels of incoming damage, Greater Heal becomes the more mana efficient way to heal at all levels of damage.

This may seem counter-intuitive at first, so I'll talk a little bit about how this happens. I can understand if your intuition tells you that if one thing is more mana efficient, then it should still be more mana efficient when used as a gap filler. The reason this doesn't end up being true is that Train of Thought is doing more work for us when we are casting other spells. While we still have the same number of real seconds between Greater Heal casts, many of those seconds are used up casting things we are going to cast regardless of what else is going on. At most we are going to fill around 30.86 of our 47.28 seconds with Heal or Greater Heal. The rest of the time we will be using more power heals with better efficiency. This allows us to increase the time between Greater Heals to Heal the same amount of damage. More time between heals means a higher ratio of free heals.

At around 8400 incoming dps - the highest incoming dps that Heal can take care of in my current gear - using Heal and the more efficient spells requires 1010 mana per second. At the same damage level, using Greater Heal instead requires only 900 mana per second. So Train of Thought is saving us 110 mana per second for two points, which is a lot more respectable than 17. Of course it's also important to consider that healing with Greater Heal instead of Heal saves us time, so we have more leeway to fit more heals in if we need them. While at this level of healing casting Heal means casting pretty much 100% of the time, using Greater Heal means only casting 70% of the time. Train of Thought is buying us a lot of time to move around and do other things, in addition to mana savings.

I haven't had Heal on my bar this entire expansion and this analysis just helps to reinforce that decision. While there would still be substantially more modelling to do to quantify the benefit of Train of Thought against the benefit of another talent, such as Divine Aegis, it is pretty safe to say that Train of Thought should be regarded as a "must have" talent rather than an optional one and that Greater Heal should be the staple spammable single target heal for Discipline priests.

1. I chose this number because it is the casting time of greater heal given that you have just enough haste rating to make sure your Renews get a fifth tick when hasted by Borrowed Time. Until throughput becomes more of an issue than mana, I will regard this as the "correct" amount of haste to have.

2. I'm assuming shielding a single target and casting one Renew per shield.

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