## Mana Regen in Mists of Pandaria – Part 2

This post was originally going to be awesome, but then reality hit home and I found a factor of five error in my calculations – that ended up proving that Spirit isn’t worth stacking as a throughput stat. In retrospect, that’s not surprising.

Either way, here’s the (not quite finished) post in its entirety just in case you want to see an interesting exercise in theorycrafting. I know that my equations only take into account the case of an exact 1:1 ratio between phrases – this was something I was working on fixing thanks to @TheckPhd ‘s comments (sadly unnecessary now).

This is following on from my last post on the subject of healing manapools in Mists of Pandaria and the consequences of some of the changes for Shaman and for the theorycrafting community. I did receive some criticism that the first in the series was less idle fact and more cold hard speculation, so I hope to remedy this today.

The problem arises that we want to find a good way of quantifying something which is known as “pseudo throughput”, which is the throughput that you gain from the regen your character gains from Spirit (and possibly other sources). We want to be able to express this regen in a form which we can compare directly to the healing done by other stats. In other words:

We want to express the healing that regen effectively gives you so that we can compare its power with that of other stats.

I think that if we were to find a way of expressing this, we could therefore safely call it “Healing Effective Power“, or HEP for short. Once we can express this HEP, we can use it to calculate the worth of Spirit as compared to other stats and define a “best” level of Spirit for a healer to have for the right amount of throughput. We might then be able to extend this into stat weights and reforging priorities. First, though, let’s talk about some limitations and assumptions we will have to put up with.

Defining the Limitations

HEP will obviously have some limitations in both how easily we can apply it and how many assumptions we have to make to obtain a number. With that in mind, let’s start by defining some reasonable starting assumptions for applying such a quantity:

• We can rely on healers using a set of combinations of spells together, which varies depending upon the situation.
• These sets of spells can be thought of as “phrases” (short combinations of spells which are used situationally, much like with words).
• We can assume that for a large amount of a reasonable fight, the healing will be fairly predictable in terms of spell/phrase distribution.

Furthermore, there will certainly be some intrinsic limits on any attempt to quantify HEP:

• HEP will depend on the spell distribution used throughout any fight. Hence, HEP is something that must be calculated on a fight basis.
• HEP depends on an individual’s healing values, and hence their own stat distribution. This means that HEP must be looked at on an individual basis.
• HEP will depend also on the consistency of the healing distribution used. Since the spell distribution is never ideal, we must somehow account for this or else use caution when making use of HEP.

Having defined those, I think we can now move on to an attempt to define something we can use given the above circumstances.

A Simple Model

So initially, we want a simple way of looking at the healing per second that our mana regen gives us. We can estimate that with the following expression;

HEP = (Spell HPS) * (Mana Regen per second) / (Spell Cost per second)

This takes a single heal type, and knowing the HPS done by the spell it shows you how much of the HPS your regen makes essentially free. If you were to calculate this version of HEP for your character, you could say something like:

Every time I use [this spell], I have got [HEP] healing per second just from the fact that I have some regen stats.

So this model of HEP is “spell specific”, and “player specific” meaning that this so far gives you no information about the general value of Spirit yet. We can call this “spell specific” HEP.

An example calculation: Healing Wave and Greater Healing Wave.*

HEP(Healing Wave) = (HPS/MPS) * regen per sec = 8344/(0.099*60000) * regen per sec = 1.4 * regen per sec.

HEP(Greater Healing Wave) = 15181/(0.363*60000) * regen per sec = 0.70 * regen per sec

Accounting for Spellpower:

HEP(HW) = (1.4+sp*0.00013) * regen per sec

HEP(GHW) = (0.70+sp*0.00006) * regen per sec

From this we can see that Healing Wave will scale better with Spirit no matter what our gear level is. This is largely unsurprising. An important thing to note here is that we can calculate two numbers to define the HEP for a spell; its base and spellpower coefficient! The spellpower coefficient is simply the spell’s normal spellpower coefficient divided by the mana cost of the spell. We can make similar calculations later on.

Now we can look at something a little more realistic. If we use the assumption that heals are used in small combinations or “phrases”, we can look at the HEP of different phrases instead of the individual heals. We can then apply the numbers to different situations (we can further extend this to model an entire encounter).

If we calculate the total mana cost (per second) of a specific phrase and its HPS based on cast times, we can look at the HEP for a phrase;

HEP = (Phrase HPS) * (Mana Regen per second) / (Phrase Mana Cost per second)

We can now use the number in another easy way of putting what it means;

Every time I use this specific combination of spells, [HEP] is sustainable due to my mana regen.

This is obviously a useful thing to know. We can make some easy calculations of HEP for two different phrases (disregarding spellpower for this one);

Phrase 1 – Riptide, Healing Wave, Healing Wave, Healing Wave

Phrase 2 – Riptide, GHW, GHW, GHW

These are common healing distributions and so by knowing their HEP, we can say that we arrive at a reasonable understanding of the healing gained from our regen. I calculated the numbers for a sample of 6000 Spirit and found:

 Phrase # Length Healing Mana Cost Regen HEP 1 7.5 39353 27420 1880.4 2698.703 2 7.5 59862 74940 1880.4 1502.062

So what happens when you take your spirit over the limit of what you need to indefinitely sustain a phrase? Well the trivial case is your HEP goes above the HPS of that phrase. Since your regen isn’t actually paying for that extra hps right there, it would be better to put that regen into another phrase. Hence, we need a way of accounting for this.

Over the LimitThis shows that (again, unsurprisingly) your regen will benefit your more efficient healing phrases more.

First, then, let me draw your attention to a quantity my fellow Physicist Aiwendeth suggested – Relative Mana Efficiency or RME. This is just the quotient of your mana cost per second divided by your regen per second;

RME = (mana cost per second) / (regen per second)

This is obviously related directly to HEP, but its importance here is that RME scales based on your mana regen – the more regen you have, the lower the number. When RME =1, you are at perfectly offset mana costs. As the mana cost increases, RME goes up – RME above 1 is not as good a place to be. When RME is below one, you get more mana than you spend – this is good!

How RME Scales with Spirit

So we can define the point of going “over the limit” when HEP isn’t useful to describe a rotation as where RME becomes 1 or less. Then, you have to start considering what else you’re spending your mana on! When RME < 1, you want to find the Spirit value where RME = 1 and any excess Spirit can be carried over to a different phrase. Let’s call HEP calculated taking excess regen into account “Excess HEP” (eHEP for short?) – add your eHEP to the normal HEP to your more expensive phrase and you get a better idea of how much stacking Spirit over the limit will get you;

[needs calculations – can do on current spreadsheet]

Considering that I calculated the Spirit level required for RME=1 to be just over 235000, I suspect that this won’t be reached on Tier 14 – rather, this may become an interesting feature when healing and item levels become very high in later tiers!

Differential HEP

This is the last extension I’ll talk about in this edition (I realise the words have gone on a long way – there is a lot more I want to talk about but there’s only so long I can delay this post in the name of completeness).

The idea of comparing other stats directly to Spirit is intriguing, especially with the loss of Intellect as a regen stat – Shaman now want to compare the healing gain of Crit directly to that of Spirit. I think that this will be easiest when considering the increase in HEP per point of Spirit increase. For the mathematically minded, that means we’re going to be taking the derivative of HEP (you can extend this to eHEP, which will also be interesting) with respect to Spirit! Going by the naming convention I’ve inadvertently set up, I’ll call this dHEP. Showing first the equation for regen per second in terms of Spirit;

Regen = 1200 + 0.11*Spirit

We see that for one point increase in Spirit, we get an 0.11 increase in regen per second. Thus we can say that the differential regen is 0.11. Substituting this into our equation for HEP, we find a formula for dHEP;

dHEP = (HPS) * 0.11 / (Mana Cost per second)

This quantity tells you exactly how much HEP you get out of a point of Spirit – it is now possible to make a similar calculation with Crit’s regen mechanic and compare dHEP to Crit’s healing increase plus it’s dHEP. However, I believe that the point where Crit will be interesting as a regen mechanic is where you already have enough regen to sustain Phrase 1 (that is, your RME < 1 for Phrase 1) – here we have to mix dHEP and differential eHEP! This is getting a little ugly, but bear with me. Once your RME > 1 for Phrase 1, your scaling with Phrase 2 increases at the rate of;

dHEP + d(eHEP) = dHEP + d(1200 + 0.11*Spirit)*HPS/MCps = dHEP + dHEP = 2*dHEP!

So in theory, once the point has passed where your filler Phrase is mana positive, your HEP scales twice as well with Spirit as it did before – this backs up my theory that the interesting applications of HEP will come about on the T14-15 threshold (where Spirit could compete with Crit for the best “pseudo-throughput” stat.

And Finally

I’ve not even covered half of the things I think this could be applied to yet! I’d love to go on, but I’m scared I’ll bore someone to death.

One important thing at the start which I stated was that we must somehow account for [variability] or else use caution when making use of HEP. I’ve not been able to do that as of now, and I’m not entirely sure that it’s possible to account for heal distribution variation in a simple HEP model – it may be that this is possible in some kind of simulation or in some kind of statistical way. In the mean time, caution is best when applying HEP – better to think hard about the limitations and have a vague idea of what’s good than not think and turn out wrong.

I think I’ll finish with a list of interesting or important problems that HEP could potentially solve with some clever application;

• Is Crit or Spirit better for throughput, overall, on a specific fight?
• What is the actual benefit in healing done (or HPS) of Mana Tide, and how important is it for Shaman to stack Spirit now it’s the only gear-scaling mana cooldown?
• What point should I stack Spirit to in order to ensure some spare mana for emergencies, but not to finish with too much?
• Is switching to a different Phrase rotation better for me overall? Do I have more room to breathe in terms of secondary stats, and by how much does that help?

In another post, I’ll be trying to tackle some of those questions. In the mean time, let me know if I’ve made any errors or if something needs updating. I’d love to give this a test somehow – anyone in the Beta who wants to help (I’m not in the Beta, sadly) can find me on Twitter. Go go, Healers! =3

Sources for healing formulae:

http://www.totemspot.com/wiki/index.php?title=Main_Page