Elitist Jerks Estimating the Value of Glyph of Starsurge

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# Estimating the Value of Glyph of Starsurge

Posted 10/19/10 at 5:33 PM by Hamlet
Updated 10/20/10 at 10:27 AM by Hamlet
The value of Glyph of Starsurge is a common topic of discussion among Druids lately. Since its DPS contribution is nontrivial to estimate, I'll give a quick walkthrough of the basic calculation. For actual results and final DPS comparison, as always, look at my spreadsheet, but this post explains the way the spreadsheet models Glyph of Starsurge, and contains some general mathematical points that are useful for theorycrafting.

First, we need to know the frequency of Starsurge casts. In the absence of Shooting Stars, we'd get one Starsurge roughly every 17 seconds (15 seconds cooldown, plus the Starsurge cast time itself, plus around a GCD of fudge factor to account for the fact that you typically have to wait for a current cast to finish before you recast it). Call this T_CD. We also need know how often we get Shooting Stars procs. Assuming 100% IS and MF uptime (the sheet does something slight more complex than this, but since we generally maintain high DoT uptime the difference isn't important), the Shooting Stars mean proc time is given by:

$\frac{1}{T_{SS}}= (\frac{1}{T_{IS}} + \frac{1}{T_{MF}}) * 0.04$,
Where T_IS and T_MF are the tick times of IS and MF. Currenly, both of those are equal to 2/(TotalHaste), ignoring the Sunfire tick time bug which should fixed at some point. On live right now with my total raid-buff haste of around 48% and NG uptime of around 75%, this is roughly 15s between Shooting Stars procs.

So how do we find our actual mean time between Starsurge casts? Remember we cannot use the same trick as above and say:
$\frac{1}{T_{Total}}= \frac{1}{T_{CD}} + \frac{1}{T_{SS}}$,
because ordinary Starsurge casts and Shooting Stars procs are not independent. Using a Shooting Stars proc resets the 15s cooldown for normal Starsurges. How do we deal with this?

First, assume that Shooting Stars procs obey a Poisson distribution - Wikipedia, the free encyclopedia . This is means that in a given time interval t, where the expected number of procs is t/T_SS, the chance of having 0 procs is:
$e^{-\frac{t}{T_{SS}}}$. Specifically, if t = T_CD, this tells us the chance of a normal Starsurge cooldown going by without a proc.

This is what we need to know because it gives us the chance that, between two Shooting Stars procs, there will be enough time for a normal Starsurge. So we're going to start with the Shooting Stars proc only, and then add in the normal Starsurge casts that might occur in between. Before we finish however, we should consider the small chance of multiple normal Starsurge casts between procs. Each successive period of T_CD seconds brings an independent chance of proccing or not proccing another Shooting Stars. Therefore the actual expected number of normal Starsurges between procs is given by the sum of a geometric series:
$\frac{e^{-\frac{T_{CD}}{T_{SS}}}}{1 - e^{-\frac{T_{CD}}{T_{SS}}}}$.

We now finally have our final expected mean time between Starsurge casts given by:
$\frac{1}{T_{Total}}= \frac{1}{T_{SS}} * (1 + \frac{e^{-\frac{T_{CD}}{T_{SS}}}}{1 - e^{-\frac{T_{CD}}{T_{SS}}}})$
$T_{Total}= T_{SS} * (1 - e^{-\frac{T_{CD}}{T_{SS}}})$

With my values of around T_CD = 17s and T_SS = 15s, this gives a T_Total of around 10.3s.

But we still have to see how this affects Starfall cooldown. Call T_Total the final Starsurge proc time from above, and using 60s for the Starfall cooldown (can use the same equation for the 90s cooldown Starfall if you want). Be careful not to assume that the new cooldown is
$T_{SF}= 60 - 5*(\frac{60}{T_{Total}}) = 60*(1-\frac{1}{T_{{Total}}})$.
This assumes that you get to apply 60 seconds' worth of Surges to each Starfall. Instead, we need to use:
$T_{SF}= 60 - 5*(\frac{T_{SF}}{T_{Total}})$.

Solving for T_SF:
$T_{SF}= \frac{60}{1 + \frac{5}{T_{Total}}}$

And that's our final Starfall cooldown that we plug back into the rotation. If T_Total is around 10s, this gives a Starfall cooldown of just about 40 seconds.

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Brief postscript: To actually get a DPS value from this, you need to insert that cooldown into whatever rotation model you're using. Remember to account for the added cast time spend on Starfall in addition to the added damage done. In WrathCalcs, I find the damage value of this Glyph against a single target is less than that of Glyph of Wrath or Glyph of Moonfire. However, one possible scenario is that the Glyph will allow you to cast one Starfall every Lunar Eclipse where you otherwise couldn't. This would make the Glyph a DPS increase, but whether it happens or not depends on many details of the fight.
Posted in Math, Druids

 Very insightful entry. I'm interested to see how multiple target dotting, thus increasing the number of ticks per unit time, will change the value of this glyph and how this will alter the output of your calculation. For example, HLK phase 3, spirits come out and you start dotting (assuming your SS is on CD) 4-5 of them and then lunar showering (which provides from my tests the max dmg output); in this example, I'd assume that glyph of starsurge will come out stronger than any of the other two since the multiple dots ticking has two things as a result: a) Higher chance of Shooting stars proccing, so more instant casts and therefore more damage b) Lower CD of starfall which will make it be up roughly every time the spirits come out for increased dmg (at least during my tests an average time of 40s as you mentioned is expected, thus with multiple targets I'd assume a bit lower) I've verified after rigorous single target testing (for my value of haste), that the glyphs IS/MF/Wrath indeed come out stronger since the starfall isn't always up (using IS/MF/SS or IS/Wrath/SS) under lunar so it decreases the glyph's value. Posted 10/19/10 at 7:45 PM by Tassdrummer
 Good post. I think it is worth considering that most fights aren't Patchwerk. 6s moving every minute is four or five more MF casts per minute (and 2 or 3 less Wrath casts/minute). If MF DD can proc shooting stars (I have not tested that, need to start nuking critters someplace), you'd get maybe 8% more shooting stars procs, and a few more seconds, on average off that Starfall cooldown. Posted 10/19/10 at 9:51 PM by Erdluf
 Impressive math, I've been raiding as much as possible on beta so far and I can throw some anecdotal evidence that the glyph does indeed allow you to keep Starfall in the Lunar eclipse. Not so much on a fight like Conclave, but for Chimaeron it was a substantial boost. Plus as both previous responses already said, any fight where there's adds or multiple targets increases the value of the glyph to a certain degree (though the benefit is capped, because pushing the CD faster than you can reach Lunar again doesn't help). But it does allow you to catch up rotation-wise if movement prevented your hard casting more than usual. Posted 10/20/10 at 2:44 AM by Faveokatro
 Right, my guide already mentioned that GoSS is situationally good, particularly with multiple targets. This is was more in response to people asking about how to estimate the DPS gain of the glyph properly, and just in general to demonstrate some math tricks. Posted 10/20/10 at 2:54 AM by Hamlet
 Question, if we decided to use the starsurge glyph, which glyph would we drop given the current values in patch 4.0.1? Wrath, moonfire or insect swarm? Posted 10/21/10 at 4:21 AM by Kizmin
 If you drop anything it would HAVE to be wrath, given the ridiculous damage DOTs do now, I would think. Then again, I went to law school and therefore, forgot all mathematics... >.> Posted 10/25/10 at 1:17 PM by cloudspc
 A funny comment, I just finished my JD and am doing an LLM now :P . Wrath would be the Glyph to drop. It's close to Moonfire, but Moonfire spreadsheets a bit better is also stronger against multiple targets. Posted 10/25/10 at 2:58 PM by Hamlet