Elitist Jerks A calculation of expected Ignite duration

09/20/06, 2:31 PM   #1
Friedrich
his surgical quality

Tauren Warrior

Maelstrom
Hi all,

sorry to make yet another ignite thread, but I think this is interesting, and I haven't seen it yet around here.

(Note: a LaTeX'd .pdf of this text is available here: http://astro.physics.upenn.edu/~hsta...e_halflife.pdf, which should be more readable.

Inspired by the interest in rolling ignites, I decided to calculate the expected duration of an ignite, assuming your mages are all spamming scorch once the initial stack is set up.

Let P(crit) be the chance to score a crit on a single scorch. Let P(hit) = 1 - P(crit). Let there be N mages each with the same %crit chance. I will use ** to denote exponentiation and ! to denote factorial.

In a given 4-second interval, a mage will either land 2 or 3 scorch spells: two thirds of the time, there will be 3 scorches, one third of the time, the mage will land 2. Let's assume then, that due to lag etc. the offsets are random and that, for a given mage, her contribution will be
P(2s) = 1/3
P(3s) = 2/3

The number of ways to land M scorches in 4 seconds with N mages is [N choose (M-2N)]

# [top] N!/((3N-M)!(M-2N)!). You can verify this by considering the first few cases with N

1,2,3,4. (It makes a Pascal's triangle).

So, in a 4-sec interval, the probability that M scorches will land is:
P(M scorches) = {[N choose (M-2N)] * P(3s)**(M-2N) * P(2s)**(3N-M)} if 2N<=M<=3N

# [top] 0 otherwise So, the probability that with you do not crit in 4 secs is: P(no crit given M scorches)

P(M scorches) * P(hit)**(M)

P(no crit in 4sec given n mages) = Sum_i P(i scorches)*P(hit)**i, for 2N<=i<=3N

# [top] P(fall)

1-P(stay).

So then the length of the ignite should follow a [a href="http://en.wikipedia.org/wiki/Geometric_distribution"]geometric distribution[/a]: the expectation value for the number of 4sec intervals that the ignite lasts is
<T_ignite> = P(stay)/P(fall).
The probability that it lasts exactly n intervals is
P(ignite lasts 4n seconds) = P(stay)**n * P(fall).

A perl program (ignite_time.pl) to calculate these quantities is available at
http://astro.physics.upenn.edu/~hstabena/

Some numbers:

N_mages critrate P(fall) <T_ignite> (sec)

3 0.15 0.27 10.5
3 0.2 0.17 19.4
3 0.25 0.10 34.8
4 0.15 0.18 18.4
4 0.2 0.095 38.3
4 0.25 0.048 78.9
5 0.15 0.12 30.4
5 0.2 0.052 72.2
5 0.25 0.023 173.

I would be interested to see if anyone can check these results in a real raid. There are several factors I haven't taken into account, of course: lag is the most notable I think.

 09/20/06, 5:54 PM #2 • Hamlet     Hamlet Tauren Druid   Mal'Ganis Here are my first attempts at it (bottom of first page): http://forums.elitistjerks.com/viewt...hp?id=7742&p=1
 09/20/06, 6:54 PM #3 Boevis Bald Bull     Ex-Boevis Tauren Druid   Lightbringer I'm not too bad at math, but I'm a bit confused what C and i are, since you make no mention of what they represent ...
 09/20/06, 9:15 PM #4 Moleva Von Kaiser   Moleva Human Warrior   Durotan C looks like short hand for Choose and i is a dummy variable for the sum that he does.
09/20/06, 10:09 PM   #5
Friedrich
his surgical quality

Tauren Warrior

Maelstrom
 Originally Posted by Moleva C looks like short hand for Choose and i is a dummy variable for the sum that he does.
yes, sorry - I'll edit my post to make that shorthand explicit

 09/21/06, 5:59 AM #6 zepi Miekkamies     Clodia Human Mage   Darksorrow (EU) Great, too bad that I can't run perl :/ One thing to note: 2.66666 scorches inside the 4s timer is only possible for those mages who don't refresh the timer. The one who refreshes, can never get more than 2 scorches. Reason should be fairly obvious, as your own scorch timer starts from the worst possible point after crit.
 09/21/06, 9:26 AM #7 Friedrich his surgical quality     Vronsky Tauren Warrior   Maelstrom hey zepi: if you want to run perl on your windows machine, a good option is Cygwin, which emulates a UNIX environment under windows. It is quite easy to install the perl package and then you are all set. Either that, or you can get a native windows perl here: http://www.perl.com/download.csp#win32 Alternatively, someone could incorporate this into a spreadsheet, but I don't have access to excel atm so not me. It is true that one of the mages will definitely only get 2 scorches, rather than 2 or 3 with some probability. However, if the casting times between the mages are uncorrelated, what I've done should hopefully still be an OK approach. I also stuck a LaTeX .pdf version of the above math text into the same web directory, which should make it a little more readable.
 09/21/06, 11:17 AM #8 Quantuvis Von Kaiser   Quantuvis Troll Mage   Kazzak (EU) Nice theory crafting for sure, but with only 16 debuff slots, Ignite just gets pushed off too often to really test your findings.
 09/25/06, 6:47 PM #9 Evice Glass Joe   Evice Undead Mage   Illidan If everyone is spamming scorch it is really really low probability that it will be dropped.

 Elitist Jerks A calculation of expected Ignite duration