Estimating the Damage dealt by purely Auto Attack reliant ADC's
The passive damage from Blade of the Ruined King ( a percentage of a champions remaining health ) greatly convoluted the estimation of the time it takes an auto attack reliant champion to completely kill an idle opponent, ignoring health regeneration. Below is the method I have utilized so far, I do seek to make my method more robust and account for the time it takes to stack up a Guinsoos Rageblade during combat, among other things.
** 1) Assignment of Variables**
let the following be taken as definitions Z = Total health of the enemy champion X = Remaining health of the Enemy Champion Y = Missing health of the Enemy Champion h = The number of auto attacks executed a = The on hit flat damage dealt to the enemy as true damage b = The % max health damage dealt to the enemy as true damage c = The % Remaining health damage dealt to the enemy as true damage
Thus this is the equation for a single auto attack,
http://mathurl.com/h9e9d5q.png
This means that,
http://mathurl.com/gr7m8fu.png
** 2) Estimating The integral**
let y(n) = be the damage dealt by nth auto attack
thus y(1) = a + bZ + cZ y(2) = a + bZ + c(Z-y(1))
and in general,
http://mathurl.com/jerubg8.png
thus,
http://mathurl.com/zspjpse.png
** 3) Estimating h**
Now rather than treat h as one of a set of discrete values and have to evaluate the sum by hand, an approximation will be used.
http://mathurl.com/znvtfjm.png
Thus,
http://mathurl.com/zc63ego.png
Then we evaluate the integral of X over h finding,
http://mathurl.com/gttqc94.png
Please note, this is a transcendental Equation, however if the Lambert W function is utilized h can be found to be.
http://mathurl.com/zro3fqc.png
** 4) Hits to Kill a Damaged Target **
Please note that, although we have a closed form value for h, actually evaluating W requires an iterative process.
Similarly, using the same process we can find the number of hits it would take to fell a damaged enemy.
First we find the hit equivalent for their current health yielding a slightly more complicated equation.
http://mathurl.com/hmhp57w.png
where A is the hit equivalent to the amount of damage taken by a given champion.
After solving for A and h numerically, the number of hits to kill a damaged enemy, H, becomes
http://mathurl.com/zz7ryvn.png
** 5) Time Taken to Kill a Target **
Let S = Attack speed of the ADC T = Time required to Kill the Target
