WoW Mathematics: Sharing the (DPS) load

In any (conventional) boss encounter, a certain number of damage dealers (dps) need to whittle down the boss' health to zero over a certain time period. Barring any extraordinary tinkering by Blizzard, the health level remains constant between attempts.

This fixed DPS burden needs to be shared between all dps. I will examine the DPS burden per dps and how it changes with the number of dps that happen to die during the encounter.

Summary:

• Given a total DPS burden (total raid DPS) of D and a total number of dps of d, the average DPS burden per dps B, where x players are dead, is B = ^D/_{d - x}.
  
• The trend is a rectangular hyperbola centered at (x = d , B = 0) and positive in the relevant range 0 < x < d .
• As more dps die, the DPS burden per dps increases at an increasing rate, even though the total dps burden still stays the same. As a result, the difficulty of downing the boss increases disproportionately fast.

The given:

• The total DPS burden is split evenly among the dps still alive.
• The healers do not dps and tanks deal negligible damage.

The detail:
Given x dps that have died from a total of d, the number of dps still alive is:

d - x

The total DPS burden is to be split between these dps:

B = ^D/_{d - x}

As an added note, to calculate the total DPS burden, the level H of the boss' health needs to be divided between the number of seconds of time t still remaining:

D = ^H/_t

To analyse how B changes with respect to x, the formula would need to be differentiated under calculus. Differentiating B with respect to x:

^dB/_dx = ^D/_{(d - x)²}. Since it is always positive (you cannot have a negative number of dps alive, nor more dps than are available) in the relevant range (which is technically called the domain in mathematics), the burden is always increasing as more dps die.

Differentiating ^dB/_dx with respect to x:

^d²B/_dx² = ^2B/_{(d - x)³}. This tells us that, since it is always positive in the relevant range, the burden is increasing at an increasing rate.

Example: Normal 25-man Blood Queen Lana'thel
Now to plug in some numbers. The Blood Queen has approximately 58.7M HP in health and a 5 minute (300 second) Enrage timer. This raid is walking in with 2 tanks, 5 healers and 18 dps:

H = 58700000 , t = 300
D = ^H/_t = ^58700000/₃₀₀ ~ 195667 HP
B = ^D/_{d - x} = ¹⁹⁵⁶⁶⁷/_{18 - x}

In the best case scenario, no dps are dead. In this case, the DPS burden per dps is:

B = ¹⁹⁵⁶⁶⁷/_{18 - 0} ~ 10870 HP/sec

Evaluating for all possible numbers of dead dps, the following graph can be drawn:

Originally posted here.

EDIT 22Apr10: Amended description of variable x in Summary, courtesy of LeCanard.