Don't slack off during the Thaddius encounter!

Aim: To calculate the drop in DPS of a single death in Phase 2 of the Heroic Thaddius encounter, subject to certain assumptions. To observe the change in total raid DPS as additional dps die.

Summary:

• The minimum personal DPS needed to down Thaddius in time is 2186 DPS.
• The total personal DPS is 100% average DPS + 115% bonus DPS from charge buff OR 215% of average DPS.
• A 1-stack of the charge buff grants 218.6 DPS.
• A single death of a healer drops total raid DPS by 1748.8 DPS, while a single death of a dps drops total raid DPS by 6230.2 DPS.
• While DPS continues dropping as additional dps die, its rate of decline is decreasing.

The given:

• Raid composition: 2 tanks, 7 healers, 16 dps.
• Raid positioning: 12.5 players on each side of Thaddius, each side having 8 dps.
• Thaddius' health: 27 600 000 HP.
• Enrage timer: 360 seconds (6 minutes)
• Average size of stacking group: 12.5 players.
• Average stack of charge buff among dps: 11.5 stacks. This gives a damage boost of 115% of average personal DPS.
• Healers do not dps, tanks do minimal damage (in this scenario, VERY minimal).

Detail:

The figure that will be used as the average personal DPS, fully buffed, is the minimum personal DPS needed to down Thaddius before the enrage. The total raid DPS is:

27 600 000/360 = 76 666.6666... DPS

The total personal DPS is:

7666.6666.../16 = 4791.6666... ~ 4700 DPS (attribute the extra 91.6666... DPS to the tanks :))

Total personal DPS is:

100% average DPS + 115% bonus DPS from charge buff (decimal 1.15) = 215% of average personal DPS (decimal 2.15)

The average personal DPS is:

4700/2.15 ~ 2186 DPS

That means the bonus DPS from the charge buff is:

4700 - 2186 = 2514 DPS

A 1-stack of the charge buff grants:

2514/11.5 ~ 218.6 DPS

Consider the death of a healer. That removes a 1-stack of the charge buff off the 8 dps on their side. So, the drop in DPS from that death is:

218.6 x 8 = 1748.8 DPS. That's about 524.6k damage over 5 minutes lost!

Consider the death of a dps. That removes the dps' total personal DPS plus a 1-stack of the charge buff from each of the 7 dps on their side. So, the drop in DPS from that death is:

7 x 218.6 + 4700 = 1530.2 + 4700 = 6230.2 DPS. That's about 1.87M damage over 5 minutes lost!

To observe the change in DPS using calculus, additional deaths will need to be considered in pairs (one dps from each group will drop total raid DPS by equal amounts) and we will assume that only the dps die. The change in total raid DPS from the first pair of deaths is:

-6230.2 x 2 = -12 460.4 DPS

Calculating the incremental changes in DPS from additional pairs of deaths, we get the following progression of values:

-12 460.4, -12 023.2, -11 586, -11 148.8, ...

This is in fact an arithmetic progression with common difference of 437.2 and first term of -12 460.4. Total DPS lost from n pairs of deaths will be the following sum of values:

S(n) = 218.6n^2 - 12 679n

Differentiating S(n) with respect to n:

dS(n)/dn = 437.2n - 12679. Since the first derivative is negative over the domain 0 < n < 8, S(n) will continue dropping.

Differentiating dS(n)/dn with respect to n:

d2S(n)/dn2 = 437.2. Since the second derivative is always positive, S(n) is decreasing at a decreasing rate.

We have the following trend:


Originally posted here.