WoW Mathematics: Steady-State Health Level

In the final phase of the Anub'arak encounter in the Trial of the Crusader (ToC), healers must heal through a variety of sources of damage, including Penetrating Cold and Leeching Swarm. While the former deals constant damage over time, the latter does more damage the higher players' health levels are! The raid might be interested in knowing the optimal health level the healer(s) can keep the players at.

At the steady-state health level, DPS of Anub'arak against the raid equals the HPS of the healers for the raid. From here, health levels do not tend to move either way; they are stable, and this determines whether the healers can provide enough healing.

Summary:

• Where h is the total HPS, r is the proportion of current health Leeching Swarm does as damage, p is the proportion of players affected by Penetrating Cold at cast and P is the tick size of Penetrating Cold, the steady-state health level H* is H* = ¹/_r(h - ³/₁₀pP). It is a good idea to incorporate a buffer against the randomness of the damage.
• For Heroic 10-man, the steady-state health level is H* = 5h - 450, and for Normal 25-man, it is H* ~ 10h - 228.
• As long as the steady-state health level is more than zero, the healing the healers provide is sufficient.

The given:

• On average, raid members have similar health. A more reasonable average would be of the non-tanks only, which can be used if the raid healers will not need to worry about the tanks. The latter will be used here.
• On average, healers' HPS is constant.
• Total HPS never falls below 2000 in 10-man or 5750 in 25-man (the minimum DPS inflicted by Leeching Swarm). (If it does, immediately consider the healing insufficient.)

The detail:
Since Leeching Swarm ticks once every second, the DPS d₁ it does equals its tick size, a certain proportion r of current health H:

d₁ = rH

Since Penetrating Cold ticks for P damage once every 3 seconds on a certain proportion p of players for ⁹/₁₀ of the time (18 seconds out of every 20 seconds), the DPS d₂ it does equals:

d₂ = ⁹/₁₀ × ¹/₃pP = ³/₁₀pP

The values of p and P depend on the type of the encounter.

The total DPS Anub'arak deals is:

d = d₁ + d₂ = rH + ³/₁₀pP

The healer(s) do(es) a constant HPS h. Given the definition of the steady state:

d = h
rH* + ³/₁₀pP = h

Rearranging for H*, the current health level at the steady state:

rH* = h - ³/₁₀pP
H* = ¹/_r(h - ³/₁₀pP)

The following table gives the values of r, p and P:

Type of encounter	r	p	P
Normal 10-man	0.1	1/4 (= 2/8)	3500
Heroic 10-man	0.2	1/4	6000
Normal 25-man	0.1	5/23	3500
Heroic 25-man	0.3	5/23	6000

If current health is not at the steady state, it will tend towards it. On close inspection of the formula, the steady-state health level can only be less than zero if the DPS of Penetrating Cold exceeds the HPS, since it is impossible to do negative healing to an ally. If the steady-state health level is less than zero, the healing provided by the healer(s) is insufficient.