‘Full’ Equilibrium Optimisation of HPM for Discipline Myllior, US-Frostmourne A healing model for Discipline’s spells may be developed by considering a weighted average of the different spell types. The equations representing the contributions of all relevant statistics to the different spell types, when overhealing is neglected, are as shown in Eqs. (1-5). In this model, Spirit Shell is not included, and Haste is also not considered as HPM is being evaluated (note that Haste has an HPM benefit for ticks of Power Word: Solace or Holy Fire, however the effect of these is so small that it may be neglected). ( (
)(
(
)( ),
)( (
(
)(
), ),
)(
)(
(
)- (
( (
),
)(
)( )(
(
)
)( )- (
( ) )
)(
)- (
)(
)- (
( )
)(
)
( )
)
( )
)
( )
The spells that behave according to each of the above equations are largely made clear by the names to the left of them; Eq. (5) refers to either Prayer of Healing or Flash Heal cast to consume the Empowered Archangel buff, while the spells that follow Eq. (2) are essentially all other Divine Aegis producing spells (except Prayer of Healing). In Eq. (3), is the increased Critical Strike chance due to Enlightenment ( ), while in Eq. (4), represents Critical Strike suppression ( ), since Atonement spells do their healing via damage. The factors out the front of each spell type represent the proportion of pre-secondary healing obtained from that spell type. The sum of these factors is unity; i.e., ( ) In order to attempt a more detailed description, additional factors may be introduced so that the different overhealing behaviours of different healing components may be assessed. Introducing all of these, Eqs. (1-5) then become, ( ( (
)( )(
)(
)(
)(
)(
)[ )[
( (
)
( )
)( )(
)] ( )(
) )] (
( ) )
( )
( (
)[ )(
(
)(
)[
)] ( (
)( )(
) )] (
)
(
)
(
)
The purpose of the factor in Eq. (7) is to represent the overhealing of Power Word: Shield and/or Clarity of Will. In Eqs. (8-9, 11), two factors are introduced to represent the overhealing of base healing ( ) and multistrikes ( ), while in Eq. (10) a single factor is used to represent the combined healing of both base healing and multistrikes ( ). This single factor is used as Multistrikes are registered under damage, not healing, for Atonement spells, so the different overhealing behaviour of base healing and multistrikes cannot be discerned for Atonement spells (at least, not by simply examining the overall healing breakdown). The third factor used in Eqs. (8-11) represents the overhealing of Divine Aegis ( ), and this factor is the same for all four spell types since Divine Aegis is not broken down by the spells that created it; it also does not separate Divine Aegis due to critical strikes from Divine Aegis due to critical multistrikes. Summing Eqs. (7-11) then gives a representative model of Discipline healing, as per Eq. (12), from which stat calculations may be performed. (
)
(
){ (
(
)(
)(
)[ (
)[ (
[ (
)[
)(
) (
)(
(
)(
)] (
)] )(
)]
) (
)(
)]} (
)
(
)
Each of the statistics may be expressed as a function of the number of points allocated to them by a function of the form, (
)
where is the number of points required in the stat to achieve an increase of (or 100%), and represents any constants in that stat (e.g. racials, raid buffs). For a function of variables, ( ), equilibrium between its statistics is expressed as in Eq. (14), so the first step is to evaluate each of these equilibrium components. (
)
Ignoring Spellpower, as it is not a secondary statistic in the sense of the others, the equilibrium equations for each of the statistics in Eq. (12) are as expressed in Eqs. (15-18). (Note: In some of the following equations, such as Eq. (15) and Eq. (17) below, the terms corresponding to the different spell types have been shuffled around in order to make the equations more presentable).
(
{
)(
(
) (
)(
[
(
)(
[
(
)(
[
*
,(
)(
(
{
(
[ (
(
)(
)[
(
(
{
)
)(
)(
( (
)]
)]} (
)
)(
)(
)[
)]
(
)(
)]}(
( )[
)
(
)
(
)
(
)
(
)
)] (
)]
)
)
)[
[
)(
(
)[ (
)
)]
-+(
)
)(
)(
)(
(
)(
(
)(
)] (
)[
)] )(
)]
) (
)(
)]}
As is currently Discipline’s best statistic, and in realistic situations where it is not, takes over as the best statistic, equilibrium of the remaining stats should all be evaluated with respect to either or . The first step is then to evaluate equilibrium between and . By equating Eq. (15) to Eq. (16) and cancelling out the like terms, {
(
)(
(
)
)(
)
(
)(
[
(
)(
[
(
)(
[
* As is nonlinear in Eq. (19) while the subject of Eq. (19) gives,
,(
)(
)
)] )(
)]
)]} -+(
is linear, it is preferable to express
)(
)
as a function of
(
)
. Making
*
( {
,( (
)
)(
.
0
)(
)
/
.
)
/ *
-+(
. ,
(
)(
).
/ /1 }
)-+ .
(
)
/
Next, evaluate equilibrium between and by equating Eq. (17) to Eq. (15). Equilibrium is evaluated in this manner as Eq. (16) for is not a function of (at least, not directly), so in the unrealistic event that , this would create a zero denominator, the possibility of which should be avoided. Following the same process as before, the result is, ( )[
( (
)(
)[
(
(
{
)(
Finally evaluate equilibrium between
and
)
[
{
)(
(
).
{
).
)]
).
(
)
(
)
}
).
)] .
/
(
/1
). ).
)( (
/1
).
(
/
/1
)(
(
(
0
)]
/
)(
0 0
)- } )
(
(
( )0
)) ]
by equating Eq. (19) to Eq. (15) to give,
(
(
(
[
)0 [
) )
(
)0
(
,
)]
( (
( (
{
)(
) ( (
[
)] } (
(
[
,
)]
( )
(
)] ) ]
)
)[ (
{
( (
[
) ( (
/ /1
). ).
}
/1 /1
}
In their above forms, the equilibrium equations cannot be utilised. This is because the equation for contains , whereas the equation for contains . It is possible to resolve this issue, by substituting the equation for one of and into the other, and rearranging to again make or the subject. This would reduce the rearranged equation to become only a function of , and the rest of the solution process could then be carried on from there. Note that, in the event that , must be explicitly enforced, to avoid division by zero.
