Extended Equilibrium Optimisation of HPM for Holy Myllior, US-Frostmourne The model for the average healing model for Holy’s spells may be represented by a weighted average of the different spell types. The equations representing the contributions of all relevant statistics to the different spell types are as shown in Eqs. (1-3). (
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The factors out the front of each spell type represent the proportion of pre-secondary healing obtained from that spell type, and so the sum of these factors is unity; i.e., ( ) However, in order to attempt a considerably more detailed description, a number of additional factors may be introduced, so that the different behaviours of different healing components may be assessed, to an extent. Introducing all of these, Eqs. (1-3) then become, ( (
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The purpose of the factor in Eq. (5) is to represent the overhealing of Power Word: Shield; the factor in Eq. (7) serves the same purpose for heal over time effects (i.e. Renew ticks, Lightwell and Holy Word: Sanctuary). In Eq. (6), overhealing of direct heals is represented via , while the overhealing of Echo of Light (which is generated from such heals) is represented by . Summing Eqs. (5-7) then gives a representative model of Holy healing, as per Eq. (8), from which optimal stat distributions may be calculated. (
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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 , and represents any constants in that stat (e.g. racials, raid buffs, etc.). For a function of variables, ( ), equilibrium between its statistics is expressed as in Eq. (10), so the first step is to evaluate each of these equilibrium components. (
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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. (8) are as expressed in Eqs. (11-15). ( (
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Before continuing, there is an issue. Note that neither Eq. (11) nor Eq. (12) are functions of either nor , whereas Eqs. (13-15) are all functions of both! As such, Eq. (10) is not sufficient to evaluate the equilibrium between and , and so more a more fundamental approach is required. Consider the sub-function of Eq. (8) which incorporates and ; i.e., [
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This equation is linear in both and , without any coupling between the two. Such a linear function is maximised at its boundaries, which are where all of the stat points are allocated to either only or only. By substituting the relevant forms of Eq. (9) into Eq. (16), the values of Eq. (16) at the boundaries are as below, where is the rating allocated to be spent in either Haste or Mastery. [
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The greater of Eqs. (17a, b) is the maximum of Eq. (16); subtracting Eq. (17b) from Eq. (17a) results in Eq. (18). If Eq. (18) is positive, is superior; if negative, is superior; if the equation is equal to zero, then they are equally beneficial. (
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So the problem to be solved depends on which of or is greater. If is superior, then Eqs. (12-15), with , are utilised to find the optimal distribution of secondaries; if is superior, then Eqs. (11, 13-15), with , are utilised. As is Holy’s best secondary statistic, equilibrium of the remaining stats should all be evaluated with respect to . The equilibrium values for and are independent of which of or is superior, and are evaluated by equating Eq. (13) and Eq. (14) (for ) and Eq. (14) and Eq. (15) (for ), resulting in Eq. (19) and Eq. (20) respectively.
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Rearranging to make
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With the equations determined, the equilibrium values may be calculated using Eqs. (19, 20) and Eq. (22) (if Eq. (18) is positive) or Eq. (23) (if Eq. (18) is negative).
