Modelling borrowed time

[Deleted]

Borrowed time is not quite as straightforward as it might seem and it can sometimes provide results which are a bit counterintuitive.

A simple way to tell whether a particular spell selection will benefit from borrowed time is the following:

Since casting a PWS costs a GCD but reduces the time to cast the next few spells, which approximately model the process as borrowed time "borrowing" some of the time needed to cast PWS from the subsequent spells. I.e. you would get the exact same HPS is you subtracted the time you "pretended" that it was the cast time of PWS that was shortened rather than the cast time of the other spells.

Therefore the requirement for a rotation which in the absence of any haste takes ta seconds to cast and produces HPS = HPSa.

If your haste is H% then the time taken to cast this sequence will actually be ta/(1+H) and it will produce HPSa*(1+H). With borrowed time it will take ta/(1+1.4*H). Thus the difference between these two is dt = 0.4*H*ta/[(1+H)*(1+1.4*H)] = ta/(1+H)*0.4*H/(1+1.4*H)

Thus in order for PWS to be a HPS increase the following condition needs to be satisfied:

Healing(PWS)/(1.5/1+H - ta/(1+H)*0.4*H/(1+1.4*H) > HPSa*(1+H)

or
H(PWS)/td > HPSa, where H(PWS) is the healing done by PWS and td = 1.5 - ta*0.4*H/(1+1.4*H)

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Calculating a formula for the total HPS for borrowed time:

HPS_total = HPS(PWS)*t_pws/T + HPS*t/T
,where t_pws is the time you spend casting PWS,
HPS(PWS) is the HPS of PWS,
t is the sum of the cast times of all the other spells in the absence of all haste (so its easy to calculate),
HPSa is the HPS of all the other spells in the absence of all haste (so its easy to calculate),
T is the total time and
H is your haste without borrowed time.

from the results above T = t/(1+H)+1.5/(1+H) - 0.4*H*ta/[(1+H)*(1+1.4*H)] = (t + 1.5 - 0.4*H*t/(1+1.4*H))/(1+H).

Since you only cast 1 PWS per borrowed time buff t_PWS = 1.5/(1+H)

Thus

HPS_total = (HPS(PWS)*1.5 + HPS*t*(1+H))/(1.5 + F(H)*t), where F(H) = 1-0.4*H/(1+1,4.H)

simplifying this gives:

HPS_total = (healing_PWS + HPS*t)*(1+H)/(1.5 + F(H)*t)

[Well]

Yeah, so? What would your suggestion be?

[Deleted]

i need 2 add the final bit

[Well]

i need 2 add the final bit

That's good. Looking foward to reading it.

[Deleted]

Lets look at the final formula one more time.

HPS_total = (healing_PWS + HPS*t)*(1+H)/(1.5 + F(H)*t)

As one could derive from elementary arguments this is simply healing done by PWS + healing done by all the other spells cast in the borrowed time window divided by the time it takes to cast the whole sequence. The (1+H) factor is the scaling of spell HPS with haste and the factor the decides the final HPS of a borrowed time sequence is F(H)*t, which is the factor that determines the cast time shortening.

This is a table that shows the value of F(H) at different haste values.

Haste	F(H)
0.000	1.000
0.050	0.981
0.100	0.965
0.150	0.950
0.200	0.938
0.250	0.926
0.300	0.915
0.350	0.906
0.400	0.897
0.450	0.890
0.500	0.882
0.550	0.876
0.600	0.870

t is the time it takes to cast all the spells you can cram into the borrowd time window at 0% haste. So t gets larger with haste, but in a discontinuous fashion, since a cast started within the borrowed time window will be hasted even if it ends when BT has expired. Thus t increases by a set amount at specific breakpoints. Here is an example of spamming holy nova after borrowed time:

No HN per BT	Haste
3	0
4	0.05
4	0.1
4	0.15
4	0.2
5	0.25
5	0.3
5	0.35
6	0.4
6	0.45
6	0.5
6	0.55
7	0.6

These breakpoints create a sew like appearance on the HPS vs haste graph but the variations are actually quite small, because both the discontinuous bits are added to quite a large constant and there are discontinuities both on the nominator and the denominator.

Now that we laid down the groundwork lets use the formulas to obtain a quick comparison of crit versus haste for holy nova spam.

PWS is now just slightly lower HPS than holy nova, though that depends on whether you are stacking crit or multistrike. That means PWS is always going to be a benefit when using holy nova regardless of how much haste rating you have. So it makes sense that you will weave PWS between holy novas.

So first lets use the following values: 5000sp 0.3 mastery, 0.05 haste, 0.05 versatility, 0.05 multistrike and lets obtain the HPS of a PWS-HN spam sequence as it ramps with crit.

At this value of haste you can already squeeze 4 HNs into a single borrowed time. Lets verify that this is true: At 5% haste GCD is 1.429s. so you will have 4.671s if casting at 7% haste which makes the GCD 1.402s and you can squeeze 3.26 HNs 0.26 of a holy nova is 400ms so plenty of time to account for latency and you can definitely get 4 hasted HNs per PWS.

t will be 1.5*4 = 6 and F(H) at this haste amount will be 0.981. Putting it all in the formula gives us

HN1x	PWS	Crit	HPS_tot	HN HPS
7153	39859	0.05	25996	25034
7589	41757	0.1	27507	26562
8026	43655	0.15	29018	28090
8462	45553	0.2	30528	29618
8899	47451	0.25	32039	31146
9335	49349	0.3	33550	32674
9772	51247	0.35	35060	34202
10208	53145	0.4	36571	35730
10645	55043	0.45	38082	37258
11082	56941	0.5	39592	38786

The benefit is minor, but it is there.

Now lets do the same for haste:

HN	PWS	Haste	HPS_tot	HN HPS
7153	39859	0.05	25996	25034
7153	39859	0.1	27602	26226
7153	39859	0.15	29205	27419
7153	39859	0.2	30806	28611
7153	39859	0.25	32370	29803
7153	39859	0.3	33979	30995
7153	39859	0.35	35588	32187
7153	39859	0.4	37195	33379
7153	39859	0.45	38810	34571
7153	39859	0.5	40425	35763

As you can see there is indeed a slight benefit to stacking haste instead of crit for holy nova spam so you can benefit from borrowed time, but it is absolutely minor. The results are fairly similar for practically every spell sequence you can come up with. So yes haste is actually more valuable than crit but the interaction of crit with aegis, makes this a moot point. It is just not big enough to matter. BT as it is now just means having some left over haste isn't entirely a bad thing. The borrowed time mechanic itself is rather tiny and people can ignore it without making any real impact.

In my opinion borrowed time as it is now is merely a gimmick. It is not a significant mechanic. I don't disagree with it being an improved scaling factor for haste, but it should be something significantly bigger than this to make us ignore the massive HPM and absorb loss by not going for crit all the way. I think it is rather sad to introduce a trivial mechanic just so people don't cry if there is haste in their gear.

[Sillychan]

Thanks for this Havoc, very useful breakdown.