Norwest

For each individual shot, iAotH is either on or off, so I tried thinking about the problem shot-by-shot instead of per-second. What I did was calculate the average number of shots a proc lasts, counting reprocs and everything. I also calculated the number a shots a 'no-proc' streak lasts.

If we know the average number of shots in a proc chain and the average number of shots between procs we can easily find the average number of hastened shots.

No-proc Streak:
Pretty simple, Hawk has a 1 / 10 chance to proc, the average length of a 'no-proc' streak is 10 slow shots. The fact that Quick Shots doesn't add haste to the shot it procs off of is accounted for.

Proc-Streak
The length of a chain of procs is a bit harder to model. But it can be done by calculating the number of fast shots inside a proc, and calculating the chance that one of those shots will chain the proc a second/third/fourth time. Each extra chained proc adds (on average) half of the duration of the proc. I modelled up to 15 procs in a row.

One small twist, is that the first shot is not hastened when iAotH procs. That makes 1st time procs and chained procs behave slightly differently depending on the bow speed.



After modelling it, I recalculated the probabilities using a Taylor Series and got the same answer. If you don't know what that is, don't worry, because, I don't really know either I just know it works.

Here's my calculation Spreadsheet:
http://www.mediafire.com/?8dnxqmdqddw

If people like my approach to this, I'll work it into the original spreadsheet. I could get the whole thing down to 2-3 lines easily, but I don't want to do that before I think people agree with my approach. I'm always very bad at explaining my mathematical thinking to people, I apologize I'm sure this is confusing.

I'll be on tomorrow to answer any questions etc.

 

◊ Lactose

I'm not sure what you mean, actually.
Do you mean:
A -- With setup 1, will my new pair of gloves (gloveX) be an upgrade to what I have in setup 1?
B -- With setup 1, what will be the best upgrade, gloveX or gloveY?

A should be possible already, current / new item(s), B isn't done as of now, and would require some rearranging.

Look, Lactose, we'd rather you didn't eradicate the whole human race.
- Sam & Max

 

Nightarcher

Ok let me get it straight. Lactose your table is while buggy in some places if you ignore that its more accurate for calculating what is better 1agi or 2 AP?

Cheecky your table is good for the big picture of dps but for fine tuning is not really good since using 50 shot sim instead of actual formulas?

In summary I should use the ratio values of Lactose atm to compare items with very minor difference? Like which gem to use 8 agi or 16 ap?

 

Dibbler

I use it for example A mostly but I can wiggle the information to fit example B.

 

Cheeky

I guess it all dpends with what you are looking for. I'm going to remove some of the explicit rounding in mine and just scale what the cell shows in terms of precision displayed. Excel keeps all the true values through the calculations to the end. That should eliminate the discrepencies between the output of Lactose's and mine.

I never really bothered to provide much of a 1 more X or X is equal to so much Y function to mine. I figured you could just make the change at the item/enchant/gem/talent level and see what the result is. If you are comparing 3-4 different item sets of gear I think mine is a bit easier to use, but for heads-up compares Lactose provides both sets of results simultaneously.

While mine requires me to keep on top of gear stats, it does have the advantage of not requiring you to have Armory, Wowhead or your character available to use. Just pick the gear by name and you're done.

I have also used my shot rotation to figure out exactly what the best way to use trinkets, racials, rapid fire, etc is. It's nice to be able to say in 12 seconds I can deal X amount of damage every 3 minutes. Without having to hand count shots and total damage up on my own.

Cheeky of Havoc

 

Aesa

Cheeky, I think the inaccuracy in the 50 shot method of dps calculation arises from the fact that right now it assumes that the 50 shot sequence will restart at the time that shot #50 finishes, which almost always is not true.

If you let the user select a 51st shot that would theoretically restart the whole rotation, you could use the starting time of that shot for your Total Time value. The 51st shot would not have its damage added to the Total Damage value, however.

With that minor tweak, it shouldn't matter if you only select one Auto Shot and one Steady Shot in your rotation, provided rounding errors aren't an issue.

 

Kolusius

I made a 3D graph at one point of crit vs rap at one point, with damage as the z axis, its hard to really infer anything through it though. There is one for auto shot, multi shot, steady shot, and one for arcane shot, and I used weighting with the results of each to get a combined (rough) "shot rotation" type graph, (4 auto, 3 steady, 2 arcane, 1 multishot) based off the rotation I generally repeat. I suppose I can divide by a time element to get it for dps as well.

It showed a definate "sweet spot" in the ap/crit ratio, but by the time I was finished, I wasnt really sure what I was looking for. I can see where my current stats are on the graph, and decide oh, if I add 40 attack power, and take away 4 crit rating, I'll be right on the sweet spot. But how do you accomplish that? I guess you could to a small degree with gems, but what does that accomplish?

Adding crit always increases damage. Adding rap always increases damage. Moving towards the sweet spot is fine, but moving away from it will increase your damage as well (as long as its a positive change to ap or crit, depending on which side you fall).

If I could figure out an exact equation for the crest of the curve, it would be useful, at least for getting an ap/crit ratio, and all the math/calculus I did in college was theoretical, not practical. Too many Number Theory classes, not enough applied math classes. Give me real numbers and I'm at a loss.

Anyways, I can check if I still have them laying around, and post them if you're interested. They're pretty rudimentary, and I've yet to really find a use, other than looking at pretty colors. I'll dig around and see if I still have them.

 

Cheeky

That's a damned good idea. I'm going to add that now and see if I can get it to match the theoretical DPS. You'll still run into issues with Multi-Shot and Arcane Shot Cooldowns if you haven't taken them into account for your "finishing" shot, but that becomes user error then.

Doing that and using a Auto/Steady rotation gets my DPS calculated to 922.14 vs. Lactose's 921.88. I think 0.03% difference is pretty negligable. I'm going to look into tweaking the code that builds the priority rotation to try and find Steady Shot patterns between Multi and Arcane, maybe I can have it stop early when it has completed a cycle.

Cheeky of Havoc

 

Cheeky

Norwest, I think you may be complicating it a bit more than you need to. [later - but after working through the numbers, maybe not.]

Rather than solve for the uptime, lets solve for the chance on any on shot that you are not in Quick Shots.

To do this you need to figure out how many shots have happened that you know it hasn't proc'd. This would be floor(12/quick speed), quick speed for a 2.8s weapon with quiver and SS is 1.76, so 6 shots.

Then we know that the chance all of those 6 shots did not proc Quick Shots is (1-chance for QS)^(shots), or .9^6 = 53.14%. That means there is 46.86% chance that one of those 6 proc'd Quick Shots. This chance is true for any shot S in an infinite series of shots. So we can generalize the up time to 46.86%

Using a 2.7s weapon like in your spreadsheet yields an uptime of 1 - (.9)^floor(12/(2.7/(1.15 * 1.3 * 1.2))) = 52.17% uptime.

The only difference is that you are trying to account for the fact that for some weapons there is a difference in the number of shots that can fit into the 12 second window when it is Quick and when it is normal. I think we can account for this by averaging the two values, which while not exact is pretty easy to do. Doing this for a 2.7s weapon yields 49.58%, about 2% off from your calculations. This methodology fails in assuming half the time QS won't be up before our shot and 1/2 the time it will be, when in truth it's the same ratio as QS itself.

In the interests of accuracy we should try using your calculations, but they make an assumption too, they assume there is 0 auto shot clipping in the shot rotation. We can account for this by using the Shot Rotations matrix to compute autoshots/second, which works even if there is clipping.

This is very interesting, and good work on your part. I'm going to think about this a bit more, but so far I think your procedure is superior to what I am using now. (Maybe it's worth looking at Frenzy the same way.)

Cheeky of Havoc

 

The Iron Colonel

For the sake of discussion, I'll mention that you can use a very similar method (a power series, that is) to describe the behavior of the new Darkmoon card (wrath). More often than not a series probably describes the proc functions more accurately than simple probability, but they also lead to more headaches due to their relative complexity.

 

Jezele

I've played around with both, but ultimately decided to create my own based on your formulas.

I've primarily used Cheeky's - not only because it was easier to "steal" formulas from, but I liked the way that everything was laid out so that you could see the "guts" of how various changes in gear would affect different variables of your play. I've been going with simple rotations, so the ability to model a more complex rotation is a non-factor for myself, but that's also a great tool (though I think I would have gone with a time cut-off rather than an arbitrary number of shots). I also liked how easy it was to compare various talent setups.

Overall, I think both are great tools, but what I found lacking in both was an easy way to compare multiple pieces of gear to determine how much of an upgrade various options are. Lactose's allows a small comparison sample, whereas it's very cumbersome with Cheeky's as you have to write down the DPS values for each individual piece of gear if you want to compare them. This is actually why I created my own, which lists a fair number of potential upgrades, then calculates the resulting change in DPS for each one so that I can prioritize which pieces of gear to go after. I didn't bother putting in a lot of options relating to talent setup or rotations, so it's a lot simpler for me to do, but it filled the need of being able to compare multiple options at a glance.

So basically, I think both are great tools, but since they didn't quite fit my own personal needs, I ended up crafting my own. But I couldn't have done it without the guidance of both of your great spreadsheets.

 

Trieste

I have a problem to open your spreadsheet cheeky!
It sais that there's a password but in a language which is not taken into account...

need help

 

The Iron Colonel

Easiest way That I've found to compare gear sets in Cheeky's sheet is to simply make several copies of the sheet. That way you can directly compare gear sets side by side. Not an elegant solution, but certainly workable.

 

Cheeky

That's odd. It only works with Microsoft Excel, not any other viewer. The whole thing uses Arial as the font, which I think is pretty common.

What application are you using to open it?

Cheeky of Havoc

 

Trieste

I use microsoft excel lol
First time I have this problem. It's really weird.

I just get openoffice and it works...

I don't really understand lol...

 

Cheeky

I'd like to come back to this idea. I've thought about it more, and I like it a lot. Has anyone found any flaw in the reasoning or math behind it. I've added it into my spreadsheet (to go along with the new Quick Shots rotation and DPS ratio idea.) It's kind of difficult to describe some of the terms easily, but I don't think that's a big deal. I'm much more interested in accuracy of information.

Norwest, would you have any issue with my using this as part of the next release? I will, of course, give you full credit for the calculations.

Cheeky of Havoc

 

The Iron Colonel

I understand it conceptually, I haven't gone through the math to check for errors. Conceptually it makes sense to me, but calculating the length of a streak that falls within a proc is a little tricky. The obvious way to do that would be to calculate the chance of NOT getting a proc in a streak for an arbitrarily long string of attacks. However, the time issue makes this tricky. You have to figure out if and by how much the time under effect from procs overlaps.

From the start of the first buff you'll have a 1-0.10^(floor(12/attack speed)) to refresh the buff. As getting a new buff overwrites the old one with a fresh duration, the likelihood of getting a string of new buffs that are contiguous would be [1-0.10^(floor(12/attack speed))]^n, where n would be the nth buff in the string. This can be made into a sequence like
This assumes that if you proc on the last shot of the buff the buffs would be overlap enough to be continuous, but this is not TOO unrealistic.

The problem I'm having, and I think Norwest addresses it better than I do, is determining how many shots fall within a string of hasted shots. I'm having a fair amount of difficulty digging through the spreadsheet (norwest's, that is) because things just aren't labeled in a fashion in which I can make sense. There's a column labeled "chance of ending quick shots" in which the probability has a limit of 0 as the number of extra procs gets higher. I assume he means this is the likelihood of getting a second proc during the duration of QS and how this effects the extension of the time under effect, but it's unclear to me from the spreadsheet.

So, here's my request: can you (or someone who understands your work) more clearly articulate what you're doing? I can sort of see the direction, but the spreadsheet really isn't aiding me in understanding your model. The Taylor Series you have (by the by, I'm not sure it's actually a Taylor Series, but rather just a series (correct me if I'm wrong). Purely a terminology thing, but precision in terminology will help in this conversation) looks to be, from cell b18,
which is essentially the same as
Can you explain from whence this comes? I know it's Monday, but I'm having a hard time figuring out how you derived that equation for the series. Understanding where this is coming from would really aid me in participating in this discussion (maybe I'm the only one who's not following your math, but could you humor me?).

As for the uptime calculation itself, it looks like it's the number of shots from a single proc plus the product of the probability of a second proc and the number of shots from a second proc is being used as the average length of a proc streak. You're dividing this by the sum of the length of a no-proc streak and the average length of a proc streak (that is, you're determining what percentage of the aggregate of your shots are in a proc streak). This stuff all makes sense to me (everything in this paragraph).

Anyway, having someone clearly articulate what the rationale behind each step would be great, I'm just not able to follow it from the spreadsheet.

Also, a nitpick: you use roundup for determining reproc chance rather than rounddown. Either is valid, however I'm inclined to use rounddown as a more conservative estimate. I changed it in my copy of the spreadsheet, but this is incredibly minor.

edited to remove erroneous information (such as to not spread misinformation)

 

Cheeky

Only auto shots can trigger Improved Aspect of the Hawk.

I believe this is because Norwest subtracts off one shot's worth of time from the 12 seconds. By rounding up he adds this shot back in. (Although it may cause errors where you have an exactly evenly divisible number of shots.)

[is "exactly evenly divisible" poor grammer? ]

Cheeky of Havoc

 

The Iron Colonel

I'm a moron who can't read tooltips - I just copied an equation I have in a spreadsheet for determing up time on non-internal cooldown proc-based effects. Thanks for correcting that.
I see why he subtracted out the first shot (has to do with when the time under effect starts), so that makes sense. However, I'm not really clear on why you'd need to add that shot back in. That shot is the shot that procs iAotH (ergo it can't proc it a second time), so it doesn't need to be included in the calculation. Am I missing something? Do I just have a bad case of the stupids today? There's no reason I can see to add that shot back into the equation for up time calculations.

edit: By the way I just realized I had the duration of iAotH incorrectly listed as 10 seconds instead of 12 seconds in my post. It's just not my day, sorry for lowering the tone of this thread with bad information.

 

Cheeky