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THEORY
Using an Aimed + Multi-Shot cycle.
Assume Multi-Shot is 100% instant.
Disregarding mana issues.
Disregarding haste buffs (Quick Shots, Rapid Fire, Berserking, etc).
Crits are disregarded. Ranged weapons with crit, either directly (+crit%) or indirectly (+agility) will be a bit lower here than they should be, if you compare different weapons.


Every cycle consists roughly of:
1 Aimed Shot
1 Multi-Shot
x Auto Shot

With most weapons, you'll find the following happening:
Aimed Shot finishes cooldown. Auto Shot fired ~1 sec ago.

This makes you choose between:
A Use Aimed Shot when its cooldown is finished.
B Use Aimed Shot when your next Auto Shot fires.

A makes you lose an Auto Shot, but your cycle time is shorter than in B.
B adds an Auto Shot, but extends your cycle time.

B will do more damage per cycle than A, that's fairly obvious. However, B will also take longer, so you can't automatically say that B gives more DPS than A.

Since Multi-Shot has the longest cooldown (10 sec), it would be logical to set a default cycle to 10 sec. However, since we stated Multi-Shot is an instant, the damage can easily be converted into any cycle lenght.
Roughly: If Multi-Shot does 1000 damage every 10 seconds, and the cycle is 5 seconds, the cycle damage for Multi-Shot becomes 500.
Likewise, if the cycle lenght is 20 seconds, the cycle damage would be 2000.
Basically, convert Multi-Shot damage to DPS (Multi-Shot / 10), multiplied by cycle lenght.
More correct, if a cycle is 10-13 seconds, you can only have 1 Multi-Shot per cycle, else it will crash with Aimed Shot cooldown.
If the cycle is between 9 (minimum cycle length) and 10, you have to include damage lost by conflicting cooldowns.
I won't go into the details here, ask if you're interested.

With that in mind, I'll give numbers for setting cycle default both to 10 seconds (Multi-Shot), and to 9 seconds (Aimed Shot).


AIMED SHOT CYCLE (9 SECOND DEFAULT)

The cooldown of Aimed Shot is 6 seconds, casting time is 3 seconds. During casting, Auto Shot cools down. This means that basically every weapon will get an instant Auto Shot after Aimed Shot. A 3.4 weapon becomes a 2.96 one with a 15% quiver. With a 13% quiver, it's 3.01
During the casting of Aimed Shot, no Auto Shots are fired.

This leads to the fact that a default cycle (9 seconds) has 6 seconds of Auto Shot, with a "free" Auto Shot following the Aimed Shot.

Example for a default cycle for a 1.0 speed weapon after quiver, Multi-Shot left out since it's instant and can be fitted anywhere (except during Aimed Shot casting):
1.0 Auto Shot fires
2.0 Auto Shot fires
3.0 Auto Shot fires
4.0 Auto Shot fires
5.0 Auto Shot fires
6.0 Auto Shot fires
6.0 Start Aimed Shot
9.0 Aimed Shot fires
9.0 Auto Shot fires (cooled down during Aimed Shot)
+ Multi-Shot

Reason for not including 0.0 Auto Shot fires: If you repeat it, you'll see the next Auto Shot fire at 10.0, which is correct, since one fired at 9.0. You could rewrite it and include the Auto Shot at 0.0, but then you couldn't include the "free" Auto Shot after Aimed Shot, since they're essentially the same shot.

CycleLength = 9.0 seconds
First, we convert Multi-Shot damage to cycle damage.
CycleDamage = 7*AutoShot + AimedShot + 0.86*MultiShot
CycleDPS = CycleDamage / CycleLength
CycleDPS = (7*AutoShot + AimedShot + *0.86MultiShot) / 9.0

That should take care of weapons fitting perfectly with the default cycle.

Next, weapons which don't fit perfectly:
For these weapons, you'll have the scenario as described earlier, Aimed Shot will be ready, Auto Shot will be in the middle somewhere. For example, with a 2.3 speed weapon after quivers:
2.3 Auto Shot fires
4.6 Auto Shot fires
6.0 Aimed Shot cooled down
6.9 Auto Shot ready to fire

Now, let's look at the cycles using our options from above.

Converting both into CycleDPS:
A (3*AutoShot + AimedShot + 0.86*MultiShot) / 9.0
B (4*AutoShot + AimedShot + 0.98*MultiShot) / 9.9

The best cycle in terms of damage would then be the one with highest CycleDPS.

Note that using a Aimed cycle, your Aimed Shot will catch up to your Multi-Shot. Using Aimed Shot first will yield the most DPS. The crashing Multi-Shots have been added in the formulae, so it should be as correct as possible.


RHOK'DELAR
15% quiver, speed = 2.52
Set AutoShot to 517
Set AimedShot to 1152
Set MultiShot to 753
Average damage against a target, before armor is applied.

MULTI-SHOT CYCLE (10 SECONDS DEFAULT)

In this case, we set the default cycle to 10 sec. This allows for a more static and predictable cycle, with Aimed Shot and Multi-Shot always appearing at the same time each cycle. Since we fire Aimed Shot directly after Multi-Shot, we have 7 seconds of Auto Shot in the default cycle.
Again, since we stated Multi-Shot is an instant, the damage can be converted into any cycle lenght.

Again, a "free" Auto Shot follows the Aimed Shot. The 1 second pause after Multi-Shot is the global cooldown.

Example for a default cycle for a 1.75 speed weapon after quiver:
1.75 Auto Shot fires
3.50 Auto Shot fires
5.25 Auto Shot fires
6.00 Multi-Shot fires
7.00 Auto Shot fires
7.00 Start Aimed Shot
10.0 Aimed Shot fires
10.0 Auto Shot fires (cooled down during Aimed Shot)

CycleLength = 10.0 seconds
First, we convert Multi-Shot damage to cycle damage.
CycleDamage = 5*AutoShot + AimedShot + MultiShot
CycleDPS = CycleDamage / CycleLength
CycleDPS = (5*AutoShot + AimedShot + *MultiShot) / 10.0

That should take care of weapons fitting perfectly with the default cycle.

Next, weapons which don't fit perfectly.
For these weapons, you'll have the scenario as described earlier, Aimed Shot will be ready, Auto Shot will be in the middle somewhere. For example, with a 2.0 speed weapon after quivers:
2.0 Auto Shot fires
4.0 Auto Shot fires
6.0 Auto Shot fires
6.0 Multi-Shot fires
7.0 Aimed Shot ready
8.0 Auto Shot ready to fire

Converting both into CycleDPS:
A (4*AutoShot + AimedShot + MultiShot) / 10.0
B (5*AutoShot + AimedShot + MultiShot) / 11

Again, keep in mind that using B, Aimed Shot would catch up with Multi-Shot if adding an additional Auto Shot would take more than 1 second (or global cooldown).

The best cycle in terms of damage would be the one with highest CycleDPS.


RHOK'DELAR

15% quiver, speed = 2.52
Set AutoShot to 517
Set AimedShot to 1152
Set MultiShot to 753
Average damage against a target, before armor is applied.


Since adding another Auto Shot here only takes 0.56 seconds, the cycle will remain constant, since we basically have a 1 second buffer due to the global cooldown. If you examine this cycle, you'll see it's the same as the Aimed Shot (default 9 second) B cycle.


Now... Should you always use cycle B? Should you always delay your Aimed Shot so you can squeeze in an additional Auto Shot?


ASHJRE'THUL

15% quiver, speed = 2.96
Set AutoShot to 616
Set AimedShot to 1186
Set MultiShot to 792
Average damage against a target, before armor is applied. These numbers are with the exact same gear as in the Rhok'delar example above, ranged weapon is the only thing changed.


Aimed + Multi-Shot (9 sec default)


In this case, A is better than B. This leads to the conclusion that your choice of cycle should be dependant on your weapon's speed.

Now, let's check with the...


Multi + Aimed cycle (10 sec default)


NOTE: Using the optimal Multi-Shot rotation is worse (by a fair margin) than using the optimal Aimed Shot rotation for Ashjre'thul.

Feel free to ask me to run through the cycles for other weapons, and I'll post them in a reply as fast as possible.

Oh, and and European servers already ;)
Talnivarr EU

Edit: Should be a bit more readable now ;)
Edit 2: Updated with corrections to Multi-Shot damage per cycle.

HUHURAN'S STINGER

15% quiver. Speed = 2.35
Set AutoShot to 493
Set AimedShot to 1154
Set MultiShot to 755
Average damage against a target, before armor is applied. These numbers are with the exact same gear as in the Rhok'delar example above, ranged weapon is the only thing changed.


Aimed + Multi (9 sec default)

Multi + Aimed cycle (10 sec default)


BLESSED QIRAJI MUSKET

15% quiver. Speed = 2.26
Set AutoShot to 477
Set AimedShot to 1150
Set MultiShot to 752
Average damage against a target, before armor is applied. These numbers are with the exact same gear as in the Rhok'delar example above, ranged weapon is the only thing changed.

Aimed + Multi (9 sec default)

Multi + Aimed cycle (10 sec default)


Edit: Updated with conflicting timers taken into consideration.

Toxin Injector.

As an additional point, going with Cycle B will abuse the 5 second rule more efficiently and get a slightly better mana regen rate, since you continue to regen mana during the 3 second cast of Aimed Shot.

Aimed Shot immediately followed by Multishot gives you only one interuption in your mana regen cycle. (Time from Aimed Shot going off until Multishot goes off, then the 5 second rule, and then you get regen tics until the next Aimed Shot.

Multishot followed by Aimed shot "wastes" the time between the Multishot and Aimed Shot (in regards to mana regen) since the Multishot breaks mana regen, and nothing happens during the 3 seconds Aimed Shot is casting, and then once it casts, the 5 second rule resets.

Heh, just came from reading the thread about that rogue spreadsheet...and gawd damn, when you compare hunter mechanics to rogue mechanics it sometimes feels like going from arithmatic to linear algebra. =P So much more going on there.

So, I shouldn't point out that to properly derive the crit/AP/hit/skill equivalence I should have derived an equation for raw DPS as a function of our melee variables, then taken the respective partial derivatives w/r/t crit, AP, hit, and skill, and setting them equal given the formulas for checking your equivalence? ;)

Hunter mechanics seem fairly clean, but how does Rapid Fire, for example, affect all this? How do you modify it when you get a Quick Shots proc? Etc.

Same mechanics, just using different times.

For example, an Aimed (default 9) cycle is composed by the 3 sec casting time, and the 6 sec cooldown. Auto Shot during cooldown (+ possible additional time to get the last one in). With Rapid Fire, your weapon speed is divided by 1.40, as is the casting time of Aimed Shot. So, still the same time for Auto Shot (although you'll most likely get in more Auto Shots now). The thing to remember is the fact that the default cycle time is changed from 9 to 8.14, when you come to the stage where you convert Multi-Shot to cycleDPS.

Same with Quick Shots, just replace 1.4 with 1.3 ;)

1493 RAP
5/5 Ranged Weapon Specialization
3/3 Barrage
3/3 in Slaying talent vs target
+7 scope

Crits aren't included in this. As far as I can tell, with this setup, unbuffed (25% ranged crit), you'll have an average DPS of 549 before armor mitigation.

Edit: With Ashjre'thul, that is.
Oh, and this is using Jagged Arrows btw. I knew there was something I forgot :P

Well, to be fair, isn't this exactly what you did? You just did it numerically. But you're still basically trying to find the gradient of the DPS surface in a phase space of all stats.

The sad part is that doing the above would be trivial for Mage DPS--everything is linear.

This is great stuff, by the way.

A question:
If you're using a cycle of length not equal to 10 seconds, won't Multi-shot "precess" around the cycle, eventually landing in the middle of an Aimed Shot? You can't realistically claim to fire Multi every 10 seconds when you occasionally have to wait out an Aimed cast time. With a cycle of length just over 10, it would seem your Multi-shot cooldown will effectively be equal to the cycle length.

Well, for a cycle length greater than 10 seconds, it seems like a decent approximation is to assume that your effective Multi-shot cooldown is equal to the cycle length. With a length of, say, 11s, you can't get more than one MS off per cycle anyway, so the coefficient in front of the Multi-shot term should be 1.0, not 1.1.

For cycles shorter than 10s, it's more complicated, but the effect is much smaller. All that will happen is that you occasionally have to wait 3s for your Multishot.

Actually, it's not all that bad. Look your 9s ideal Ashjre'Thul cycle, for example. Multi-shot precesses 1s forward every cycle. If on cycle #1 you use it immediately after Aimed shot #1, then on cycle #7 it will interfere with Aimed Shot #8. Overall, you get to cast 6 Multi-Shots every 7 cycles, rather than 9 every 10 cycles. So the coefficient in front of the Multi-shot term should be 0.857, not 0.9.