How much CDR is required for permanent Vengeance uptime?
TL;DR
Without Dawn, you need ~77.8% CDR.
With Dawn, you need ~36.5% CDR.
(This is the number you see in your inventory, not the sum of CDR on your gear. See below for explanation and how to achieve this).
Maths
How CDR stacks
As briefly explained in this answer, Cooldown Reduction (CDR) stacks multiplicatively. This often leads to confusion among newer players, as the final number is smaller than the sum of CDR on your equipment.
For example:
Say you have two items that reduce the cooldown of a skill by 50 % and 20 % respectively. A naive interpretation is that the resulting cooldown will be reduced by 50 % + 20 % = 70 %. But the cooldown is only reduced by 60%. Why?
The formula for cooldown reduction goes like this:
[CD] = [BCD] * (1 - CDR1) * (1 - CDR2) * (1 - CDR3) * ...,
where CDR1, CDR2, ... are the CDR numbers for each individual contributor, expressed as a fraction (so 20 % = 0.2) and [BCD] is the base cooldown of the skill.
So for the example above (50 % and 20 %), we get:
[CD] = [BCD] * (1 - 0.5) * (1 - 0.2) = [BCD] * 0.5 * 0.8 = [BCD] * 0.4
So the resulting cooldown is 40 % of the base cooldown.
Is this 'diminishing returns'?
No. There is no reduced effectiveness of CDR as more is added. Adding 20 % CDR will increase the amount of casts possible in a given time by 1 / (1 - 0.2) = 1.25, or 25 %. The apparent diminishing is just an effect of using percentage-based cooldown reduction (fixed time reduction would be a different manner).
You can think of it as follows:
Any subsequent CDR effect reduces the remaining cooldown by exactly how much it says. Since earlier CDR reduces the cooldown, there is less cooldown to be reduced, and thus the effect of subsequent CDR appears smaller (as measured in seconds).
Permanent Vengeance
Vengeance has a 90 second cooldown and a 20 second duration. To maintain permanent uptime, we need to reduce the cooldown below 20 seconds.
We then have the following relation:
20 > 90 * (1 - CDR),
which when solved for CDR becomes (given CDR < 1):
CDR > 1 - 20/90 which approximates to CDR > 0.778.
We thus need at least 77.8% CDR for permanent uptime.
Keep in mind that CDR here represents the resulting combined cooldown reduction from all your gear, subject to the stacking detailed above. This is in most cases unrealistic to achieve with CDR from gear, paragons and gem. Instead, we can look to Dawn.
Permanent Vengeance with Dawn
Now we have Dawn, which reduces the cooldown of Vengeance by 50-65 %. We can plug this into the CDR formula as one of the factors:
20 > 90 * (1 - 0.65) * (1 - CDR),
which when solved for CDR gives (again, given CRD < 1):
CDR > 1 - 20 / (90 * (1 - 0.65)) or approximately CDR > 0.365.
Which tells us we need at least 36.5 % CDR from gear, paragon and/or gems.
This assumes that your Dawn rolled a perfect 65 %. This is a fair assumption, considering most people (as of 2.4) use Dawn in Kanai's Cube rather than actually equip it (it automatically gets the best possible roll when in the cube).
Ways of getting enough CDR for permanent uptime
We have the following commonly obtainable sources of CDR. I will assume Dawn is being used with a 65 % roll, as is the most common case:
- Paragon - up to 10 %
- Weapons - up to 10 %
- Armor, Quiver, Jewelry - up to 8 %
- Diamond in head slot socket - up to 12.5 % (25 % with Leoric's Crown)
- Gogoc of Swiftness - 15 % (but not fully consistent)
The most common suggestion is to use Paragon (10 %), Diamond (12.5 %) and three pieces of equipment (8 %). This brings it to a total of:
[CD] = [BCD] * (1 - 0.1) * (1 - 0.125) * (1 - 0.08)^3,
which becomes:
[CD] ~ [BCD] * 0.613, which equates to CDR ~ 38.7 %.
This is slightly over what is needed, but that is a good thing. Not only does it give you margin of error during gameplay (overlap), but it also allows you to keep permanent uptime even if your gear doesn't roll perfectly.
You can mix and match any sources of CDR to arrive at the 36.5 % required, but in my personal opinion, I suggest using Paragon, Diamond in head, Shoulders, and then choosing between Quiver, Gloves and your Rings, depending on your preferred balance of toughness and damage (and what you happen to find).