Procs

Definition
A procs is an effect that has a chance of occurring from certain qualifying events. This includes effects that proc off of physical attacks, spell casts, and certain rare conditions such as being attacked, being crit, or taking a certain type of damage.  Procs are of two general types, static-chance and proc-per-minute (PPM).  Procs may also have what's know as an Internal Cooldown (ICD), which indicates the period of time after a proc occurs in which the proc cannot occur again.  In general, static-chance attacks tend to have ICDs, while PPM effects tend not to.

Proc Per Minute
Effects that run on a PPM, system occur almost exclusively as procs from weapon attacks, and are designed to give approximately the same proc rate regardless of the speed of the weapon.  Each proc is given a specific PPM number (for example, 2 PPM for Rune of the Fallen Crusader).  This number is then multiplied by the base speed of the weapon divided by 60 to give the percentage chance per attack of the proc occurring:

P = Proc chance per swing
M = Proc per minute value
S = Base speed of the weapon


Excluding a few BC-era items, the proc chance is calculated from the weapon's base speed (in other words, the speed listed on the tooltip).  Thus any haste effects will increase the actual proc rate of the effect, as the character will swing more often, but at the same percentage proc chance.

In addition, most PPM attacks are capable of procing off special attacks made by the character, though it may be limited to weapon-based specials (in other words, special attacks that involve using your weapon to attack, such as Heart Strike or Crusader Strike, but not Bloodthirst or Judgement).  In these cases, the special attacks use the same percentage chance as the melee attacks.

This means that for PPM effects, the chance per swing from white attacks is irrespective of weapon swing, but the proc chance per special increases as base weapon speed increases.  Assuming two players, one with a fast weapon, one with a slow weapon, with the same amount of haste from buffs and gear, using the same number of procable special attacks per period time, the player with the slower weapon will experience more procs on average over a given interval due to the higher proc chance per special attack.

Static Chance
Static chance procs function just as they sound:  they have a fixed percentage chance per qualifying attack to proc.  Otherwise, they function just as PPM effects.

Uptime and Time Between Procs
Calculating the uptime (for a duration proc) or time between procs (for instant effect procs, such as damage), the method is strikingly similar.  One needs to know 4 values: the duration of the proc (only applicable to procs with duration.  Instant effects have an effective duration of 1 second), the internal cooldown (if any), the chance to proc (either static or the percentage calculated from PPM), and the average time between procable attacks.  These are noted as follows in all following formulae:

U = Percentage uptime
D = Proc duration (D = 1 if the effect is instant)
T = Time between procable attacks
P = Proc chance per attack
I = Internal Cooldown

The following formulae will give the uptime for any duration proc.  If seeking the time between procs, whether for an instant effect or duration proc, simply take D / U (remember, D = 1 for instantaneous effects).

Instant Effects and Long-ICD Effects
If the effect has an ICD at least as long as the duration or if the effect is instantaneous (most trinkets), the uptime is as follows:

As an example, the ability to use Deep Freeze for a frost mage has a 15% proc per cast, and an effective internal cooldown of 30 seconds (the cooldown on Deep Freeze).  Assuming the mage has an average net haste of 15%, the mage will be casting a Frost Bolt every 2 seconds.  Based on the above, the mage will then have an average time between Deep Freezes of 43.3 seconds.

No-ICD Duration Effects
Duration procs without an internal cooldown function differently, as overlapping procs will reduce uptime below what one would expect intuitively.  Uptime for a proc without an internal cooldown is as follows:

As an example, Fallen Crusader with a 3.0 speed polearm has precisely a 10% chance to proc per swing, with a 15 second duration and no ICD.  Assuming no haste and no special attacks, FC will show a 41% uptime.  Compare this with the 50% uptime one would estimate if one simply took the PPM times the duration and divided by 60.

Short ICD Duration Effects
Procs with an Internal cooldown shorter than their duration use a complex amalgam of these two.  The formula is as follows:

As an example, the Totem of Electrifying Wind has a 70% proc chance per Lightning Bolt, with a 12 second duration and a 6 second ICD.  Assuming 33% overall haste (reduces Lightning Bolt to a 1.5 second cast), and assuming the shaman is spamming nothing but Lightning Bolt, the totem will have an uptime of 99.6%.