I’ll skip describing the proc on pull portion because it’s not extremely relevant but in a gist, if you recall the equation for the chance to proc equation where t is the last time since a proc:
By setting this equation to 1 (definite chance to proc) and knowing your RPPM and Haste, you can calculate t so that you are guaranteed a proc on pull.
The next equation approximates the maximum time before a proc during combat. This is different from the above approximation because the change in time term is no longer 10 seconds (simulating no attacks) and you are not scaling the approximation by any kind of attack factor. The 10 seconds is the maximum time which was provided by Blizzard to count between attacks meaning even if you were not in combat, the time between attacks would max at 10 seconds in their calculations.
In this equation, we assume similarly that tmax, the maximum possible theoretical time before the RPPM mechanics has 100% chance to proc is greater than 3/2 * lambda. This value was not pulled out of a hat. Recall the first part of this analysis where I talked about what time would be necessary for the penalty factor, the one which accounts for the last time the mechanic procced, would be above 1.
For this value to be above 1, t must be greater than the value 3/2*lambda. This is a valid assumption since otherwise, the maximum value in the above max() expression would be 1 and nothing would change with the proc chance. We’re assuming there’s a reason for the penalty factor to be implemented in that the maximum time before a proc is going to cause the second penalty term above to increase greater than 1.
With the same reasoning as the proc on pull mechanic, the following equation is derived:
I was confused about the reasoning behind this equation. Since they want the chance to proc to be 100%, I thought the second part of the equation above would already be one. Furthermore, I don’t see any kind of summarization term which combines all attacks. To make an educated guess at the reasoning behind this equation, based on the attacks per second, you want to find the probability of the proc occurring with 100% certainty. You must include the ‘attacks per second’ term because your 100% certainty must include this scaling factor based on attack speed.
By plugging our equations into the above expression (we have everything in these equations from parts 1 and 2) we find that the maximum time before a proc is proportional to:
The author demonstrates that the RPPM constant has limits (can not go below a certain amount) for there to be a 100% chance to proc. In other words, if the RPPM constant was under 0.019, you might never get a single proc. This is calculated by setting t (time since last proc) to the Blizzard stated limit of 1000 seconds and finding the proc rate lambda (Blizzard’s equation).
Average Proc Rates (Single Proc)
The author here defines some initial terms.
This term represents the total number of events over a period a-b. This essentially integrates the function for proc rate based on the time since last proc. Integrating the function allows you to find the area under the curve, the sum of a number of different proc chances. It is entirely possible that this value will not add to 1, but if it does it means that at least one event will occur in this time period a to b. To average this over a time interval [0,t) we divide the above equation substituting a by 0 and b by t and divide it over the time range t, which is exactly how you would calculate a rate (number of events over time). If we identify an equation for the average proc rate at time t over interval (0,t) we get:
This is essentially a modified form of one of the first equations we derived and the one we were referring to in the previous second:
We have substituted the integral above into the first term in this equation. We have replaced a chance to proc (based on last time since proc value) which is essentially a rate value with an average proc rate value. With some math, this turns out to be”
This part is confusing to me because we are replacing one kind of term with a different kind of term. The first term in the new equation is a proc rate, and the first term in the original equation is a probability term. We can look at the two symbols and thing well they’re both lambda but they actually represent what is in my opinion different things. In fact, the term in the new equation is actually the integral of the term in the second equation which makes the reasoning behind the substitution even more dodgy. Lets just give the author the benefit of the doubt though.
Let me say a few sentence about the graph. First, we notice the protection doesn’t kick in until the time between procs reaches above 80 seconds. After a certain point, the protection system will actually return less proc chance than without the protection system. It is likely however, that the time between procs will never reach this period but that still needs to be determined.
Overall, this part of the analysis is at times confusing to me, not in the mathematical sense but in the logical reasoning sense. Approximations can only be made if the base considerations follow what we would expect in real life. For example, you can not approximate the speed of a car if you only take its mileage for example and how long you’ve owned the car. These are not related to the speed. Regardless, perhaps the reasoning for these is beyond by level of understanding and you can in fact use these equations to calculate timings. These calculations might be a little more complicated since you will be forced to take the integrals yourself. If you have access to Mathematica or even some online websites such as Wolfram Alpha, you can use those to calculate you integrals. In this section, the authors have included a graph which allows the analogy behind all of this math and the real word to be discerned. You can easily use a plot like this to identify your proc rate after knowing a period of no procs and see for example, if your trinkets (by either looking at world of logs or other things) is a following this approximation.
Remember the words of our good friend George E. P. Box, “All models are false but some models are useful.”