Beginning from where we left off last time:
Single Proc Trinket, no ICD
Recall this equation for the chance of n events in time t:
Recall that the lambda term as derived in the paper comes from this term representing the total number of events over a time period [a,b]:
By plugging in a with 0 and b with t, we will get the number of events in a period of time [0,t]. Dividing this value by the time period will get the rate of events, the lambda-t term in the first equation. This is because rate is defined as the number of occurrences over time. This term can then be plugged into the lambda terms in the first equation.
To find the uptime of a trinket, first we need find the average time to proc by multiplying the above formula by t to find the average number of procs and integrating the time from 0 to infinity to find the overall average time to proc. Very little explanation is given for the reasoning of this equation. In a sense, the equation above representing the chance of n events in timeframe [0,t] is multiplied by t.
In the formula above, to represent the chance of 1 event, n is taken to be 1. Thus we multiply the chance of one event in time t, by t. This is approximated as the average time between procs given a certain time period, t. By integrating the expression over infinity, we sum up all possibilities for different time periods. I’m not quite sure where the average part comes in to all of this as I don’t see a normalizing term to number of events or something similar to that.
Assumptions (which are not mentioned) are made to arrive at the following equation after plugging in the equation P above into the equation for t. I do not own Mathematica, but if you do, you can plug the code given into it and evaluate the result to see exactly what assumptions were made.
The uptime given the duration of the proc will be defined then as followed:
It is clear where uptime is calculated this way as the bottom term represents the average time between single procs and the term on the top represents the duration of the proc. In terms of healing trinkets however for example for Horridon’s Last Gasp with a stacking debuff the uptime will be calculated differently.
Single Proc Trinket, with ICD
This equation is exactly the same as above, the only difference is the times are adjusted for the duration of the ICD and changing any probabilities for procs in the ICD duration to be zero. For example, the first two equations (which are actually the same with one generalized so the time factor is not negative which is impossible) adds a case where if the time period is under ICD, the probability of a proc is zero.
The second equation which is the average time to proc rather than integrating from 0 to infinity, is now integrated from the ICD to infinity. This returns a very complicated variation of the above equation for the time between procs. The t(ICD) term is the same equation for t with the integral above, taking the ICD to be the time value at the beginning.
Further variations of this formula can be changed around to find the average time between procs for different situations. I won’t go into them here but the following are calculated:
- Time to first proc give a time delay factor (time since last proc) less than limit of the time in the max equation required for the largest term to go above 1
- Time to first proc give a time delay factor greater than the limit, note that the time must go from the beginning of the time delay factor to infinity in calculating the time between procs
The last equation seeks to find the time to first proc which makes the equation below equal to 1 (one event must occur).
To do this we simply change the time limits, a to the time delay (last time since proc) and b to time at first proc).
The term in the integral comes from the second point of this old equation:
By solving for the time at first proc the following equation is derived:
By plugging in the time delay (typically the time since last wipe) we can calculated to an approximation how much time into the fight until your first proc will happen.
I hope these analyses have at least been an enjoyable read for everyone and not too confusing. Essentially all distributions in mathematics can be approximated with certain functions, in this case exponential functions. What I hope to do after this is to use the derive equations and put in known values for the healing trinkets to see how close the approximation is to the real values. Speaking of real values, I’ve been gathering a database of world of logs posts for people who have used the healing trinkets in fights. I plan to summarize what hopefully will be a larger data set than the initial analysis and see where they are at. I learned a lot about what to look for in analyzing the trinket logs and my new analysis should be a lot cleaner and answer exactly the questions proposed.
Also, I hope to talk about and simulate some common problems or questions that always go unanswered, including topics such as EF blanketing and when to switch tier 14 to tier 15.