How do you account for the effect of Jump?

Modeling the effect of Jump on damage rate isn't too bad provided that you invoke the following major simplifications: let both Jump and High Jump have the same amount of merit upgrades, and treat TP from Jumps as accumulating toward a weapon skill independently of TP from auto-attack. In this way, we can estimate the proportion of attack rounds that Jumps contribute to the average number of attack rounds required to accumulate 100 TP (for a weapon skill). We need this proportion to estimate the time savings from using Jumps that contribute to increasing WS frequency.

Suppose that there are 5 merits both in Jump and High Jump. This means that in a 150-second time frame, two Jumps and one High Jump can occur, for a total of three jumps. Also, do not (yet) assume that attack rounds from Jumps are equivalent to those from auto-attack in terms of multi-hit "capability" (from double attack, multi-hit weapons, etc.). It is then possible to obtain a general expression for the denominator required to obtain the respective proportions of attack rounds that auto-attack, Jump, and High Jump contribute to the average number of attack rounds to 100 TP:


The implied units for this denominator are rounds per WS (with spamming of TP after 100 TP is achieved).

The first term (factors specific to it denoted with the subscript 1) in the expression accounts for how many weapon skills from auto-attack can occur in 150 seconds when accounting for a weapon skill delay of two seconds. T₁ denotes the time per attack round at 0% haste, and H denotes the haste level as an integer. E[R] in general denotes the average number of attack rounds to 100 TP, and usually, E[R₁] = E[R₂] = E[R₃] except in the case of virtue weapons, apparently (the only reason the equality wouldn't hold because virtue weapons apparently do not work with Jumps).

The second term accounts for how many weapon skills from Jump (two Jumps in 150 seconds, remember) can occur in the previously specified 150-second time frame (necessarily a fraction), and the third term accounts for how many weapon skills from High Jump can occur in 150 seconds.

With this denominator expression, it should then be obvious how to obtain the actual proportions of attack rounds that each of auto-attack, Jump, and High Jump contribute to the average number of attack rounds to 100 TP. For example, the proportion of attack rounds that High Jump contributes to the average number of attack rounds to 100 TP is


These proportions can then be used to obtain an estimate of the adjusted average of the number of attack rounds to 100 TP accounting for Jump effects (this is a weighted average). Of course, if E[R₁] = E[R₂] = E[R₃] = E[R], then the weighted average simplifies to E[R].

However, the adjusted average of attack rounds cannot be multiplied by a simple "time per attack round" conversion factor to get the average time to 100 TP. Recall that TP from Jumps is treated as independent of TP from auto-attack as a simplifying assumption. Instead, the aforementioned proportions must be used to obtain a weighted average of the time "per cycle" of 100 TP generated, with 2E[R₂] seconds for Jump and 2E[R₃] seconds for High Jump (ignoring stacking of Jump and High Jump; the units of E[R] are attack rounds "per cycle" of 100 TP generated) and E[R₁]T₁(100-H)/100 seconds for auto-attack.

Mechanistically, we should already recognize before doing modeling that the dominant effect of Jumps is to increase WS frequency by reducing the time required to generate 100 TP, except when T₁(100-H)/100 < 2 seconds. With modeling, it is possible to estimate the reduction (both absolute and relative) in average time to generate 100 TP from Jumps. From modeling, it is also possible to account for differences in damage between auto-attack hits, Jump, and High Jump (you don't use haste equipment for Jumps, right?), but this effect is slight compared to the effect on WS frequency and will not be accounted for in future posts.