# Devoted

Not to be confused with Devotion.
For the improved perk, see Enhanced devoted.
 This article has a calculator here.Calculators determine experience and costs based on real-time prices from the Grand Exchange Market Watch.

Devoted is an Invention perk that has a chance each hit of replicating the Devotion ability for 3 seconds. This effect does not put the ability itself on cooldown. Devoted has an effective level requirement of 74 as the materials required can only be unlocked at that level; however, enhanced devoted can be utilised at any Invention level for the cost of 2 perk slots. It can be created in armour gizmos.

Any type of damage, including typeless damage and self-inflicting damage, can activate the perk. The hit that activates Devoted will also be affected by the perk's effect. Devoted will play the same animation as Devotion whenever it activates, but uses its own icon in the buffs/debuffs interface. In player-versus-player situations the icon does not appear.

Having Devoted and Enhanced Devoted on one's gear has a benefit[1]. Enhanced Devoted takes priority over Devoted. However, as Enhanced Devoted is unable to proc while it is currently active, this allows for a small window of time for Devoted to proc. Both Devoted and Enhanced Devoted, on their own, last 5 game ticks (3 seconds). This also means that it is possible (though unlikely) to have continuous procs of Enhanced Devoted and Devoted. The chance of continuous procs occurring is highly dependent on the attack rate of the enemy. Further information about the combination can be found here.

An example of Enhanced Devoted and Devoted proccing together to increase damage reduction.
Activation chance
Rank Chance (level 20)
1 3% (3.3%)
2 6% (6.6%)
3 9% (9.9%)
PvM average damage reduction
Rank ${\displaystyle t_{AR}=1}$ ${\displaystyle t_{AR}=2}$ ${\displaystyle t_{AR}=3}$ or ${\displaystyle 4}$ ${\displaystyle t_{AR}=5+}$
1 13.39% (14.58%) 8.49% (9.29%) 5.83% (6.39%) 3.00% (3.30%)
2 24.19% (26.11%) 16.07% (17.49%) 11.32% (12.38%) 6.00% (6.60%)
3 33.09% (35.46%) 22.88% (24.79%) 16.51% (18.02%) 9.00% (9.90%)
PvP average damage reduction
Rank ${\displaystyle t_{AR}=1}$ ${\displaystyle t_{AR}=2}$ ${\displaystyle t_{AR}=3}$ or ${\displaystyle 4}$ ${\displaystyle t_{AR}=5+}$
1 6.70% (7.29%) 4.25% (4.64%) 2.91% (3.19%) 1.50% (1.65%)
2 12.10% (13.05%) 8.04% (8.75%) 5.66% (6.19%) 3.00% (3.30%)
3 16.54% (17.73%) 11.44% (12.40%) 8.26% (9.01%) 4.50% (4.95%)
• ${\displaystyle t_{AR}}$ is the attack rate of the enemy in game ticks.
• All numbers in parentheses refer to level 20 gear.
Calculations

For PvM :

• The average ratio of damage taken, ${\displaystyle r_{avg}}$, from Devoted to that of without Devoted is
${\displaystyle r_{avg}={\frac {\sum \limits _{n=0}^{\infty }\left(1-p_{R}\right)^{n}p_{R}\left[\left(\sum \limits _{i=1}^{n}d_{n,i}^{(0)}\right)+\left\lceil {\frac {t_{d}}{t_{AR}}}\right\rceil \right]}{\sum \limits _{n=0}^{\infty }\left(1-p_{R}\right)^{n}p_{R}\left[\sum \limits _{i=1}^{n}d_{n,i}^{(0)}+\sum \limits _{i=1}^{\left\lceil {\frac {t_{d}}{t_{AR}}}\right\rceil }d_{n,i}^{(1)}\right]}}}$
• The average damage reduction is then, after some rearranging,
${\displaystyle 1-r_{avg}={\frac {\sum \limits _{n=0}^{\infty }\left(1-p_{R}\right)^{n}p_{R}\sum \limits _{i=1}^{\left\lceil {\frac {t_{d}}{t_{AR}}}\right\rceil }\left(d_{n,i}^{(1)}-1\right)}{\sum \limits _{n=0}^{\infty }\left(1-p_{R}\right)^{n}p_{R}\left[\sum \limits _{i=1}^{n}d_{n,i}^{(0)}+\sum \limits _{i=1}^{\left\lceil {\frac {t_{d}}{t_{AR}}}\right\rceil }d_{n,i}^{(1)}\right]}}}$

For PvP :

• The average ratio of damage taken, ${\displaystyle r_{avg}}$, from Devoted to that of without Devoted is
${\displaystyle r_{avg}={\frac {\sum \limits _{n=0}^{\infty }\left(1-p_{R}\right)^{n}p_{R}\left[\sum \limits _{i=1}^{n}d_{n,i}^{(0)}+\sum \limits _{i=1}^{\left\lceil {\frac {t_{d}}{t_{AR}}}\right\rceil }\left\lfloor {.5\times d_{n,i}^{(1)}}\right\rfloor \right]}{\sum \limits _{n=0}^{\infty }\left(1-p_{R}\right)^{n}p_{R}\left[\sum \limits _{i=1}^{n}d_{n,i}^{(0)}+\sum \limits _{i=1}^{\left\lceil {\frac {t_{d}}{t_{AR}}}\right\rceil }d_{n,i}^{(1)}\right]}}}$
• The average damage reduction is then, after some rearranging,
${\displaystyle 1-r_{avg}={\frac {\sum \limits _{n=0}^{\infty }\left(1-p_{R}\right)^{n}p_{R}\sum \limits _{i=1}^{\left\lceil {\frac {t_{d}}{t_{AR}}}\right\rceil }\left(d_{n,i}^{(1)}-\left\lfloor {.5\times d_{n,i}^{(1)}}\right\rfloor \right)}{\sum \limits _{n=0}^{\infty }\left(1-p_{R}\right)^{n}p_{R}\left[\sum \limits _{i=1}^{n}d_{n,i}^{(0)}+\sum \limits _{i=1}^{\left\lceil {\frac {t_{d}}{t_{AR}}}\right\rceil }d_{n,i}^{(1)}\right]}}}$
Notes
• A calculator for this is at the top of this page.
• Assumptions :
• The enemy is always attacking with the same style that matches the protect prayer (deflect curse) used.
• The enemy is always attacking at the same attack rate.
• ${\displaystyle R}$ is the rank of the Devoted perk.
• ${\displaystyle p_{R}}$ is the proc chance of Devoted to activate (if the perk is on level 20 gear, then this value is multiplied by 1.1).
• ${\displaystyle \textstyle \sum _{n=0}^{\infty }\left(1-p\right)^{n}p=1}$ when ${\displaystyle 0\leq p<1}$ or ${\displaystyle \left|1-p\right|<1}$. This represents summing over all possibilities where integer ${\displaystyle n\in [0,\infty )}$ represents the amount of hits prior to the proc of Devoted. The probability of Devoted proccing on hit ${\displaystyle {n+1}}$ is therefore ${\displaystyle \left(1-p\right)^{n}p}$.
• ${\displaystyle t_{d}}$ is the time that Devoted is active. This is taken to be 5 game ticks.
• ${\displaystyle t_{AR}}$ is the attack rate of the enemy in game ticks.
• ${\displaystyle d_{n,i}^{(j)}}$ is the ${\displaystyle i^{th}}$ random value uniformly sampled between the enemy's minimum hit and maximum hit. The ${\displaystyle j}$ describes different sets of hits. The sets are regenerated for every new value of ${\displaystyle n}$.
• ${\displaystyle d_{n,i}^{(0)}}$ has ${\displaystyle n}$ elements with integer ${\displaystyle i\in [1,n]}$. This is the damage taken before the perk procs.
• Clarification : This does mean that for ${\displaystyle n=0}$ that there are no elements in this set.
• ${\displaystyle d_{n,i}^{(1)}}$ has ${\displaystyle \left\lceil {\frac {t_{d}}{t_{AR}}}\right\rceil }$ elements with integer ${\displaystyle i\in \left[1,\left\lceil {\frac {t_{d}}{t_{AR}}}\right\rceil \right]}$. This is the damage taken during the time where Devoted procced.
• The set of values in ${\displaystyle d_{n}^{(j)}}$ for integer ${\displaystyle j\in [0,1]}$ is the same in both the numerator and denominator of the ratios for any given ${\displaystyle n}$.

Simplifications
• This can be simplified if the assumption is that every value of ${\displaystyle d_{n,i}^{(j)}}$ ${\displaystyle \forall }$ integer ${\displaystyle i\in [1,n]}$, integer ${\displaystyle n\in [0,\infty )}$, integer ${\displaystyle j\in [0,1]}$, is taken to be the same ${\displaystyle \left(d_{n,i}^{(j)}\rightarrow d\right)}$. In this scenario, there is no random element and the above ${\displaystyle r_{avg}}$ is then only dependent on the proc chance and the attack rate of the enemy. This simplification is increasingly accurate in the limit of large ${\displaystyle d}$. All examples are using large enough values of ${\displaystyle d}$ such that the values should not change to the first two decimal places in percent. Two tables are provided at the top of the page. The first covers all values of ${\displaystyle t_{AR}}$ for PvM situations. The second table is for PvP for all values of ${\displaystyle t_{AR}}$ (normal attack rate of player is ${\displaystyle t_{AR}=3}$ in non-legacy mode , but all values of ${\displaystyle t_{AR}}$ have been provided to account for proccing on damage-over-time abilities, 4 tick auto attack, etc., as well as the varied attack rates of weapons if using legacy combat).

Using these simplifications, the damage reduction reduces to

For PvM :

${\displaystyle 1-r_{avg}={\frac {\left\lceil {\frac {t_{d}}{t_{AR}}}\right\rceil p_{R}}{\left\lceil {\frac {t_{d}}{t_{AR}}}\right\rceil p_{R}+\left(1-p_{R}\right)}}}$

For PvP :

${\displaystyle 1-r_{avg}={\frac {.5\left\lceil {\frac {t_{d}}{t_{AR}}}\right\rceil p_{R}}{\left\lceil {\frac {t_{d}}{t_{RS}}}\right\rceil p_{R}+\left(1-p_{R}\right)}}}$

## Sources

MaterialRarityPerk ranks with X materials
12345
Zamorak componentsRare01222–3
Bandos componentsRare01222–3

### Suggested gizmos

Gizmo layout Possible perks

## Update history

The update history project is a work-in-progress – not all updates to this topic may be covered below. See here for how to help out!
• hotfix 18 September 2017 (Update):
• An issue with the Devotion[sic] perk was fixed.

## References

1. ^ Mod Pi. Devoted and enhanced devoted stack. 20 November 2018.*