WoW Mathematics: Scavenging for Resources

Garrison missions are a great way to earn Garrison Resources, of which rewards may be further boosted by assigning followers with the Scavenger trait. Unfortunately, followers with Scavenger may not also have the required counters for the mission, and so a question arises: maximise the success chance of the mission, or boost the potential rewards up for grabs?

This commander has received the mission Out of His Element, which offers 48 Garrison Resources as a reward. He or she has the choice of assigning Auriel Brightsong, whom has a needed Group Damage counter for a 100% success chance, and Greatmother Geyah, whom has Scavenger, adding a further 200% Garrison Resources for a successful mission. Whom should he or she assign?

Treasure awaits!

Summary:

• To maximise Garrison Resources over the long term, the commander should assign Greatmother Geyah to Out of His Element.

WoW Mathematics: Heavy Breathing

When Valiona breathes fire, she breathes lots of it. Not only does it reach the very edges of the room she is in, it also has a 180° angular spread.

Given that most fires tend to be bad, it is usually prudent to run out of it. There are, of course, a multitude of possible directions to run, which leave the player roasting for different amounts of time. Which path is the quickest?

Summary:

• The quickest path out of Devouring Flames is a straight line perpendicular (at right angles) to the boundary of the cone.
• If Valiona is facing the player, the quickest path out is through Valiona.
• Given a Cartesian coordinate system, the starting position (x₁,y₁) and the equation of the boundary ax + by + c = 0 ,the distance d to run is:

WoW Mathematics: Server Loading

Hosts of MMORPGs face several dilemmas regarding the servers they operate. One of these, by far a significant one, is the maximum capacity the servers are able to take. Too low a capacity and players will complain about latency and other performance problems, while too high a capacity and the investment in the hardware will not be cost-effective. This problem is exacerbated by the tendency of the load to change cyclically throughout the day.

Deciding on the servers' capabilities is not a simple matter of allowing for a specific number of players. As the mathematics that follows shows, the throughput load created by those players will change according to the tendency of players to cluster together in the game world.

WoW Mathematics: Steady-State Health Level

In the final phase of the Anub'arak encounter in the Trial of the Crusader (ToC), healers must heal through a variety of sources of damage, including Penetrating Cold and Leeching Swarm. While the former deals constant damage over time, the latter does more damage the higher players' health levels are! The raid might be interested in knowing the optimal health level the healer(s) can keep the players at.

At the steady-state health level, DPS of Anub'arak against the raid equals the HPS of the healers for the raid. From here, health levels do not tend to move either way; they are stable, and this determines whether the healers can provide enough healing.

Summary:

• Where h is the total HPS, r is the proportion of current health Leeching Swarm does as damage, p is the proportion of players affected by Penetrating Cold at cast and P is the tick size of Penetrating Cold, the steady-state health level H* is H* = ¹/_r(h - ³/₁₀pP). It is a good idea to incorporate a buffer against the randomness of the damage.
• For Heroic 10-man, the steady-state health level is H* = 5h - 450, and for Normal 25-man, it is H* ~ 10h - 228.
• As long as the steady-state health level is more than zero, the healing the healers provide is sufficient.

Volumes of Tori and Pyramids

The volume of some solids can be found intuitively (e.g. for a right rectangular prism, V = lbh). However, some more exotic solids will require calculus for their volumes to be found.

Two such solids are the torus (donut) and rectangular pyramid. The former's volume can be found, possibly among others, using the technique of volume by rotation, and the latter's with volume by cross-section. These techniques will be used for deriving the formulae for the volumes of the respective solids. In summary:

• Volume of torus = 2cπ²r²
• Volume of rectangular pyramid = ¹/₃abh

Halian Geometry

The Halion encounter in the Ruby Sanctum requires careful positioning and movement in order to conquer it. Certain principles of geometry can help greatly with this endeavour, so it helps to be acquainted with them! Even if they will not be used in actually formulating strategies (since strategies will conveniently be recommended here), it is good to see how they were derived.

Circles and their radii
A circle is a (plane) shape where all points on its perimeter (the circumference) are the same distance from a single point (the centre). A radius is any straight line drawn from the centre to the circumference. Because of the definition of a circle, all radii are of the same length.

Direct Costing (with Defects)

In the ideal world, every bit of material and labour allocated to a product is used, and no waste is generated. Unfortunately, materials can have defects and precious labour is spent on dealing with it. This means that more input than ideally required is actually used.

In developing standard costs for products, management accountants must take defects into account. To do so, they need to have a good grasp of division and multiplication.

If C is the theoretical amount of material or labour and r is its defect rate, the total amount of material/labour actually used (T) is:

T = ^C/_{1 - r}

WoW Mathematics: Breach of Contract

Ever signed a contract later regretted? Unfortunately, they are all too common during the Blood-Queen Lana'thel encounter, as she regularly binds raid members into Pacts of the Darkfallen. The law in Icecrown Citadel cares not of requirements of intention and genuine consent, so surely it will not mind a breach? What is the most expedient and mathematically sensible way to do so?

Summary:

• To remove Pact of the Darkfallen the quickest way possible in the 10-man Blood-Queen Lana'thel Encounter, the raid members should run to the midpoint of the interval they form.
• To remove Pact of the Darkfallen in the quickest way possible in the 25-man Blood-Queen Lana'thel encounter:
• If the raid members trace an acute triangle, they should run to its circumcentre.
• If the raid members trace a right-angled or obtuse triangle, they should run to the midpoint of the longest side.

WoW Mathematics: Sharing the (DPS) load

In any (conventional) boss encounter, a certain number of damage dealers (dps) need to whittle down the boss' health to zero over a certain time period. Barring any extraordinary tinkering by Blizzard, the health level remains constant between attempts.

This fixed DPS burden needs to be shared between all dps. I will examine the DPS burden per dps and how it changes with the number of dps that happen to die during the encounter.

Summary:

• Given a total DPS burden (total raid DPS) of D and a total number of dps of d, the average DPS burden per dps B, where x players are dead, is B = ^D/_{d - x}.
  

WoW Mathematics: Destructive Shadow interference

During the Blood Prince Council encounter in Icecrown Citadel, players may find a number of Dark Nuclei floating around. These are vital for the tank tanking Prince Keleseth, for they each offer a stacking Shadow damage reduction. Against conventional wisdom, this damage reduction stacks multiplicatively, so full immunity cannot be achieved from a stack of three (as is the case with additive stacking).

Aim: To calculate the damage taken from Empowered Shadow Lance at a particular level of Shadow Resonance.

Summary:

• With Shadow Resonance of an order of a stack of n, the Keleseth tank will take 80,000(0.65)ⁿ damage from an Empowered Shadow Lance.

WoW Mathematics: Fatally Attracted to You

Back in the Burning Crusade era, Mother Shahraz was notorious for the encounter's high Shadow Resistance requirement and her femininity. She would also, at random times, cast Fatal Attraction on three random non-tank players.

Despite having Shadow Resistance at the cap, the damage is still great, and the unlucky trio need to separate promptly if they are to survive. It may be useful to know, given a certain set of vectors and using calculus, how fast they are moving away from each other...

Names of Raid Target Icons

Each of the eight raid target icons (also raid icons or lucky charms) are referred to with common (and usually standard) names. However, for some of them, the names used in mathematics are different.

A few definitions

• Shape: An object occupying space, distinguished by its perimeter (outer boundary).
• Plane figure: A shape occupying a two-dimensional plane.
• Quadrilateral: A plane figure with four straight sides
• Parallelogram: A quadrilateral with opposite sides parallel to each other respectively.
• Rectangle: A parallelogram with at least one vertex (corner) a right angle.

A simple gear score

There exists several ways to rate the items a player has equipped (examples include gear scores from Be Imba, Armory Light and the GearScore addon. Each method places varying importances on different characteristics of the gear. While changes in single gear scores can track the player's gear progression, different gear scores cannot be compared with each other, or to any other benchmark.

There is another, simpler way to rate the quality of gear. Every item has an item level (ilvl), which Blizzard calculates from from the statistics (stats) they assign to it. Generally, the more stats there are on the item, the higher the ilvl. Why not base a gear score on a stat sanctioned by Blizzard?

WoW Mathematics: Programming Your Warlock Tank

In a few rare NPC encounters, a warlock may be needed to tank a boss. Examples include Mimiron's Aerial Command Unit and Leotheras the Blind in Demon Form. While the focus of the warlock is to maximise threat, they may be interested in maximising DPS while generating the minimum amount of threat needed to hold aggro.

Fortunately, they have the mathematical tool of programming (not to be confused with programming computers) to work with. In its simplest form, linear programming, a certain value (e.g. profit, costs, or in this case a warlock tank's DPS) will need to be optimised (maximised or minimised). Of course, two chosen factors will influence that value (for this case, the number of casts of Shadow Bolt and Searing Pain), and, subject to several constraints, the right combination of both will need to be found.

This exercise gives a little insight to the workings of stat optimising utilities like Rawr.

The Time-Value of Health

This blog post builds on the concepts suggested in 'Not All Hitpoints are Created Equal', helping to adjust healing for the health level of the recipient.

Experts in finance speak of the time-value of money. Under this concept, the value of a certain quantity of dollars (or other currency) will decrease as time passes. Many attribute this to the fact that people want money now rather than later. Therefore, to keep this time-value the same, interest must be paid on top of this quantity of dollars.

Hitpoints restored by healing also seem to have this time-value, since a heal of the same size will be more valued the sooner it is received (considering that people wish to keep their health at a maximum). How should such heal weighting be done?

WoW Mathematics: My Large Brown Sack

To help find items easily, players often organise them in their bags in a certain way. It might be interesting to know just how many ways those items can be arranged, and what happens of those arrangements are restricted in some way.

Permutations are very useful in this process. It is a special case of the fundamental counting technique where an object, once picked, cannot be picked again, and so the available pool of objects decreases by one each time. Consider a certain 10-slot bag, the Large Brown Sack...

There's Something About Lady RNG

Computerised environments intrinsically lack randomness, so a random number generator (RNG) must be used where an unpredictable result is required. In effect, the RNG works on randomly generated data provided to generate a seemingly random number. Since this piece of programming code decides your fate and destiny in WoW in several cases (just like how Fortuna (Lady Luck) decides your fate and destiny in real life), it helps to know a bit about probability!

The golden rule of probability
Probability is that branch of mathematics that deals with random phenomena to (among other things) determine how likely it will act in a certain way. It is important to know that, for truly random phenomena:

Random does not necessarily mean even or fair!

WoW Mathematics: Your guild bank donation

Aim: To calculate the value of a deposit of gold to a guild bank (gbank) that discounts sales at a given rate.

Summary:

• A donation of G gold to a gbank with a decimal discount rate of r will be worth G/r gold. This means that G is multiplied by 1/r, the reciprocal of the discount rate. Keep in mind that Full price = Discounted price + Discount.
• As long as the discount rate is less than 100%, the donation is worth more than the amount donated (even in pure money terms)!
• A gbank donation's impact will grow according to the sum of a geometric series and approach a limiting value.

The given:

• The discount rate is fixed.
• There is no cost in performing transactions and the full proceeds of a gbank sale are deposited back in.
• All gold is used in buying other resources for resale. No gold is dispensed as a repair allowance (though it can be factored into the discount rate).

The detail:
Consider a gbank deposit of G gold. It is used in buying (for example) enchanting materials worth G gold. They are sold to a guild member at a discount rate of r, so the undiscounted proportion of the price is:

1 - r

The proceeds from that sale are:

G × (1 - r) = G(1 - r) gold

The proceeds are then used for buying potions worth G(1 - r) gold. There potions are sold at a discounted price of:

G(1 - r) × (1 - r) = G(1 - r)² gold

Continuing this process for n terms, we get the following progression of values:

G, G(1 - r), G(1 - r)², G(1 - r)³, ... , G(1 - r)^{n - 1}

The geometric series 1 + 1/2 + 1/4 + ... , with its value represented by area.

This is in fact a geometric progression with common ratio (1 - r) and first term B. Note that, while the amount deposited diminishes, it keeps making an impact with each gbank sale. The total impact for n sales is the following sum of the corresponding geometric series:

S(n) = G(1 - (1 - r)ⁿ) ÷ (1 - (1 - r)) = G(1 - (1 - r)ⁿ)/r gold

As n » ∞ , (1 - r)ⁿ » 0 , and (1 - (1 - r)ⁿ) » 1 . So, the total impact approaches the following limiting value as resales are conducted repeatedly with that amount:

S(∞) = G/r gold

Here is an example. Say 100 gold is donated to a gbank with a discount rate of 25% or 1/4. That donation will be worth:

100 ÷ 1/4 = 100 × 4 = 400 gold

Don't slack off during the Thaddius encounter!

Aim: To calculate the drop in DPS of a single death in Phase 2 of the Heroic Thaddius encounter, subject to certain assumptions. To observe the change in total raid DPS as additional dps die.

Summary:

• The minimum personal DPS needed to down Thaddius in time is 2186 DPS.
• The total personal DPS is 100% average DPS + 115% bonus DPS from charge buff OR 215% of average DPS.
• A 1-stack of the charge buff grants 218.6 DPS.
• A single death of a healer drops total raid DPS by 1748.8 DPS, while a single death of a dps drops total raid DPS by 6230.2 DPS.
• While DPS continues dropping as additional dps die, its rate of decline is decreasing.

The given:

• Raid composition: 2 tanks, 7 healers, 16 dps.
• Raid positioning: 12.5 players on each side of Thaddius, each side having 8 dps.
• Thaddius' health: 27 600 000 HP.
• Enrage timer: 360 seconds (6 minutes)
• Average size of stacking group: 12.5 players.
• Average stack of charge buff among dps: 11.5 stacks. This gives a damage boost of 115% of average personal DPS.
• Healers do not dps, tanks do minimal damage (in this scenario, VERY minimal).

Detail:

The figure that will be used as the average personal DPS, fully buffed, is the minimum personal DPS needed to down Thaddius before the enrage. The total raid DPS is:

27 600 000/360 = 76 666.6666... DPS

The total personal DPS is:

7666.6666.../16 = 4791.6666... ~ 4700 DPS (attribute the extra 91.6666... DPS to the tanks :))

Total personal DPS is:

100% average DPS + 115% bonus DPS from charge buff (decimal 1.15) = 215% of average personal DPS (decimal 2.15)

The average personal DPS is:

4700/2.15 ~ 2186 DPS

That means the bonus DPS from the charge buff is:

4700 - 2186 = 2514 DPS

A 1-stack of the charge buff grants:

2514/11.5 ~ 218.6 DPS

Consider the death of a healer. That removes a 1-stack of the charge buff off the 8 dps on their side. So, the drop in DPS from that death is:

218.6 x 8 = 1748.8 DPS. That's about 524.6k damage over 5 minutes lost!

Consider the death of a dps. That removes the dps' total personal DPS plus a 1-stack of the charge buff from each of the 7 dps on their side. So, the drop in DPS from that death is:

7 x 218.6 + 4700 = 1530.2 + 4700 = 6230.2 DPS. That's about 1.87M damage over 5 minutes lost!

To observe the change in DPS using calculus, additional deaths will need to be considered in pairs (one dps from each group will drop total raid DPS by equal amounts) and we will assume that only the dps die. The change in total raid DPS from the first pair of deaths is:

-6230.2 x 2 = -12 460.4 DPS

Calculating the incremental changes in DPS from additional pairs of deaths, we get the following progression of values:

-12 460.4, -12 023.2, -11 586, -11 148.8, ...

This is in fact an arithmetic progression with common difference of 437.2 and first term of -12 460.4. Total DPS lost from n pairs of deaths will be the following sum of values:

S(n) = 218.6n^2 - 12 679n

Differentiating S(n) with respect to n:

dS(n)/dn = 437.2n - 12679. Since the first derivative is negative over the domain 0 < n < 8, S(n) will continue dropping.

Differentiating dS(n)/dn with respect to n:

d2S(n)/dn2 = 437.2. Since the second derivative is always positive, S(n) is decreasing at a decreasing rate.

We have the following trend:


Originally posted here.