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.
Showing posts with label mathematics-arithmetic.progression. Show all posts
Showing posts with label mathematics-arithmetic.progression. Show all posts
31 January 2011
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.
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.
30 May 2009
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 given:
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:
The total personal DPS is:
Total personal DPS is:
The average personal DPS is:
That means the bonus DPS from the charge buff is:
A 1-stack of the charge buff grants:
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:
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:
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:
Calculating the incremental changes in DPS from additional pairs of deaths, we get the following progression of values:
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:
Differentiating S(n) with respect to n:
Differentiating dS(n)/dn with respect to n:
We have the following trend:
Originally posted here.
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).
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.
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