Optimizing performance in heavy-tailed system: a case study Lyubov - - PowerPoint PPT Presentation
Optimizing performance in heavy-tailed system: a case study Lyubov V. Potakhina Alexander S. Rumyantsev Petrozavodsk State University IAMR Karelian Research Center RAS April, 2011 Heavy-tailed distributions Optimizing Heavy-tails have been
Lyubov V. Potakhina Alexander S. Rumyantsev
Petrozavodsk State University IAMR Karelian Research Center RAS
April, 2011
Optimizing performance in heavy-tailed system: a case study Lyubov V. Potakhina Alexander S. Rumyantsev
Heavy-tails have been observed in: hydrology geology insurance risk analysis network analysis and computer science and others
Optimizing performance in heavy-tailed system: a case study Lyubov V. Potakhina Alexander S. Rumyantsev
Pareto distribution P(X > x) = x−α, x > 1, α > 1 Some key properties: Pareto law (20 / 80) Infinite variance and (if α < 1) infinite mean are possible Heavy tails can cause burstiness
Optimizing performance in heavy-tailed system: a case study Lyubov V. Potakhina Alexander S. Rumyantsev
M/G/1 system, service times S The tail distribution ¯ B(x) = P{S > x} – heavy tail The integrated tail distribution is ¯ Br(x) =
1 ES
x
0 ¯
B(y)dy How we can reduce a negative influence of heavy tails? Practice recommendations:
1 Choose a service discipline 2 Choose a server architecture 3 Choose a task assignment policy
Optimizing performance in heavy-tailed system: a case study Lyubov V. Potakhina Alexander S. Rumyantsev
W – waiting time, V – sojourn time1 First Come First Served P{W > x} ∼ ρ 1 − ρ ¯ Br(x), x → ∞ Processor Sharing P{V > x} ∼ ¯ B
Last Come First Served Preemptive-Resume P{V > x} ∼ 1 1 − ρ ¯ B
1The source is [1]
Optimizing performance in heavy-tailed system: a case study Lyubov V. Potakhina Alexander S. Rumyantsev
Last Come First Served Non-Preemptive P{W > x} ∼ ρ¯ Br
Foreground-Background Processor Sharing P{V > x} ∼ ¯ B
Shortest Remaining Process Time First P{V > x} ∼ ¯ B
Optimizing performance in heavy-tailed system: a case study Lyubov V. Potakhina Alexander S. Rumyantsev
Simulating waiting time in M/G/1 system
500 1000 1500 2000 2500 3000 3500 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000 ’out.txt’
Optimizing performance in heavy-tailed system: a case study Lyubov V. Potakhina Alexander S. Rumyantsev
Simulating waiting time in M/G/2 system
50 100 150 200 250 300 350 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000 ’out.txt’
Optimizing performance in heavy-tailed system: a case study Lyubov V. Potakhina Alexander S. Rumyantsev
Simulating waiting time in M/G/4 system
5 10 15 20 25 30 35 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000 ’out.txt’
Optimizing performance in heavy-tailed system: a case study Lyubov V. Potakhina Alexander S. Rumyantsev
Simulating waiting time in M/G/8 system
0.5 1 1.5 2 2.5 3 3.5 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000 ’out.txt’
Optimizing performance in heavy-tailed system: a case study Lyubov V. Potakhina Alexander S. Rumyantsev
M/G/n system, task sizes are bounded. Bounded Pareto distribution B(k, p, α) f (x) = αkα 1 − (k/p)α x−α−1, k ≤ x ≤ p Task assignment policies: Random: a choice with equal probability Round-Robin: a cyclical order Dynamic: a core with the smallest amount of remaining work is selected Size-based: SITA-E defines the size range associated with each core
Optimizing performance in heavy-tailed system: a case study Lyubov V. Potakhina Alexander S. Rumyantsev
SITA-E — Size Interval Task Assignment with Equal Load algorithm
Optimizing performance in heavy-tailed system: a case study Lyubov V. Potakhina Alexander S. Rumyantsev
SITA-E — Size Interval Task Assignment with Equal Load algorithm The total work of each core is the same => mean waiting time decrease
Optimizing performance in heavy-tailed system: a case study Lyubov V. Potakhina Alexander S. Rumyantsev
SITA-E — Size Interval Task Assignment with Equal Load algorithm The total work of each core is the same => mean waiting time decrease If B(x) – the distribution function, M – mean tasks size; “Cutoff points” xi, i = 0..n, x0 = k, xn = p are defined by: x1
x0
xdB(x) = x2
x1
xdB(x) = ... = xn
xn−1
xdB(x) = M n
Optimizing performance in heavy-tailed system: a case study Lyubov V. Potakhina Alexander S. Rumyantsev
Simulated mean waiting time in M/G/n system2
2The source is [2]
Optimizing performance in heavy-tailed system: a case study Lyubov V. Potakhina Alexander S. Rumyantsev
Simulated standard deviation of waiting time in M/G/n system
Optimizing performance in heavy-tailed system: a case study Lyubov V. Potakhina Alexander S. Rumyantsev
[1] Borst S.C., Boxma O.J., Nunez-Queija R. Heavy Tails: The Effect of the Service Discipline. 2002. [2] Harchol-Balter M. The Effect of Heavy-Tailed Job Size Distributions on Computer System Design. 1999. [3] Morozov E., Pagano M., Rumyantsev A. Heavy-tailed distributions with applications to broadband communication
[4] Samorodnitsky G. Long Range Dependence, Heavy Tails and Rare Events. 2002. [5] Zwart A. Queueing Systems with Heavy Tails. 2002.
Optimizing performance in heavy-tailed system: a case study Lyubov V. Potakhina Alexander S. Rumyantsev