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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 observed in: performance in hydrology heavy-tailed system: a case study geology Lyubov V. insurance Potakhina Alexander S. risk analysis Rumyantsev network analysis and computer science and others

Pareto distribution Optimizing performance Pareto distribution in heavy-tailed system: P ( X > x ) = x − α , x > 1 , α > 1 a case study Lyubov V. Some key properties: Potakhina Alexander S. Rumyantsev Pareto law (20 / 80) Infinite variance and (if α < 1) infinite mean are possible Heavy tails can cause burstiness

Model and problem formulation Optimizing M/G/1 system, service times S performance in The tail distribution ¯ B ( x ) = P { S > x } – heavy tail heavy-tailed system: � x The integrated tail distribution is ¯ 0 ¯ 1 a case study B r ( x ) = B ( y ) dy ES Lyubov V. Potakhina Alexander S. How we can reduce a negative influence of heavy tails? Rumyantsev Practice recommendations: 1 Choose a service discipline 2 Choose a server architecture 3 Choose a task assignment policy

Case study 1: Choosing a service discipline Optimizing W – waiting time, V – sojourn time 1 performance in First Come First Served heavy-tailed system: a case study ρ ¯ P { W > x } ∼ B r ( x ) , x → ∞ Lyubov V. 1 − ρ Potakhina Alexander S. Rumyantsev Processor Sharing P { V > x } ∼ ¯ � � B ( 1 − ρ ) x , x → ∞ Last Come First Served Preemptive-Resume 1 ¯ � � P { V > x } ∼ B ( 1 − ρ ) x , x → ∞ 1 − ρ 1 The source is [1]

Case study 1: Choosing a service discipline Optimizing Last Come First Served Non-Preemptive performance in heavy-tailed P { W > x } ∼ ρ ¯ � � system: B r ( 1 − ρ ) x , x → ∞ a case study Lyubov V. Foreground-Background Processor Sharing Potakhina Alexander S. Rumyantsev P { V > x } ∼ ¯ � � B ( 1 − ρ ) x , x → ∞ Shortest Remaining Process Time First P { V > x } ∼ ¯ � � B ( 1 − ρ ) x , x → ∞

Case study 2: Choosing a server architecture Optimizing Simulating waiting time in M/G/1 system performance in 3500 heavy-tailed ’out.txt’ system: a case study 3000 Lyubov V. Potakhina Alexander S. 2500 Rumyantsev 2000 1500 1000 500 0 0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000

Case study 2: Choosing a server architecture Optimizing Simulating waiting time in M/G/2 system performance in 350 heavy-tailed ’out.txt’ system: a case study 300 Lyubov V. Potakhina Alexander S. 250 Rumyantsev 200 150 100 50 0 0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000

Case study 2: Choosing a server architecture Optimizing Simulating waiting time in M/G/4 system performance in 35 heavy-tailed ’out.txt’ system: a case study 30 Lyubov V. Potakhina Alexander S. 25 Rumyantsev 20 15 10 5 0 0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000

Case study 2: Choosing a server architecture Optimizing Simulating waiting time in M/G/8 system performance in 3.5 heavy-tailed ’out.txt’ system: a case study 3 Lyubov V. Potakhina Alexander S. 2.5 Rumyantsev 2 1.5 1 0.5 0 0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000

Case study 3: Choosing a task assignment policy Optimizing M/G/n system, task sizes are bounded. performance in heavy-tailed Bounded Pareto distribution B ( k , p , α ) system: a case study α k α 1 − ( k / p ) α x − α − 1 , k ≤ x ≤ p Lyubov V. f ( x ) = Potakhina Alexander S. Rumyantsev 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

Case study 3: Choosing a task assignment policy Optimizing SITA-E — Size Interval Task Assignment with Equal Load performance in algorithm heavy-tailed system: a case study Lyubov V. Potakhina Alexander S. Rumyantsev

Case study 3: Choosing a task assignment policy Optimizing SITA-E — Size Interval Task Assignment with Equal Load performance in algorithm heavy-tailed system: The total work of each core is the same => mean waiting a case study time decrease Lyubov V. Potakhina Alexander S. Rumyantsev

Case study 3: Choosing a task assignment policy Optimizing SITA-E — Size Interval Task Assignment with Equal Load performance in algorithm heavy-tailed system: The total work of each core is the same => mean waiting a case study time decrease Lyubov V. Potakhina Alexander S. If B ( x ) – the distribution function, M – mean tasks size; Rumyantsev “Cutoff points” x i , i = 0 .. n , x 0 = k , x n = p are defined by: � x 1 � x 2 � x n xdB ( x ) = M xdB ( x ) = xdB ( x ) = ... = n x 0 x 1 x n − 1

Case study 3: Choosing a task assignment policy Optimizing Simulated mean waiting time in M/G/n system 2 performance in heavy-tailed system: a case study Lyubov V. Potakhina Alexander S. Rumyantsev 2 The source is [2]

Case study 3: Choosing a task assignment policy Optimizing Simulated standard deviation of waiting time in M/G/n system performance in heavy-tailed system: a case study Lyubov V. Potakhina Alexander S. Rumyantsev

References Optimizing performance [1] Borst S.C., Boxma O.J., Nunez-Queija R. Heavy Tails: in The Effect of the Service Discipline . 2002. heavy-tailed system: a case study [2] Harchol-Balter M. The Effect of Heavy-Tailed Job Size Lyubov V. Distributions on Computer System Design . 1999. Potakhina Alexander S. Rumyantsev [3] Morozov E., Pagano M., Rumyantsev A. Heavy-tailed distributions with applications to broadband communication systems . 2008. [4] Samorodnitsky G. Long Range Dependence, Heavy Tails and Rare Events . 2002. [5] Zwart A. Queueing Systems with Heavy Tails . 2002.

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