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Analysis of Parallel-Server Systems with Dynamic Affinity Scheduling and Load Balancing

Mark S. Squillante Mathematical Sciences Department April 18, 2004. Analysis of Parallel-Server Systems with Dynamic Affinity Scheduling and Load Balancing. Problem Motivation. Cache Affinity [SquillanteLazowska90]. Problem Motivation. Cache Affinity [SquillanteLazowska90].

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Analysis of Parallel-Server Systems with Dynamic Affinity Scheduling and Load Balancing

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  1. Mark S. Squillante Mathematical Sciences Department April 18, 2004 Analysis of Parallel-Server Systems with Dynamic Affinity Scheduling and Load Balancing

  2. Problem Motivation Cache Affinity [SquillanteLazowska90]

  3. Problem Motivation Cache Affinity [SquillanteLazowska90] Key Points of Fundamental Tradeoff • Customers can be served on any server of a parallel-server queueing system • Each customer is served most efficiently on one the servers • Load imbalance among queues occurs due to stochastic properties of system

  4. General Overview • Optimal Dynamic Threshold Scheduling Policy • Fluid limits • Diffusion limits • Analysis of Dynamic Threshold Scheduling • Consider generalized threshold scheduling policy • Matrix-analytic analysis and fix-point solution, asymptotically exact • Numerical experiments • Optimal settings of dynamic scheduling policy thresholds • Stochastic Derivative-Free Optimization

  5. () 1  () P Scheduling Policy

  6. () 1  ()  () P Scheduling Model

  7. (1-ps) … … …       0,1,0 1,0,1 Tru,1,0 Tru+1,0,1 Tsu,1,0 Tsu+1,0,1 prg r r r … … … pr (1-pr)g       s (1-ps) … … …       0,0,0 1,0,0 Tru,0,0 Tru+1,0,0 Tsu,0,0 Tsu+1,0,0 +r +r +r m m m (1-pr) Mathematical Analysis State Vector ( i, j, v ): • i: total number of customers waiting or receiving service at the processor of interest • j: number of customers in the process of being migrated to the processor of interest • v: K-bit vector denoting customer type of up to the first K customers at the processor

  8. (1-ps) … … …       0,1,0 1,0,1 Tru,1,0 Tru+1,0,1 Tsu,1,0 Tsu+1,0,1 prg r r r … … … pr (1-pr)g       s (1-ps) … … …       0,0,0 1,0,0 Tru,0,0 Tru+1,0,0 Tsu,0,0 Tsu+1,0,0 +r +r +r m m m (1-pr) Mathematical Analysis

  9. (1-ps) … … …       0,1,0 1,0,1 Tru,1,0 Tru+1,0,1 Tsu,1,0 Tsu+1,0,1 prg r r r … … … pr (1-pr)g       s (1-ps) … … …       0,0,0 1,0,0 Tru,0,0 Tru+1,0,0 Tsu,0,0 Tsu+1,0,0 +r +r +r m m m (1-pr) Mathematical Analysis

  10. (1-ps) … … …       0,1,0 1,0,1 Tru,1,0 Tru+1,0,1 Tsu,1,0 Tsu+1,0,1 prg r r r … … … pr (1-pr)g       s (1-ps) … … …       0,0,0 1,0,0 Tru,0,0 Tru+1,0,0 Tsu,0,0 Tsu+1,0,0 +r +r +r m m m (1-pr) Mathematical Analysis

  11. (1-ps) … … …       0,1,0 1,0,1 Tru,1,0 Tru+1,0,1 Tsu,1,0 Tsu+1,0,1 prg r r r … … … pr (1-pr)g       s (1-ps) … … …       0,0,0 1,0,0 Tru,0,0 Tru+1,0,0 Tsu,0,0 Tsu+1,0,0 +r +r +r m m m (1-pr) Mathematical Analysis

  12. (1-ps) … … …       0,1,0 1,0,1 Tru,1,0 Tru+1,0,1 Tsu,1,0 Tsu+1,0,1 prg r r r … … … pr (1-pr)g       s (1-ps) … … …       0,0,0 1,0,0 Tru,0,0 Tru+1,0,0 Tsu,0,0 Tsu+1,0,0 +r +r +r m m m (1-pr) Mathematical Analysis

  13. (1-ps) … … …       0,1,0 1,0,1 Tru,1,0 Tru+1,0,1 Tsu,1,0 Tsu+1,0,1 prg r r r … … … pr (1-pr)g       s (1-ps) … … …       0,0,0 1,0,0 Tru,0,0 Tru+1,0,0 Tsu,0,0 Tsu+1,0,0 +r +r +r m m m (1-pr) Mathematical Analysis Final Solution Obtained via Fix-Point Iteration • Initialize (ps, s, pr, r) • Compute stationary probability vector in terms of (ps, s, pr, r) • Compute new values of (ps, s, pr, r) in terms of stationary vector • Goto 2 until differences between iteration values are arbitrarily small

  14. (1-ps) … … …       0,1,0 1,0,1 Tru,1,0 Tru+1,0,1 Tsu,1,0 Tsu+1,0,1 prg r r r … … … pr (1-pr)g       s (1-ps) … … …       0,0,0 1,0,0 Tru,0,0 Tru+1,0,0 Tsu,0,0 Tsu+1,0,0 +r +r +r m m m (1-pr) Mathematical Analysis

  15. (1-ps) … … …       0,1,0 1,0,1 Tru,1,0 Tru+1,0,1 Tsu,1,0 Tsu+1,0,1 prg r r r … … … pr (1-pr)g       s (1-ps) … … …       0,0,0 1,0,0 Tru,0,0 Tru+1,0,0 Tsu,0,0 Tsu+1,0,0 +r +r +r m m m (1-pr) Mathematical Analysis

  16. Mathematical Analysis General K, Pure Sender-Initiated Policy

  17. Mathematical Analysis General K

  18. Mathematical Analysis General K

  19. Numerical Results

  20. Numerical Results

  21. Numerical Results

  22. Numerical Results

  23. Numerical Results

  24. Stochastic Derivative-Free Optimization • Internal Model • External Model • Trust Region

  25. General Overview • Optimal Dynamic Threshold Scheduling Policy • Fluid limits • Diffusion limits • Analysis of Dynamic Threshold Scheduling • Consider generalized threshold scheduling policy • Matrix-analytic analysis and fix-point solution, asymptotically exact • Numerical experiments • Optimal settings of dynamic scheduling policy thresholds • Stochastic Derivative-Free Optimization

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