IBM Thomas J Watson Research Center Analysis of
IBM Thomas J. Watson Research Center Analysis of Parallel-Server Systems with Dynamic Affinity Scheduling and Load Balancing § Mark S. Squillante § Mathematical Sciences Department § April 18, 2004 © 2004 IBM Corporation
IBM Thomas J. Watson Research Center Problem Motivation Cache Affinity [Squillante. Lazowska 90] 2 Analysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante © 2004 IBM Corporation
IBM Thomas J. Watson Research Center Problem Motivation Cache Affinity [Squillante. Lazowska 90] 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 3 Analysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante © 2004 IBM Corporation
IBM Thomas J. Watson Research Center 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 4 Analysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante © 2004 IBM Corporation
IBM Thomas J. Watson Research Center Scheduling Policy 5 1 P ( ) Analysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante © 2004 IBM Corporation
IBM Thomas J. Watson Research Center Scheduling Model 6 1 P ( ) Analysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante © 2004 IBM Corporation
IBM Thomas J. Watson Research Center Mathematical Analysis 0, 1, 0 s pr 1, 0, 1 (1 -pr) 0, 0, 0 1, 0, 0 (1 -pr) … … … Tru, 1, 0 Tru+1, 0, 1 Tru, 0, 0 Tru+1, 0, 0 r + r … … … (1 -ps) r + r Tsu, 1, 0 r Tsu+1, 0, 1 (1 -ps) Tsu, 0, 0 + r Tsu+1, 0, 0 … … … 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 7 Analysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante © 2004 IBM Corporation
IBM Thomas J. Watson Research Center Mathematical Analysis 0, 1, 0 s pr 1, 0, 1 (1 -pr) 0, 0, 0 1, 0, 0 (1 -pr) 8 … … … Tru, 1, 0 Tru+1, 0, 1 Tru, 0, 0 Tru+1, 0, 0 r + r … … … (1 -ps) r + r Analysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante Tsu, 1, 0 r Tsu+1, 0, 1 (1 -ps) Tsu, 0, 0 + r Tsu+1, 0, 0 … … … © 2004 IBM Corporation
IBM Thomas J. Watson Research Center Mathematical Analysis 0, 1, 0 s pr 1, 0, 1 (1 -pr) 0, 0, 0 1, 0, 0 (1 -pr) 9 … … … Tru, 1, 0 Tru+1, 0, 1 Tru, 0, 0 Tru+1, 0, 0 r + r … … … (1 -ps) r + r Analysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante Tsu, 1, 0 r Tsu+1, 0, 1 (1 -ps) Tsu, 0, 0 + r Tsu+1, 0, 0 … … … © 2004 IBM Corporation
IBM Thomas J. Watson Research Center Mathematical Analysis 0, 1, 0 pr s pr 1, 0, 1 (1 -pr) 0, 0, 0 1, 0, 0 (1 -pr) 10 … … … Tru, 1, 0 Tru+1, 0, 1 Tru, 0, 0 Tru+1, 0, 0 r + r … … … (1 -ps) r + r Analysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante Tsu, 1, 0 r Tsu+1, 0, 1 (1 -ps) Tsu, 0, 0 + r Tsu+1, 0, 0 … … … © 2004 IBM Corporation
IBM Thomas J. Watson Research Center Mathematical Analysis 0, 1, 0 pr s pr 1, 0, 1 (1 -pr) 0, 0, 0 1, 0, 0 (1 -pr) 11 … … … Tru, 1, 0 Tru+1, 0, 1 Tru, 0, 0 Tru+1, 0, 0 r + r … … … (1 -ps) r + r Analysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante Tsu, 1, 0 r Tsu+1, 0, 1 (1 -ps) Tsu, 0, 0 + r Tsu+1, 0, 0 … … … © 2004 IBM Corporation
IBM Thomas J. Watson Research Center Mathematical Analysis 0, 1, 0 pr s pr 1, 0, 1 (1 -pr) 0, 0, 0 1, 0, 0 (1 -pr) 12 … … … Tru, 1, 0 Tru+1, 0, 1 Tru, 0, 0 Tru+1, 0, 0 r + r … … … (1 -ps) r + r Analysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante Tsu, 1, 0 r Tsu+1, 0, 1 (1 -ps) Tsu, 0, 0 + r Tsu+1, 0, 0 … … … © 2004 IBM Corporation
IBM Thomas J. Watson Research Center Mathematical Analysis 0, 1, 0 pr s pr 1, 0, 1 (1 -pr) 0, 0, 0 1, 0, 0 (1 -pr) … … … Tru, 1, 0 Tru+1, 0, 1 Tru, 0, 0 Tru+1, 0, 0 r + r … … … (1 -ps) r + r Tsu, 1, 0 r Tsu+1, 0, 1 (1 -ps) Tsu, 0, 0 + r Tsu+1, 0, 0 … … … Final Solution Obtained via Fix-Point Iteration 1. Initialize (ps, s, pr, r) 2. Compute stationary probability vector in terms of (ps, s, pr, r) 3. Compute new values of (ps, s, pr, r) in terms of stationary vector 4. Goto 2 until differences between iteration values are arbitrarily small 13 Analysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante © 2004 IBM Corporation
IBM Thomas J. Watson Research Center Mathematical Analysis 0, 1, 0 pr s pr 1, 0, 1 (1 -pr) 0, 0, 0 1, 0, 0 (1 -pr) 14 … … … Tru, 1, 0 Tru+1, 0, 1 Tru, 0, 0 Tru+1, 0, 0 r + r … … … (1 -ps) r + r Analysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante Tsu, 1, 0 r Tsu+1, 0, 1 (1 -ps) Tsu, 0, 0 + r Tsu+1, 0, 0 … … … © 2004 IBM Corporation
IBM Thomas J. Watson Research Center Mathematical Analysis 0, 1, 0 pr s pr 1, 0, 1 (1 -pr) 0, 0, 0 1, 0, 0 (1 -pr) 15 … … … Tru, 1, 0 Tru+1, 0, 1 Tru, 0, 0 Tru+1, 0, 0 r + r … … … (1 -ps) r + r Analysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante Tsu, 1, 0 r Tsu+1, 0, 1 (1 -ps) Tsu, 0, 0 + r Tsu+1, 0, 0 … … … © 2004 IBM Corporation
IBM Thomas J. Watson Research Center Mathematical Analysis General K, Pure Sender-Initiated Policy 16 Analysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante © 2004 IBM Corporation
IBM Thomas J. Watson Research Center Mathematical Analysis General K 17 Analysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante © 2004 IBM Corporation
IBM Thomas J. Watson Research Center Mathematical Analysis General K 18 Analysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante © 2004 IBM Corporation
IBM Thomas J. Watson Research Center Numerical Results 19 Analysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante © 2004 IBM Corporation
IBM Thomas J. Watson Research Center Numerical Results 20 Analysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante © 2004 IBM Corporation
IBM Thomas J. Watson Research Center Numerical Results 21 Analysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante © 2004 IBM Corporation
IBM Thomas J. Watson Research Center Numerical Results 22 Analysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante © 2004 IBM Corporation
IBM Thomas J. Watson Research Center Numerical Results 23 Analysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante © 2004 IBM Corporation
IBM Thomas J. Watson Research Center Stochastic Derivative-Free Optimization § Internal Model § External Model § Trust Region 24 Analysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante © 2004 IBM Corporation
IBM Thomas J. Watson Research Center 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 25 Analysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante © 2004 IBM Corporation
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