250 likes | 357 Views
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].
E N D
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] 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
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
() 1 () P Scheduling Policy
() 1 () () P Scheduling Model
(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
(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
(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
(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
(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
(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
(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
(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
(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
Mathematical Analysis General K, Pure Sender-Initiated Policy
Mathematical Analysis General K
Mathematical Analysis General K
Stochastic Derivative-Free Optimization • Internal Model • External Model • Trust Region
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