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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]. - PowerPoint PPT Presentation
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IBM Thomas J. Watson Research Center
© 2004 IBM Corporation
Analysis of Parallel-Server Systems with Dynamic Affinity Scheduling and Load Balancing
Mark S. SquillanteMathematical Sciences DepartmentApril 18, 2004
2
IBM Thomas J. Watson Research Center
© 2004 IBM CorporationAnalysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante
Problem Motivation
Cache Affinity [SquillanteLazowska90]
3
IBM Thomas J. Watson Research Center
© 2004 IBM CorporationAnalysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante
Problem Motivation
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
Cache Affinity [SquillanteLazowska90]
4
IBM Thomas J. Watson Research Center
© 2004 IBM CorporationAnalysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante
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
IBM Thomas J. Watson Research Center
© 2004 IBM CorporationAnalysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante
Scheduling Policy
()
()
1
P
6
IBM Thomas J. Watson Research Center
© 2004 IBM CorporationAnalysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante
Scheduling Model
()
()1
P
()
7
IBM Thomas J. Watson Research Center
© 2004 IBM CorporationAnalysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante
Mathematical Analysis
Ts
u,1,0
Tsu,0,0
Tsu+1,0,1
Tsu+1,0,0
r r
(1-ps)
(1-ps)
r r
s
0,1,0
0,0,0
1,0,1
1,0,0
(1-pr)
(1-pr)pr
pr
…
……
Tru,1,0
Tru,0,0
Tru+1,0,1
Tru+1,0,0
r
r
…
……
…
……
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
IBM Thomas J. Watson Research Center
© 2004 IBM CorporationAnalysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante
Mathematical Analysis
Ts
u,1,0
Tsu,0,0
Tsu+1,0,1
Tsu+1,0,0
r r
(1-ps)
(1-ps)
r r
s
0,1,0
0,0,0
1,0,1
1,0,0
(1-pr)
(1-pr)pr
pr
…
……
Tru,1,0
Tru,0,0
Tru+1,0,1
Tru+1,0,0
r
r
…
……
…
……
9
IBM Thomas J. Watson Research Center
© 2004 IBM CorporationAnalysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante
Mathematical Analysis
Ts
u,1,0
Tsu,0,0
Tsu+1,0,1
Tsu+1,0,0
r r
(1-ps)
(1-ps)
r r
s
0,1,0
0,0,0
1,0,1
1,0,0
(1-pr)
(1-pr)pr
pr
…
……
Tru,1,0
Tru,0,0
Tru+1,0,1
Tru+1,0,0
r
r
…
……
…
……
10
IBM Thomas J. Watson Research Center
© 2004 IBM CorporationAnalysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante
Mathematical Analysis
Ts
u,1,0
Tsu,0,0
Tsu+1,0,1
Tsu+1,0,0
r r
(1-ps)
(1-ps)
r r
s
0,1,0
0,0,0
1,0,1
1,0,0
(1-pr)
(1-pr)pr
pr
…
……
Tru,1,0
Tru,0,0
Tru+1,0,1
Tru+1,0,0
r
r
…
……
…
……
11
IBM Thomas J. Watson Research Center
© 2004 IBM CorporationAnalysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante
Mathematical Analysis
Ts
u,1,0
Tsu,0,0
Tsu+1,0,1
Tsu+1,0,0
r r
(1-ps)
(1-ps)
r r
s
0,1,0
0,0,0
1,0,1
1,0,0
(1-pr)
(1-pr)pr
pr
…
……
Tru,1,0
Tru,0,0
Tru+1,0,1
Tru+1,0,0
r
r
…
……
…
……
12
IBM Thomas J. Watson Research Center
© 2004 IBM CorporationAnalysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante
Mathematical Analysis
Ts
u,1,0
Tsu,0,0
Tsu+1,0,1
Tsu+1,0,0
r r
(1-ps)
(1-ps)
r r
s
0,1,0
0,0,0
1,0,1
1,0,0
(1-pr)
(1-pr)pr
pr
…
……
Tru,1,0
Tru,0,0
Tru+1,0,1
Tru+1,0,0
r
r
…
……
…
……
13
IBM Thomas J. Watson Research Center
© 2004 IBM CorporationAnalysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante
Mathematical Analysis
Ts
u,1,0
Tsu,0,0
Tsu+1,0,1
Tsu+1,0,0
r r
(1-ps)
(1-ps)
r r
s
0,1,0
0,0,0
1,0,1
1,0,0
(1-pr)
(1-pr)pr
pr
…
……
Tru,1,0
Tru,0,0
Tru+1,0,1
Tru+1,0,0
r
r
…
……
…
……
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
14
IBM Thomas J. Watson Research Center
© 2004 IBM CorporationAnalysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante
Mathematical Analysis
Ts
u,1,0
Tsu,0,0
Tsu+1,0,1
Tsu+1,0,0
r r
(1-ps)
(1-ps)
r r
s
0,1,0
0,0,0
1,0,1
1,0,0
(1-pr)
(1-pr)pr
pr
…
……
Tru,1,0
Tru,0,0
Tru+1,0,1
Tru+1,0,0
r
r
…
……
…
……
15
IBM Thomas J. Watson Research Center
© 2004 IBM CorporationAnalysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante
Mathematical Analysis
Ts
u,1,0
Tsu,0,0
Tsu+1,0,1
Tsu+1,0,0
r r
(1-ps)
(1-ps)
r r
s
0,1,0
0,0,0
1,0,1
1,0,0
(1-pr)
(1-pr)pr
pr
…
……
Tru,1,0
Tru,0,0
Tru+1,0,1
Tru+1,0,0
r
r
…
……
…
……
16
IBM Thomas J. Watson Research Center
© 2004 IBM CorporationAnalysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante
Mathematical Analysis
General K, Pure Sender-Initiated Policy
17
IBM Thomas J. Watson Research Center
© 2004 IBM CorporationAnalysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante
Mathematical Analysis
General K
18
IBM Thomas J. Watson Research Center
© 2004 IBM CorporationAnalysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante
Mathematical Analysis
General K
19
IBM Thomas J. Watson Research Center
© 2004 IBM CorporationAnalysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante
Numerical Results
20
IBM Thomas J. Watson Research Center
© 2004 IBM CorporationAnalysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante
Numerical Results
21
IBM Thomas J. Watson Research Center
© 2004 IBM CorporationAnalysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante
Numerical Results
22
IBM Thomas J. Watson Research Center
© 2004 IBM CorporationAnalysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante
Numerical Results
23
IBM Thomas J. Watson Research Center
© 2004 IBM CorporationAnalysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante
Numerical Results
24
IBM Thomas J. Watson Research Center
© 2004 IBM CorporationAnalysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante
Stochastic Derivative-Free Optimization
Internal Model
External Model
Trust Region
25
IBM Thomas J. Watson Research Center
© 2004 IBM CorporationAnalysis of Dynamic Affinity Scheduling and Load Balancing | Mark S. Squillante
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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