1 Alan Scheller-Wolf Joint with: Mor Harchol-Balter, Taka Osogami, Adam Wierman, and Li Zhang. Dimensionality Reduction for the analysis of Cycle Stealing, Task Assignment, Priority Queueing, and Threshold Policies (PART 2)

2 Affinity Scheduling      

3 Prior Work: Affinity Scheduling Threshold policies Squillante, Xia, Yao and Zhang Williams Bell and Williams Harrison Harrison and Lopez Squillante, Xia, Zhang Williams Fluid or Diffusion Green Schumsky Stanford and Grassman Applications (cycle stealing) No accurate analysis for non-limiting behavior.

4 Situation 1: Self-Affinities       Optimal control policy: Cycle Stealing.

5 Situation 2:Eager to Help       If server 2 overzealous, a brake is needed.

6 Why? Potential Instability       Maybe server two is too eager to help: Take too much work from server 1,leaving her idle, Neglect own work, letting it build up. 

7 The Brake: T1 Policy Asymptotically optimal, robustness concerns. We provide first easy, accurate analysis. “Come help, but only when I call you.” N2 N1 T1 1

8 T1 Policy: Performance vs. T1 

9 T1 Performance IIT1 Policy: Performance vs.  

10 N1 N2 What is the Dream? Switching Curve Optimal?

11 New Control Policy: The ADT Policy Performs like best of T1(1) and T1(2). We propose and analyze. “Come help when I call you.” N1 N2 T1(1) “If you are very busy and I am not, do not come.” “But if I really need you, you have to come.” T1(2) T2

12 T1 Policy: Performance vs. T1

13 ADT Policy: Performance vs T1(1)

14 T1 Performance IIT1 Policy: Performance vs.  

15 ADT Policy: Performance vs  

16 Goal: Mean response time per job type. RDR and Priority Scheduling nD-infinite chain 1D-infinite chain HARDEASY Priority Scheduling in M/PH/k L H H MHL

17 Scaling as Single-server: Buzen and Bondi Aggregation into Two classes: Mitrani and King Nishida Multi-class simple approx. Two job classes, exponential Cidon and Sidi Feng el at Gail et al Miller Matrix Analytic or Gen. Functions Aggregation or Truncation Two job classes, exponential Two job classes, hyper- exponential Sleptchenko et alKao and Narayanan Kao and Wilson Kapadia et al Nishida Ngo and Lee Iterative sol to balance equations Little work for > 2 classes or non-exponential. Prior Work: Multi-Server Priority Queues

18 What’s so Hard? Low Hi Med Now chain grows infinitely in 3 dimensions!

19 Recursive Dimensionality Reduction(RDR) Apply standard dimensionality reduction (DR) to two highest classes (Mor’s talk). Aggregate these classes -- carefully -- into single higher class. Many types of busy periods. Apply DR to two-class system made up of aggregated classes and third class. Recurse. Chain for class m used to calculate busy periods for next lower class (m+1).

20 Representative Types of Busy Periods L H LLL Becomes… M H LLLL H H LLLL or M L L H L LL

21 What are these busy periods? M MM 1,00,0 M  M 3,02,0 M  M M M MM 1,10,1 M MM 3,12,1 M MM M MM M 1,2 + 0,2 + M 3,2 + 2,2 + M M H HH H HH H HH H HH H H H H BHBH BHBH BHBH BHBH Neuts[1978]

22 The Low Job Chain M MM 1,00,0 0,1 H HH

23 The Low Job Chain M MM 5,1,05,0,0 5,0,1 H HH

24 The Low Job Chain M MM 5,1,05,0,0 5,0,1 H HH

25 The Low Job Chain M MM 5,1,05,0,0 5,0,1 H HH 5,0,2 H 5,1,1 M 5,0,2 M 5,1,1 H 5,2,0 H 5,2,0 M

26 The Low Job Chain M MM 5,1,05,0,0 5,0,1 H HH 5,0,2 H 5,1,1 M 5,0,2 M 5,1,1 H 5,2,0 H 5,2,0 M

27 M/M/2 Four Priority Classes: Accuracy

28 M/M/2 Four Priority Classes: Resp.

29 M/M/2 Four Priority Classes: Perf

30 Generalizations and Extensions Phase-type service times. More classes, more servers. Number of different busy periods grows with complexity of system (service times, servers, classes). RDR-A approximation for these more complex systems, within 5% error for four class problem.

31 DR and RDR, future directions         We solve problems where one class depends on the other, but the dependencies can be solved sequentially (H,M,L). What about systems that do not decouple?