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Taka Osogami

Recursive dimensionality reduction for the analysis of multiserver scheduling policies. Taka Osogami. Joint with: . Mor Harchol-Balter (CMU, CS) Adam Wierman (CMU, CS) Alan Scheller-Wolf (CMU, Tepper School) . H. L. L. H. H. multiserver scheduling problems . Goal :

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Taka Osogami

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  1. Recursive dimensionality reduction for the analysis of multiserver scheduling policies Taka Osogami Joint with: Mor Harchol-Balter (CMU, CS) Adam Wierman (CMU, CS) Alan Scheller-Wolf (CMU, Tepper School)

  2. H L L H H multiserver scheduling problems Goal: Mean response time = f(scheduling policy, arrival processes, job size distributions)

  3. l l l 1 2 0 m m m Phase-type job size (M/PH/1) MAP/PH/1 l l l 1 2 0 m1 m1 m1 1 2 0 l1 l1 l1 m2 m2 1 2 0 m3 m3 m m m m3 1 2 1 2 0 l3 l3 l4 l4 l3 l4 l l l2 l2 l2 1 2 0 Markovian arrival process (MAP/M/1) m m m 1 2 1 2 Analysis of single-server FCFS FCFS Poisson arrival & exponential job size (M/M/1)

  4. H L L H H Goal: Mean response time = f(scheduling policy,MAP arrival processes, PH job size distributions) H has preemptive priority over L Common Problem: 2D-infinite Markov chain (or nD-infinite)

  5. #L #H lL lL lL 0,0 0,1 0,2 H mL 2mL 2mL mH mH mH lH lH L L H lH lL lL lL 1,0 1,1 1,2 H mL mL mL lH lH lH 2mH 2mH 2mH lL lL lL 2,0 2,1 2,2 lH lH lH 2mH 2mH 2mH lL lL lL 3,0 3,1 3,2 M/M/2 with 2 priority classes

  6. Prior work: 2D infinite Markov chain transform methods (’80s-’90s) Boundary value problems (’70s-’00s) matrix analytic methods (’80s-’00s) Gail Hunter Taylor Mitrani King Kao Wilson : Cohen Boxma King Mitrani Fayolle Iasnogorodski Borst Jelenkovic Uitert Nain : truncation (1D) no truncation Rao Posner Kao Narayanan Leemans Miller Ngo Lee : 2D 2D→1D nD→1D Latouche Ramaswami Bright Taylor Sleptchenko Squilante : Our approach

  7. H L L H H 2D infinite Markov chain #L L #H lL lL lL 0,0 0,1 0,2 mL 2mL 2mL L mH mH mH lH lH lH H lL lL lL 1,0 1,1 1,2 L mL mL mL L lH lH lH 2mH 2mH 2mH H lL lL lL 2,0 2,1 2,2 L L H lH lH lH 2mH 2mH 2mH lL lL lL 3,0 3,1 3,2

  8. H L L H H 2D 1D #L L #H lL lL lL 0,0 0,1 0,2 L mL 2mL 2mL L mH mH mH lH lH lH H lL lL lL 1,0 1,1 1,2 L L mL mL mL L lH b3 lH b3 lH b3 lL lL lL 2+,1 2+,2 2+,0 b2 b1 b2 b1 b2 b1 lL lL lL 2+,1 2+,2 2+,0

  9. H L L M H M/M/2 with 3 priority classes #L #H #M Now chain grows infinitely in 3 dimensions!

  10. Transitions within a level: #L = 2 Transitions within a level: #L = 3 L L L L L lM lM lH lH mM mM mH mH H H M M L L L L L L L L L L lH lH lM lM lM lM lH lH H H M L L L L L L L L L M H M H H M lL L L L L L L H M M

  11. Analysis via RDR Simulation Relative error (%) Class 1 Class 2 Class 3 Class 4 r Accuracy of recursive DR Mean delay r High prio Low prio

  12. Impact of our new analysis Evaluation of existing approximation New approximation Design heuristics for multiserver priority systems n slow servers sometimes better than 1 fast servers variability of H jobs – big impact on L jobs relative priority among higher priority jobs – little impact on L jobs

  13. H L H H nD 1D Summary Multiserver scheduling Common problem nD infinite Markov chain Recursive dimensionality reduction

  14. Thank you! Recursive dimensionality reduction for the analysis of multiserver scheduling policies Taka Osogami

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