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Recursive Dimensionality Reduction for Analyzing Multiserver Scheduling Policies

This research presents recursive dimensionality reduction (RDR) techniques to analyze multiserver scheduling policies, focusing on the impact of scheduling policy, arrival processes, and job size distributions on mean response time. It explores models such as M/PH/1 and MAP/M/1, providing insights into preemptive priority and multi-class systems. The paper discusses the challenges of infinite Markov chains and introduces innovative methods that improve the evaluation of existing approximations, offering new heuristics for better system performance in automated service environments.

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Recursive Dimensionality Reduction for Analyzing Multiserver Scheduling Policies

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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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