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FINDING THE OPTIMAL QUANTUM SIZE Revisiting the M/G/1 Round-Robin Queue

FINDING THE OPTIMAL QUANTUM SIZE Revisiting the M/G/1 Round-Robin Queue. VARUN GUPTA Carnegie Mellon University. M/G/1/RR. Incomplete jobs. Poisson arrivals. Jobs served for q units at a time. M/G/1/RR. Incomplete jobs. Poisson arrivals. Jobs served for q units at a time.

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FINDING THE OPTIMAL QUANTUM SIZE Revisiting the M/G/1 Round-Robin Queue

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  1. FINDING THE OPTIMAL QUANTUM SIZERevisiting the M/G/1 Round-Robin Queue VARUN GUPTA Carnegie Mellon University

  2. M/G/1/RR Incomplete jobs Poisson arrivals Jobs served for q units at a time

  3. M/G/1/RR Incomplete jobs Poisson arrivals Jobs served for q units at a time

  4. M/G/1/RR Incomplete jobs Completed jobs Poisson arrivals Jobs served for q units at a time • arrival rate =  • job sizes i.i.d. ~ S • load Squared coefficient of variability (SCV) of job sizes: C2 0

  5. q M/G/1/RR q→ 0 q=  M/G/1/FCFS M/G/1/PS  Preemptions cause overhead (e.g. OS scheduling)  Variable job sizes cause long delays preemption overheads job size variability small q large q

  6. GOAL Optimal operating q for M/G/1/RR with overheads and high C2 effect of preemption overheads SUBGOAL Sensitivity Analysis: Effect of q and C2 on M/G/1/RR performance (no switching overheads) • PRIOR WORK • Lots of exact analysis: [Wolff70], [Sakata et al.71], [Brown78] •  No closed-form solutions/bounds •  No simple expressions for interplay of q and C2

  7. Outline • Effect of q and C2 on mean response time • Approximate analysis • Bounds for M/G/1/RR • Choosing the optimal quantum size No preemption overheads Effect of preemption overheads

  8. Approximate sensitivity analysis of M/G/1/RR Approximation assumption 1: Service quantum ~ Exp(1/q) Approximation assumption 2: Job size distribution

  9. For high C2: E[TRR*] ≈E[TPS](1+q) Approximate sensitivity analysis of M/G/1/RR • Monotonic in q • Increases from E[TPS] →E[TFCFS] • Monotonic in C2 • Increases from E[TPS] →E[TPS](1+q)

  10. Outline • Effect of q and C2 on mean response time • Approximate analysis • Bounds for M/G/1/RR • Choosing the optimal quantum size

  11. M/G/1/RR bounds Assumption: job sizes  {0,q,…,Kq} THEOREM: Lower bound is TIGHT: E[S] = iq Upper bound is TIGHT within (1+/K): Job sizes  {0,Kq}

  12. Outline • Effect of q and C2 on mean response time • Approximate analysis • Bounds for M/G/1/RR • Choosing the optimal quantum size

  13. Optimizing q Preemption overhead = h • Minimize E[TRR] upper bound from Theorem: Common case:

  14. Mean response time h=0 service quantum (q) Job size distribution: H2 (balanced means) E[S] = 1, C2 = 19,  = 0.8

  15. approximation q* optimum q Mean response time 1% h=0 service quantum (q) Job size distribution: H2 (balanced means) E[S] = 1, C2 = 19,  = 0.8

  16. approximation q* optimum q Mean response time 2.5% 1% h=0 service quantum (q) Job size distribution: H2 (balanced means) E[S] = 1, C2 = 19,  = 0.8

  17. approximation q* optimum q Mean response time 5% 2.5% 1% h=0 service quantum (q) Job size distribution: H2 (balanced means) E[S] = 1, C2 = 19,  = 0.8

  18. approximation q* optimum q 7.5% Mean response time 5% 2.5% 1% h=0 service quantum (q) Job size distribution: H2 (balanced means) E[S] = 1, C2 = 19,  = 0.8

  19. approximation q* optimum q 10% 7.5% Mean response time 5% 2.5% 1% h=0 service quantum (q) Job size distribution: H2 (balanced means) E[S] = 1, C2 = 19,  = 0.8

  20. Outline • Effect of q and C2 on mean response time • Approximate analysis • Bounds for M/G/1/RR • Choosing the optimal quantum size

  21. q Conclusion/Contributions • Simpleapproximation and bounds for M/G/1/RR • Optimal quantum size for handling highly variable job sizes under preemption overheads

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