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SRPT for Multiserver Systems

SRPT for Multiserver Systems. Isaac Grosof Ziv Scully Mor Harchol-Balter. Carnegie Mellon University. M/G/1. Response time: T. Remaining size. Job size. Age. M/G/1. Response time: T. Remaining size. Job size. Q: Can we analyze E[T] M/G/1/SRPT ?.

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SRPT for Multiserver Systems

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  1. SRPT for Multiserver Systems Isaac Grosof Ziv Scully MorHarchol-Balter Carnegie Mellon University

  2. M/G/1 Response time: T Remaining size Job size Age

  3. M/G/1 Response time: T Remaining size Job size Q: Can we analyze E[T]M/G/1/SRPT? Q: What policy minimizes mean response time, E[T]? A: SRPT [Schrage ‘68] A: Yes! [Schrage ‘66] Age 3

  4. SRPT-k: Run k jobs w/ least remaining size M/G/k k Q: Can we analyze E[T]M/G/k/SRPT? Q: What policy is optimal? Q: What policy is optimal? Q: Is SRPT-k? ?? OPEN

  5. Our 3 Results • First analysis of M/G/k/SRPT (bound on E[T]) • Analysis is tight as (for any fixed k) • SRPT-k is OPT for M/G/k as Bonus: Technique generalizes to first analysis of M/G/k/PSJF, M/G/k/FB and others.

  6. More Specifics M/G/1 M/G/k Speed 1/k Speed 1/k Speed 1/k Speed 1/k Speed 1 E[]M/G/k/OPT E[]M/G/1/OPT E[]M/G/k/SRPT E[]M/G/k/OPTE[]M/G/1/OPT E[]M/G/k/SRPT E[]M/G/k/OPTE[]M/G/1/OPT E[]M/G/1/SRPT We show:For all k, SRPT-k is optimal

  7. Prior work on M/G/k scheduling M/G/k/FCFS • Bounds based on first two moments [Kingman ‘70], [Daley ‘97] • Arbitrarily tight bounds [Gupta, Osogami ‘11] M/PH/k/PRIORITY • Numerical solutions, limited priority classes [Sleptchenko et al ‘05], [Osogami et al ‘05] No prior work on M/G/k/SRPT, M/G/k/PSJF, M/G/k/FB, M/G/k/RS We can analyze all!

  8. Outline • Analyze mean response time of M/G/k/SRPT • Show SRPT-k is asymptotically optimal for M/G/k • Empirical evaluation

  9. Analyzing Response time in M/G/k/SRPT • We focus on a tagged job, j • j is of size x • Our goal is to bound Tj • SRPT-k implies that at any moment when j is in the system, either: • A server is working on j, or • All servers are working on jobs with less remaining size than j (relevant jobs)

  10. Example z j c b a Server 3 Increasing remaining job size Server 2 Server 1 Time

  11. Example z j c Q: What happens next? Q: What jobs are initially in service? b a c Server 3 Increasing remaining job size Increasing remaining job size b Server 2 Server 1 a Time

  12. Example z j c Q: What happens next? b a c j Server 3 Increasing remaining job size Increasing remaining job size c b Server 2 Server 1 a b Time

  13. Example z j c b a n c c j j j j z z z z Server 3 Increasing remaining job size Increasing remaining job size z Server 2 b b c c c j j j Server 1 a a b n c c j Time

  14. Response Time Irrelevant z Tagged j c Relevant b a Area n Speed 1/k c c j j j j z z z z Server 3 Increasing remaining job size Increasing remaining job size z z Server 2 Speed 1/k b b c c c j j j Speed 1/k Server 1 a a b n c c j Tj

  15. Response Time Irrelevant z Tagged j c Old Relevant Virtual work b a Area New Relevant n Tagged work c c j j j j z z z z Server 3 Old relevant work Increasing remaining job size Increasing remaining job size z Server 2 b b c c c j j j Server 1 a a b n c c j New relevant work Tj

  16. Response Time Irrelevant z Tagged j c Old Relevant Virtual work Virtual work b a Area New Relevant n Tagged work Tagged work c c j j j j z z z z Server 3 Old relevant work Old relevant work Increasing remaining job size Increasing remaining job size z z Server 2 b b c c c j j j Server 1 a a b n c c j New relevant work Relevant busy period New relevant work Tj

  17. Relevant Work Relevant work is work made up of jobs of remaining size Relevant work SRPT-k SRPT-1 SRPT-1 Time Goal: Show

  18. Relevant Work Relevant work SRPT-k SRPT-1 SRPT-1 Many jobs intervals: relevant jobs in SRPT-k Time Few jobs intervals: relevant jobs in SRPT-k

  19. Few jobs intervals Relevant work SRPT-k SRPT-1 SRPT-1 relevant jobs in SRPT-k relevant work in SRPT-k Time

  20. Many jobs intervals Relevant work SRPT-k SRPT-1 SRPT-1 relevant jobs in SRPT-k SRPT-k is completing relevant work at speed 1 is nonincreasing Time

  21. Many jobs intervals Relevant work SRPT-k SRPT-1 SRPT-1 At all times, Time

  22. Response Time Virtual work Virtual work Tagged work Area Tagged work c c j j j j z z z z Server 3 Old relevant work Old relevant work Increasing remaining job size Increasing remaining job size z z Server 2 b b c c c j j j Server 1 a a b n c c j New relevant work Relevant busy period New relevant work Tj

  23. Outline • Analyze mean response time of M/G/k/SRPT • Show SRPT-k is asymptotically optimal for M/G/k • Empirical evaluation

  24. Optimality Our result: After some algebra * * [Lin, Wierman, Zwart ‘11]: * Subject to technical assumption on job size distribution

  25. Ratio k = 10 Ratio Ratio Analysis bound Simulation 10 Analysis bound Simulation 10 Shows optimality 1 1 0 1-10-2 1-10-4 1-10-6 1-10-8 0 1-10-2 1-10-4 1-10-6 1-10-8 System load () System load () SHyperexponential, C2=10 SUniform(0, 1)

  26. Conclusion • First analysis of M/G/k/SRPT (bound on E[T]) • Analysis is tight as (for any fixed k) • SRPT-k achieves optimal E[T] for M/G/k as j j z z z z Server 3 c c j j Increasing remaining job size Increasing remaining job size z z Server 2 c j j j b b c c Server 1 n c c j a a b Tj

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