Multiserver Stochastic Scheduling Analysis

This presentation delves into the analysis and optimality of multiserver stochastic scheduling, focusing on the theory of large-scale computing systems, queueing theory, and prior work on single-server and multiserver scheduling. It explores optimizing response time and resource efficiency in modern computing systems with multiple servers.

• Scheduling
• Analysis
• Optimality
• Computing Systems
• Queueing Theory

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1. Multiserver Stochastic Scheduling: Analysis and Optimality Isaac Grosof CMU Northwestern University, Jan. 31, 2023 1 Northwestern University - Isaac Grosof

2. Theory of Large-Scale Computing Systems Datacenters, Supercomputers 1% of world s electricity 205 TWh, 140 Mt CO2, $320B. Need systems to be: Fast: low response time Efficient: minimal resources Scheduling is a key tool 2 Northwestern University - Isaac Grosof

3. Abstract Model: Queueing Theory Arrivals Response time: ? Varying job sizes (inherent work of a job) Arrivals are a stochastic process Scheduling policy preemptively determines which job to serve Modern computing systems: many servers. 3 Northwestern University - Isaac Grosof

4. Prior work: Single-server & Multiserver Scheduling 1 server Multiserver Well understood! [Pollaczek 30], [Schrage & Miller 66], [Scully 18]. Analysis only for First-Come First-Served (FCFS) [K llerstr m 74 & 79]. Nothing for complex scheduling policies. Analysis: Characterize ?[?] (mean response time) This talk: Multiserver results, building on single- server progress. Optimization: Minimize ?[?] over all scheduling policies. Entirely open. Negative worst-case results [Leonardi & Raz 97]. Shortest Remaining Processing Time (SRPT) is optimal! [Schrage 68]. 4 Northwestern University - Isaac Grosof

5. First Scheduling Analysis & Optimality 1 server/job (M/G/k) 3 Multiple servers/job (MSJ) 2 4 1 3 3 1 First analysis & optimality results for each system. 5 Northwestern University - Isaac Grosof

6. One server/job (M/G/k) model Load ? = ?? ? < 1 Size: ?, i.i.d. SRPT-k k servers Poisson(?) 6 Northwestern University - Isaac Grosof

7. One server/job (M/G/k) model Load ? = ?? ? < 1 Size: ?, i.i.d. SRPT-k k servers Poisson(?) SRPT for Multiserver Systems . Grosof, Scully, Harchol-Balter. IFIP Performance 18. Best Student Paper Award. Better scheduling: 37% higher completion rate, same resources, same response time. 7 Northwestern University - Isaac Grosof

8. Background: Relevant work x SRPT-k Work Relevant work Work ?: total remaining size of jobs Relevant job: job with remaining size ? Relevant work ??: total remaining size of relevant jobs Normalize: Work completes at rate 1 if all servers busy. 8 Northwestern University - Isaac Grosof

9. Analysis and Optimality of SRPT-k Speed 1 Speed 1/? SRPT-k SRPT-1 Speed 1/? Speed 1/? Idea: Bound SRPT-k relative to resource-pooled SRPT-1 Lemma: Bound SRPT-k relevant work relative to SRPT-1 relevant work Goal: Bound SRPT-k response time relative to SRPT-1 response time 9 Northwestern University - Isaac Grosof

10. Step 1: Bound relevant work SRPT-k SRPT-1 ??? ??1 Couple SRPT-k and SRPT-1: Same arrival sequence Consider ?(?) = ???(?) ??1(?). When # relevant jobs in SRPT-k system ?: ?(?) nonincreasing. 10 Northwestern University - Isaac Grosof

11. Step 1: Bound relevant work SRPT-k SRPT-1 ??? ??1 Couple SRPT-k and SRPT-1: Same arrival sequence Consider ?(?) = ???(?) ??1(?). When # relevant jobs in SRPT-k system ?: ?(?) nonincreasing. When # relevant jobs in SRPT-k system < ?: ???(?) ??, so ?(?) ?? For all t, ?(?) ??, so ? ??? ? ??1+ ?? 11 Northwestern University - Isaac Grosof

12. Step 2: Bound response time x SRPT-k Key idea: Bound on x s response time implied by bound on work completed while x in system. Work while x in queue Work while x in service 12 Northwestern University - Isaac Grosof

13. Step 2: Bound response time SRPT-k x Key idea: Bound on x s response time implied by bound on work completed while x in system. Relevant work seen on arrival Work while x in queue Relevant work Work while x in service 13 Northwestern University - Isaac Grosof

14. Step 2: Bound response time SRPT-k x Key idea: Bound on x s response time implied by bound on work completed while x in system. Relevant work seen on arrival Lemma: ? ??? ?[??1] + ?? Work while x in queue Relevant work Bound: Relevant work arriving after x Same as single server 2?? 1 ?? ? ??? ? ??1+ Work while x in service ?? 14 Northwestern University - Isaac Grosof

15. SRPT-k SRPT-k Analysis x 2?? 1 ?? Bound: ? ?????? ? ? ?????? 1+ Theorem: ? ????? ? ? ????? 1+ 2?? ? 1 ?ln 1 ? First analysis of SRPT-k! First analysis of any nontrivial multiserver scheduling policy! 15 Northwestern University - Isaac Grosof

16. SRPT-k SRPT-k Optimality x Theorem: ? ????? ? ? ????? 1+ 2?? ? 1 ?ln 1 ? ?(? ????? 1) as ? 1 [Lin, Wierman, & Zwart 11] ? ????? ? ? ????? 1= 1 ? ????? ? ? ???? ?= 1 Theorem: lim ? 1 lim ? 1 SRPT-k yields asymptotically optimal mean response time! First optimality result in the M/G/k! 16 Northwestern University - Isaac Grosof

17. Empirical Impact S: Hyperexponential, ?2 10. ? = 4 servers. Target: ? ? 10. FCFS-k (First-Come First-Served) SRPT-k: ? = 0.7 ? = 0.96 37% higher completion rate, same resources, same response time. Potential to save 10s TWh, 10s Mt CO2, $100Bs. 30 FCFS-k Prio-k PLCFS-k PSJF-k SRPT-k 25 20 ?[?] 15 10 5 0 0.5 0.7 0.8 0.6 0.9 1.0 Load ? 17 Northwestern University - Isaac Grosof

18. Prior Work: Single-server & Multiserver Scheduling Multiserver 1 server/job 1 server Well understood! [Pollaczek 30], [Schrage & Miller 66], [Scully 18]. SRPT-k s ?[?] bounded! Analysis: Characterize ?[?] (mean response time) SRPT is optimal! [Schrage 68]. SRPT-k is optimal! Optimization: Minimize ?[?] over all scheduling policies. 18 Northwestern University - Isaac Grosof

19. Prior Work: Single-server & Multiserver Scheduling Multiserver many servers/job Multiserver 1 server/job 1 server 3 3 2 2 4 2 1 Well understood! [Pollaczek 30], [Schrage & Miller 66], [Scully 18]. No analysis for any scheduling policy. SRPT-k s ?[?] bounded! Analysis: Characterize ?[?] (mean response time) SRPT is optimal! [Schrage 68]. Completely open SRPT-k is optimal! Optimization: Minimize ?[?] over all scheduling policies. 19 Northwestern University - Isaac Grosof

20. Multiserver-job (MSJ) Model 3 Size (height) = duration server need 4 2 2 3 3 1 Motivated by Google Borg scheduler [Tirmazi et al. 20]. Trace: Factor of 105 variation in server need. Prior scheduling: Heuristics and/or poor response time 20 Northwestern University - Isaac Grosof

21. New Multiserver-job (MSJ) Scheduling Policy Want to design SRPT-like scheduling policy. First try: 3 Greedy-SRPT 3 2 4 2 3 1 Wasted Server Want new policy: Avoid waste, SRPT-like. Goal: First ?[?] analysis & optimality. Setting for talk: ? is power of 2, all server needs are powers of 2. 21 Northwestern University - Isaac Grosof

22. New Multiserver-job (MSJ) Scheduling Policy 8 1 2 4 2 Setting for talk: ? is power of 2, all server needs are powers of 2. Optimal Scheduling in the Multiserver-job Model under Heavy Traffic , Grosof, Scully, Harchol-Balter, Scheller-Wolf. SIGMETRICS, 23. Also WCFS, Queueing Systems, 22. Better scheduling: 73% higher completion rate, same resources, same response time. 22 Northwestern University - Isaac Grosof

23. Define ServerFilling-SRPT M Least remaining size order k=8 2 4 1 2 8 1 1 4 4 1 2 1 1 1. Find minimal subset M of jobs of least rem. size with total server need ? servers 2. Among M, serve largest server need first. (tiebreak by remaining size) 23 Northwestern University - Isaac Grosof

24. Define ServerFilling-SRPT M Least remaining size order 4 k=8 2 4 1 2 8 1 1 4 2 1 2 1 1 1 1 1. Find minimal subset M of jobs of least rem. size with total server need ? servers 2. Among M, serve largest server need first. (tiebreak by remaining size) 3. When a job completes, M changes, rearrange preemptively. 24 Northwestern University - Isaac Grosof

25. Define ServerFilling-SRPT M Least remaining size order 4 k=8 2 4 1 2 8 1 1 4 2 1 1 1 1. Find minimal subset M of jobs of least rem. size with total server need ? servers 2. Among M, serve largest server need first. (tiebreak by remaining size) 3. When a job completes, M changes, rearrange preemptively. Efficiency theorem: If total server need of all jobs ?, ServerFilling-SRPT fills all servers Next: Analyze ? ? Server need is power of 2 25 Northwestern University - Isaac Grosof

26. Understand ?[?] for ServerFilling-SRPT 1/k 1/k 1/k 1/k 1/k 1/k 1/k 1/k ServerFilling-SRPT 4 SRPT-1 1 1 2 4 1 2 2 Proof plan: Compare against resource-pooled SRPT-1 Bound relevant work ? ?? Bound response time ? ? But: Obstacles at each step! 26 Northwestern University - Isaac Grosof

27. ServerFilling-SRPT: Analysis & Optimality Compare against resource-pooled SRPT-1 Bound relevant work ? ?? Bound response time ? ? ? ! ? Old ? ?? bound too weak New technique: MIAOW Bound intervals of ?, not individual ?. Old ? ? bound fails New formula: WINE ? ? =1 ? ? ! What size distribution? Size = duration server need ! ?[??] ?2 ?? 0 ?+1 ? 1 ? 1 ?+? 1 First analysis: ? ??? ???? ? ????? 1+ ? ??? ???? ? ???? ???= 1 ln ? First optimality: lim ? 1 27 Northwestern University - Isaac Grosof

28. Empirical Impact Setting: ? = 8, server need uniform [1,2,4,8], size Hyperexp, ?2 10, independent of server need. Target ?[?] = 1.5 Greedy-SRPT ServerFilling-SRPT: ? = 0.56 ? = 0.97 73% higher completion rate, same resources, same response time. Even more TWh, Mt CO2, $B. 5 4 3 ?[?] 2 1 0 0.2 1.0 0.6 0 0.4 0.8 Load ? 28 Northwestern University - Isaac Grosof

29. Theory Practical Power Management Huge Waste of Capacity & Energy Response Time Guarantees + Problem: No Preemption Current solution: Heuristics [Autopilot 20] 3 1 Wasted Servers My approach: Learning Workload Distributions Advanced Scheduling Mathematical Guarantees + 29 Northwestern University - Isaac Grosof

30. ??? Future directions: Response time tails Response time ? Problem: Practitioners care about tail of response time. Analysis only for mean ? ? . Analyze tail? Schedule to optimize tail? Current theory: Optimal worst-case max. response time: FCFS. Asymptotic tail analysis. Conjecture: FCFS optimal for light-tailed sizes. My work and future approach: 1. Overturned conjecture with Nudge [GYSH, SIGMETRICS 21]. 2. Intermediate tails? 3. Multiserver tails? 30 Northwestern University - Isaac Grosof

31. Future directions: Scheduling with Estimates Problem: Don t know sizes Current solution: Non-size-based scheduling (FCFS). Schedule based on age & size distribution: Gittins Index [Gittins & Nash 77]. My approach: 1. Bring Gittins to multiserver world: Gittins-k [SGH, 21].ServerFilling-Gittins. 2. Distribution-robust scheduling: SRPT-B [SGM, 22]. 3. Confidence bounds from learning theory 4. New theory integrating scheduling + learning. 31 Northwestern University - Isaac Grosof

32. Broader Scope: My Papers SRPT-k, Performance 18, GSH Guardrails, SIGMETRICS 19, GSH M-Gittins-k, Performance 20, SGH Gittins-k, SIGMETRICS 21, SGH Nudge, SIGMETRICS 21, GYSH SRPT-B, ITCS 22, SGM ServerFilling-FCFS, Queueing Systems 22, GHS ServerFilling-SRPT, SIGMETRICS 23, GSHS +4 non-queueing-theory papers 32 Northwestern University - Isaac Grosof

33. Conclusion: Multiserver Analysis & Optimality One server/job: First optimality result: SRPT-k Multiple servers/job: First analysis & first optimality: ServerFilling-SRPT 2 SRPT-k ServerFilling-SRPT 4 2 4 2 1 1 33 Northwestern University - Isaac Grosof

34. 34 Northwestern University - Isaac Grosof

35. Key tool: Size & Work 2 4 4 2 2 1 1 Size: Duration * server need Ex: 4hrs * 2CPUs = 8 CPU hours Work ?: Total remaining size of all jobs Normalize: Work completes at rate 1 if all servers busy. 35 Northwestern University - Isaac Grosof

36. Bound mean work ?[???] Key idea: In steady state, expected rate of increases matches expected rate of decreases. Arrivals and rate-1 work same as FCFS. Only lower-rate work differs. ? ???= E WFCFS 1+ k jobs work For reasonable S (e.g. phase type), ? ???= ? ????? 1+ ??(1) ? ???2= 0 Work at rate 1 Job arrives Work ??? Too few jobs: lower rate Time Formalize using ? 36 Northwestern University - Isaac Grosof

37. ServerFilling Analysis Lemma: ? ???= ? ????? 1+ ??(1) Theorem: ? ???= ? ????? 1+ ??(1) First analysis for any multiserver-job scheduling policy! Work at rate 1 Job arrives Work ??? Too few jobs: lower rate Time 37 Northwestern University - Isaac Grosof

38. Multiserver-Job Optimality ServerFilling: based on FCFS. For optimality: ServerFilling-SRPT. M Least remaining size order k=8 4 2 1 2 1 1 1 1 1 2 2 1 Optimal Scheduling in the Multiserver-job Model under Heavy Traffic . Grosof, Scully, Harchol-Balter, Scheller-Wolf. SIGMETRICS 23. 38 Northwestern University - Isaac Grosof