Jeff Edmonds York University Kirk Pruhs University of Pittsburgh

1. Every Deterministic Nonclairvoyant Scheduler has a Suboptimal Load Threshold 21 pages written already Open Problem Jeff Edmonds York University Kirk Pruhs University of Pittsburgh And You

2. Model • Multiprocessor (fractional, preemptive) • Jobs: • Arrival time, • work, • speedup function parallelizable – sequential. • Flow time =  completion time. • Online, nonclairvoyant • Competitive analysis, resource augmentation.

3. History [KP] Parallel Jobs:  Flow(BAL1+)/Flow(Opt1) ≤ 1/ [E] Parallel-Sequential Jobs:  Flow(Equi2+)/Flow(Opt1) ≤ 1/  Flow(LAPS1++)/Flow(Opt1) ≤ 1/ [EP]

4. Performance vs Speed ≈ 1/Load Small β can handle almost full load  Flow(LAPS1++)/Flow(Opt1) ≤ 1/ Equi (β=1) has the best performance, but it only can handle half load. but its performance degrades with 1/ L

5. Are we done? ’ ’  LAPS  Flow(LAPS1++)/Flow(Opt1) ≤ 1/ Desired result: Alg  Flow(Alg1+)/Flow(Opt1) ≤ 1/2 Obtained [EP]:  Alg Flow(Alg1+)/Flow(Opt1) ≤ 1/2 Open problem Alg  Flow(Alg1+)/Flow(Opt1) ≥ (1) Every Deterministic Nonclairvoyant Scheduler has a Suboptimal Load Threshold

6. Challenge ’ ’ Open problem Alg  Flow(Alg1+)/Flow(Opt1) ≥ (1) Easy if for each time stepwe can see what the alg does and set the speed s=1+. But if speed s=1+ is fixedand the algorithm learns it then the algorithm can run LAPS/2 and be competitive Keeping the speed a secret is hard

7. Lower Bound ti Input Opt Alg Opt ignores extra jobs& competes stream Flow = ∑ 2ti Alg attempts all& completes none Flow = ∑ (1+i) ti

8. Lower Bound l A g I t n p u O t p Alg specifies processor allocation for each jobwhen nt jobs alive Compute work wi completed on each job Eg Equi or Lapsβ Arbitrary Adv gives promises no job completes Adv gives work wito job so itdoes not complete Non trivial algebra Brouwer's fixed point theorem Time ti between jobs = wi so Opt can complete as arrive Time ti between jobs Compute competitive ratio