Online Preemptive Scheduling with Immediate Decision or Notification and Penalties

1. Online Preemptive Scheduling with Immediate Decision or Notification and Penalties Stanley P. Y. Fung

2. Motivation • Preemption is not really “free” • E.g. context switch in CPU scheduling • Introduce preemption penalty (PP): each preemption introduces a penalty proportional to value of job • Related but different scenario: we want commitment to process a job • If a job cannot be processed, better to let the client know early • Introduce non-completion penalty: if an accepted job is later not finished, gives a penalty

3. Immediate Decision/Notification • How to model “commitment”: • Immediate Notification (IMM NOTIF): • Upon a job’s arrival, have to decide immediately whether to accept the job, or reject it right away • If rejected, no value (customer goes away) • If an accepted job is eventually not completed, incurs a penalty (cannot charge customer since work not done; in addition, customer is unhappy when he finds this out) • Immediate Decision (IMM DEC): • A stronger notion where in addition, an accepted job needs to be immediately allocated a time window for execution (“your delivery will take place on Friday morning”)

4. Problem Formulation • Input: a set of jobs • Parameters: release time r(j), deadline d(j), length p(j), weight w(j) • Value of a job v(j) = p(j) x w(j) • Unit length (UL) case: p(j) = 1 for all j • Unit weight (UW) case: w(j) = 1 for all j • Objective: Find a schedule with maximum total value of completed jobs, minus any penalties • Penalty model: penalty = × v(j) • Preemptive • We mainly consider the restart model • But also some on resume model • (Doesn’t matter in IMM DEC)

5. Online Algorithms • Jobs arrive online • No information about future • Decisions irrevocable • Measure of performance: competitive ratio • An online algorithm A is c-competitive if for any input instance I, OPT(I)/A(I) ≤ c • Where OPT = offline optimal algorithm, A(I) = value of A on instance I

6. Previous and Our Results (PP)

7. Previous and Our Results (PP)

8. Previous and Our Results (IMM) • Previous results: • IMM DEC + UW: an upper bound from [TL09] • No other prior result for preemptive case • Non-preemptive case considered in various papers (usually UL, and parallel machines) • Our results: • IMM DEC (or IMM NOTIF + restart) • UW: • UL: • Lower bound 4(1+) carries over to here • Lower bound with laxity

9. UW Lower Bound (PP/IMM) • Tight jobs: PP = IMM NOTIF = IMM DEC • No algorithm gives competitive ratio better than 4(1+) • Idea: (similar to [BKM+92, Woeg94]) • If switch to short job, stop • If not preempt from vi to vi+1, v0 v1 v2

10. Lower Bound (cont’d) • Lemma: if  < 4(1+), {vi} cannot be strictly increasing forever • Let vn be the last term. Then • If ONL is about to finish vn, release another job of value vn just before it finishes vn-1 vn vn

11. PP + UW: Algorithm • 4(1+)-competitive algorithm: • Let  = 2(1+)  2,  = 2( – 1)  • Job arrival: add to pool • When idle, start the pending job with earliest expiry time • While j is running, if another job k reaches its expiry time: • If v(k) >  v(j) and j has preempted other jobs before, or • If v(k) >  v(j) and j has not preempted others before • Then preempt j and start k; j never considered again • Else continue with j (and k will expire)

12. Ideas of Proof • Basic subschedules: sequence of preemption followed by one job completion • vk-1≤ vk / , vk-2 ≤ vk-1 / , …, v1 ≤ v2 /  • Value obtained = vk – (v1+…+vk-1) • Times in OPT when ONL is busy  OK (1 to 1) • Times in OPT when ONL is idle • For each (part of) such job j executed by OPT, map it to some basic subschedule in ONL vk v1

13. Ideas of Proof • When OPT starts (part of) a job j while ONL idle: • If ONL is idle at x(j) (expiry time of j), only possible if j started/completed before its expiry  j is the first job in a basic subschedule  map to it • Otherwise, x(j) falls within some basic subschedule (v1…vk)  map to it  can show total of v(j) < vk (or v1 if k=1) • Each basic subschedule is mapped by: vk v1 v1+…+vk OPT ONL v1 vk

14. IMM DEC + UW: Algorithm • At any time, the algorithm maintains the provisional schedule of accepted jobs • Execute according to provisional schedule • When a new job j arrives: • Identify a set of jobs { ji } in the provisional schedule so that their preemption create a long enough timeslot to fit the new job, and that  v(ji) < v(j)/  • If no such set exists, reject j • Else preempt those ji, schedule j in the new empty interval (as early as possible)

15. Properties of Provisional Schedule • Theorem:-competitive • Simple but useful properties of provisional schedule: • “No gap”, i.e. no idle period between planned jobs in it • “monotone”, i.e. once a time t becomes busy, won’t become idle later on • If a preemption happens, it preempts a suffix of the provisional schedule • Only linear number of potential job sets to consider (prov. sch.)

16. Ideas of Proof • Classify jobs in OPT into four sets: • C: Completed in both OPT and ONL • R: Rejected by ONL (upon arrival) • D: Discarded by ONL (preempted before getting started) • A: Aborted by ONL (preempted while execution) • Jobs in R: • Cannot start in idle time in ONL • If start in a busy period with completed jobs j1,…,jm, then v(j) <  v(ji) <(v(j1)+v(j2)) OPT A j1 j2

17. Bounding R jobs • Suppose an R-job j starts at t when ONL is idle • St[t] is empty. By monotone property, Sr(j)[t] is also empty • By no-gap property, Sr(j)[u] is all empty for u > t • Hence j could not be rejected • Suppose j starts at t when ONL is busy • Rejected due to a set X of jobs in Sr(j)[t] s.t. v(j) < v(X) • Jobs in X may be preempted but replaced by other (higher-value) jobs in the same busy period (no-gap) • Eventually goes to one of ji’s j t <(v(j1)+v(j2)) j1 j2

18. Ideas of Proof • Consider one busy period • Let V(C) = V • Jobs in D/A: • Can prove v(D) + v(A)  V/( – 1) • Jobs in R: v(R)  L + V where L = length of b.p.  v(C) + v(A)  (1+1/( – 1))V • OPT  v(C) + v(A) + v(D) + v(R) • ONL  v(C) – (v(D)+v(A)) • Competitive ratio = OPT/ONL, choose a optimised 

19. IMM DEC + UL • Same algorithm • Provisional schedule different properties: • Each preemption preempts either one job, or two including the currently executing one • Efficient identification of candidate preemption sets • Theorem: -competitive • Proof: • Similar ideas for C, A, D • Different bound for R

20. Proof for UL Case • Each job in R is “blocked” by at most 2 jobs in provisional schedule (i.e. jobs in C/A/D) • Total value  (value of R-job)/ • Distribute the values to them • Each job in C/A/D “blocks” at most two jobs in R (i.e. in OPT) • V(R)  2( v(C)+v(A)+v(D) ) < (v1+v2) OPT v1 v2 v1 v1 Prov.sch. v1 v2 v1

21. Unit Jobs • Discrete time steps, p(j) = 1 = time step size • No issue of preemption • Without immediate decision/notification: unit job scheduling problem • Motivation: buffer management in QoS switches • Previous results: • Deterministic UB 1.828, LB 1.618 • Randomized UB 1.582, LB 1.25 • Various restricted cases, adversary models etc

22. Unit Jobs + IMM NOTIF • Algorithm: • Keep a provisional schedule S • When each job j arrives: • Find the earliest slot u before d(j) in S such that w(St[u]) < w(j)/ for some . Evict this job (call it j’) and j takes its place. If no such slot, reject j • Repeat above for the evicted job j’ (but with different condition w(St[u]) < w(j’)), the subsequently evicted j’’, … • -competitive • Lower bound?

23. Unit Jobs + IMM DEC • Algorithm: • Keep a provisional schedule S • When a new job j arrives, find any slot u such that w(St[u]) < w(j)/ for some  • Replace St[u] with j • 4(1+)-competitive • () lower bounds

24. Future Work • PP + UL: gap still exists • E.g. between (4, 4.56) for restart without penalty • Better bounds for IMM DEC + UL (=UW?) • Upper/lower bounds to separate the competitive ratio of IMM DEC / IMM NOTIF / no-commitment • Algorithms here works for IMM NOTIF, but trivially • Unit jobs lower bounds • IMM DEC but limited re-scheduling / wider interval / …