Resource augmentation and on-line scheduling on multiprocessors Phillips, Stein, Torng, and Wein. Optimal time-critical scheduling via resource augmentation. STOC (1997). Algorithmica (to appear).

Background: on-line algorithms qOptimization problems: given problem instance I, algorithm A obtains a value val A (I) -- goal is to maximize this value qOn-line algorithms vs an optimal off-line/ clairvoyant algorithm (OPT) qCompetitive ratio of on-line algorithm A: min all I ( val A (I)/ val OPT (I) ) qGoal: Design an on-line algorithm with largest competitive ratio

Background: hard-real-time scheduling qThe on-line problem: –Instance I = {J 1, J 2,..., J n } of jobs –Each job J j = (r j, p j, d j ) arrives at instant r i needs to execute for p i units... by a deadline at instant d i –Job J i is revealed at instant r i all deadlines must be met! qDifficult to formulate as an optimization problem -- all deadlines must be met! qIn uniprocessor systems, we dodged this issue –EDF/ LL are optimal algorithms (always meet all deadlines) –EDF/ LL are on-line algorithms... –... with competitive ratio one

Hard-real-time scheduling: multiprocessors qNo optimal (in the EDF/LL sense) on-line algorithm exists qMust still meet all deadlines...So, give the on-line algorithm extra resources (more/ faster processors) qThis paper asks: how much extra resources do EDF/ LL need, in order to meet all deadlines for sets of jobs known to be feasible on m processors? qThe answers: – EDF/ LL meet all deadlines if processors are (2 - 1/m) times as fast –No on-line algorithm can meet all deadlines if processors are < 1.2 times as fast –EDF cannot always meet all deadlines if processors are (2 - 1/m -  ) times as fast, for any  > 0

Why we care qOur (RTS) task systems: –usually pre-specified (e.g., periodic tasks/ sporadic tasks) –“on-line”ness usually not an issue exception: overload scheduling (later) qWe’ll do feasibility analysis (does a schedule exist?) qIf feasible, we’ll use the results in this paper –choose an algorithm (usually, EDF) –overallocate resources as mandated by these results –sleep well, knowing that the system performs as expected qWhy choose feasibility analysis (versus schedulability analysis with chosen algorithm)? –provably competitive performance translates to approximation guarantees

Model and definitions Instance I = {J 1, J 2,..., J n } of jobs Each job J j = (r j, p j, d j ) arrives at instant r i needs to execute for p i units... by a deadline at instant d i If I is feasible on m processors, an s-speed on-line algorithm will meet all deadlines on m processors each s times as fast (Thus, EDF is a (2 - 1/m)-speed algorithm)

Digression: An example of how we’d use these results

Scheduling periodic tasks - taxonomy Priorities task-level static job-level static dynamic Migration task-level fixed job-level fixed migratory Baker/ Oh (RTS98) Pfair scheduling Andersson/ Jonsson bin-packing + LL (no advantage) bin-packing + EDF RM EDF LL/ Pfair Periodic task system  = {  1,  2,...,  n };  i = (T i, C i ),

Remember this? (last class) RM-US(1/4) –all tasks  i with (T i / C i > 1/4) have highest priorities –for the remaining tasks, rate-monotonic priorities Lemma: Any task system satisfying [ (SUM  j :  j  : C i /T i )  m/4] and [ (ALL  j :  j  : C i /T i )  1/4] is successfully scheduled using RM-US(1/4) Theorem: Any task system satisfying [ (SUM  j :  j  : C i /T i )  m/4] is successfully scheduled using RM-US(1/4)

A new (job-level static priority) scheduling algorithm EDF-US(1/2): –If C i /T i  0.5, then jobs of  i get EDF priority –If C i /T i > 0.5, then jobs of  i get highest priority (EDF implementation: set deadline to -  ) Lemma: Any task system satisfying [ (SUM  j :  j  : C i /T i )  m/2] and [ (ALL  j :  j  : C i /T i )  1/2] is successfully scheduled using EDF-US(1/2) Theorem: Any task system satisfying [ (SUM  j :  j  : C i /T i )  m/2] is successfully scheduled using EDF-US(1/2)

Scheduling periodic tasks w/ migration Priorities task-level static job-level static dynamic Migration task-level fixed job-level fixed migratory Baker/ Oh (RTS98) Pfair scheduling Andersson/ Jonsson bin-packing + LL (no advantage) bin-packing + EDF RM-US(1/4) EDF-US(1/4) Pfair 25% 50% 100%

Back to the results in this paper... (faster processors)

The big insight Definitions: –A(j,t) denotes amount of execution of job j by Algorithm A until time t –A(I,t) = [SUM: j  I: A(j,t)] The crucial question: Let A be any “busy” (work-conserving) scheduling algorithm executing on m processors of speed   1. What is the smallest  such that at all times t, A(I, t)  A’(I,t) for any other algorithm A’ executing on m speed-1 processors? Lemma 2.6:  turns out to be (2 - 1/m) Use Lemma 2.6, and an individual algorithm’s scheduling rules, to draw conclusions regarding these algorithms

The oh-so-important lemma 2.6 Proof: by contradiction Suppose there are time instants at which this is not true Let  = { i |  t  A(I,t) < A’(I,t) and A(i,t) < A’(i,t) } Let j be the job with the earliest release time r j in  Let t o be the earliest time instant at which A(I,t o ) < A’(I,t o ) Eq (1) A(j,t o ) < A’(j,t o ) Eq (2) Lemma: Let I be an input instance, t  0 any time-instant. For any busy algorithm A using (2-1/m)-speed machines, A(I,t)  A’(I, t) for any algorithm A’ using 1-speed machines.

EDF is a (2 - 1/m)-speed algorithm Instance I = {J 1, J 2,..., J n }; job J j = (r j, p j, d j ) is feasible on m procs Wlog, assume that d i  d i+1 for all i Let I k = {J 1, J 2,..., J k } Proof Proof: Induction on k Base: EDF on m (1 - 2/m)-speed procs meets all deadlines for I 1,.., I m IH: EDF on m (1 - 2/m)-speed procs meets all deadlines for I 1,.., I k We’re considering I k+1. –Let Q k+1  I k+1 denote the jobs in I k+1 with deadlines at d k+1 –(I k+1 \ Q k+1 ) is I q for some q  k –By IH, EDF on m (1 - 2/m)-speed procs meets all deadlines for I q –BY definition of EDF, EDF(I k+1 ) is identical to EDF(I q ) on jobs of I q ; -- thus, all deadlines in I q are met in EDF(I k+1 ) –By Lemma 2.6, EDF(I k+1,d k+1 )  OPT(I k+1, d k+1 ) –Since OPT meets all deadlines at d k+1, so must EDF on m (1 - 2/m)-speed procs