Scheduling Parallel DAG Jobs to Minimize the Average Flow Time K. Agrawal, J. Li, K. Lu, B. Moseley

2 Client-Server Scheduling Clients send jobs to the server Server processes jobs and provides service guarantees Jobs arrive over time – online Jobs can be preempted

3 Service Guarantees  Flow time – difference between arrival and completion of a job  Common objectives in online scheduling:  Average/Sum Flow Time  Maximum Flow Time  L k Norms of Flow Time ArrivalCompletion Time Job's Flow Time

4 Many Server Models Sequential Jobs Single machine Multiple machine Identical Machines Related Machines,... Parallelizable Jobs on Identical Machines Speed-up Curves model Directed Acyclic Graph (DAG) Model

5 Parallelism Models  Speed-Up Curves Jobs have total work W divided into phases Each phase has work Phases are processed sequentially Processing rate function Γ(m) Function of number of processors given Function is usually positive sub-linear Function can be different depending on the phase the job is currently in. A Job's Phases

6 Parallelism Models  Directed-Acyclic Graph (DAG) Model  Jobs have work which correspond to a DAG Each job is modeled as a DAG Job completed when last node of its DAG is completed  Processing rate depends on the number of nodes being worked on

7 Directed Acyclic Graph Model of Parallelism Nodes represent computation Arrows represent dependencies Important parameters – Critical path length – Total work

8 Parallelism Model Differences Results do not transfer directly between models Jobs are difficult to relate:

9 Notation n – number of jobs m – total number of machines W i – total work of job i C i – critical path length of job i W(t) – work remaining at time t

10 Resource Augmentation  Strong lower bounds exist  Speed augmentation [Kalyanasundaram, Pruhs 95]  Algorithm is given processors which are faster than the optimal scheduler  s-speed c-competitive algorithms Scalable algorithms are (1+ε) speed O(1) competitive for fixed ε>0

11 Common Scheduling Algorithms Round Robin Latest Arrival Processor Sharing (LAPS) Greedy Shortest Job First Shortest Remaining Processing Time

12 Previous Work  Online on Average Flow  Sequential Jobs  LAPS, SJF, SRPT, all scalable [Torng, McCullough; Edmonds, Pruhs]  Lower bound (without resource augmentation) of Ώ (min(log P,log(n/m)) [Leonardi, Raz]  Parallel Jobs  LAPS is scalable in the speed-up curves model [Edmonds, Pruhs]  DAG Model?

13 Online Study of Models  DAG model  Well-studied offline  Only studied recently online  Naturally captures programs generated by languages and libraries such as Cilk, Cilk Plus, Intel TBB, OpenMP.  Microsoft researching  Cyber-Physical-Systems (Real-Time) community excited (Outstanding paper award ECTRS 2013, Best-Student-Paper Award RTSS 2011)

14 Results  First results for average flow in DAG model  LAPS is (1+ε) speed O(1) competitive, for fixed ε>0 [this work]  Greedy (Shortest Work First) is (2+ε) speed O(1) competitive, for fixed ε>0 [this work]  Algorithms are online and oblivious to DAG structure  Both require knowledge of the ready nodes for each job

15 Algorithm Analysis Latest Arrival Processor Sharing (LAPS) Potential function analysis Shortest Work First (SWF) Potential Function analysis for fractional flow time Conversion form fractional to integral flow

16 Fractional Flow Technique Fractional flow Integral flow: Job costs 1 per unit time step Fractional flow: Job costs W A (t)/W per unit time step Alg_Int ← Alg_Frac ← Opt Conversion argument (at a high level): If fractional cost is large, charge integral to fractional After some some point, fractional cost is small Given more speed to the algorithm, this part is finished JobTailJobTail Speed

17 Fractional Flow Problem Problem with DAG jobs

18 Remarks about SWF Analysis New charging scheme (at a high level) *Assume the faster algorithm completes more total aggregate work Charge jobs integral flow that can be bounded fractional flow directly Other jobs only charge after they wait much longer than their fractional flow time in the slower schedule Otherwise if the fast SJF completes more work than the slower then there is a mapping between alive jobs fractional jobs in the slower schedule SWF with (1+ε) more speed can fall behind SWF

19 Fast SWF Falling Behind Fast SWF falls behind in total aggregate work Fast SWF may unlock the parallel portion of a short job earlier Slow SWF may pack the jobs better Examples in paper With (2+ε) speed the faster algorithm completes more aggregate work

20 Remarks Two models are different A greedy algorithm (SWF) works in the DAG model – Practical method to schedule DAG jobs – No known greedy algorithm in speed-up curves DAG structure gives new challenges – standard techniques do not extend

21 Open Questions Other metrics in the DAG model, Max Flow, etc. A scalable greedy algorithm?

22 Thank You! Questions?