Multidimensional Scheduling (Polytope Scheduling Problem) Competitive Algorithms from Competitive Equilibria

Transcription

1 Multidimensional Scheduling (Polytope Scheduling Problem) Competitive Algorithms from Competitive Equilibria Sungjin Im University of California, Merced (UC Merced) Janardhan Kulkarni (MSR) Kamesh Munagala (Duke)

2 Competitive Algorithms from Competitive Equilibria

3 Competitive Online Scheduling Algorithms for Datacenter from Competitive Equilibria

4 Speed Competitive Online Scheduling Algorithms for Datacenter Competitive Equilibria

5 Speed Competitive Online Scheduling Algorithms for Datacenter Fairness Competitive Equilibria in Economics

6 Speed Competitive Online Scheduling Algorithms for Datacenter Fairness Competitive Equilibria in Economics

7 Outline Multidimensional Scheduling (Polytope Scheduling Problem) System Desiderata Fairness Job latency: total completion time and total flow/response time Our Results Conclusions

8 Multidimensional Scheduling System has many resources CPU, memory, disk, network, Each available in limited supply Each user runs a job with a certain resource requirement At each time instant: Allocate resources among the users Rate of processing = Function of resources allocated Utility of job = Rate of processing

9 An Example Ghodsi, Zaharia, Hindman, Kondwinski, Shenker, Stoica. NSDI 2012 Two (divisible) resources 9 CPUs; 18GB RAM Denote this as Supply vector, S = (9,18) User 1 runs a job that needs r 1 = (1,4) and size 6 Needs resources x 1 r 1 to get processing rate x 1 User 2 runs a job that needs r 2 = (3,1) and size 6 If User 1 gets rate x 1 and User 2 gets rate x 2 then: x x x 1 + x 2 18

10 For simplicity Normalize supply to equalize all dimensions Supply = (9,18) Normalized Supply S = (18,18) User 1 runs a job that needs r 1 = (2, 4) and size 6 User 2 runs a job that needs r 2 = (6, 1) and size 6 If User 1 gets rate x 1 and User 2 gets rate x 2 then: 2 x x x 1 + x 2 18 If x 1 =3 and x 2 = 2, then the two jobs complete in 6/3 and 6/2 time steps, respectively.

11 Multidimensional Scheduling n jobs D dimensions/resources (1, 1, 1,, 1) Each job j has demand vector (r j1, r j2, r j3,, r jd ) size/processing requirement p j Arrives at time a j online Feasible Schedule at time t Σ j r jd x j (t) 1 for all d = 1, 2,, D, Then, each job j gets processed at a rate of x j (t) at time t

12 Extension to Polytope Constraints n jobs in system Each job j needs to be assigned a processing rate x j (t) Set {x j (t)} needs to lie within some feasibility polytope P Typically, P is a packing (downward-closed) polytope Encodes many well-studied scheduling problems: Rate is arbitrary concave function of resource allocation [Edmonds, Pruhs 02; Zahedi, Lee 13] Unrelated machine scheduling [Chadha, Garg, Kumar, Muralidhara 09] Broadcast scheduling, switch scheduling,

13 Desiderata 1: Fair Allocations The system is shared by many clients. Fairness really matters. In the single resource/dimensional case, Max-Min Fairness is widely accepted. Max-Min Fairness: water-filling 1 1 1/6 2/3

14 Desiderata 1: Fair Allocations The system is shared by many clients. Fairness really matters. In the single resource/dimensional case, Max-Min Fairness is widely accepted Max-Min Fairness: water-filling 1 1 1/6 2/3

15 Desiderata 1: Fair Allocations The system is shared by many clients. Fairness really matters. In the single resource/dimensional case, Max-Min Fairness is widely accepted Max-Min Fairness: water-filling 1 1 1/6 2/3

16 Desiderata 1: Fair Allocations [Ghodsi et al. NSDI 2012] The notion of fairness is not clear in the multidimensional setting Pareto-optimality (PO) Cannot improve everyone s rate simultaneously Envy-freeness (EF) Users should not envy the share of other users Truthfulness (IC) Users should not misreport resource requirement Sharing Incentive (SI) Users should not be happier with 1/n share of all resources

17 Two Examples of Fair Allocations Dominant Resource Fairness [Ghodsi et al. NSDI 2012] Proportional Fairness [Nash 1950; Kelly et al. 1998] There are many other allocations that generalize max-min fairness

18 Dominant Resource Fairness (DRF) CPU Memory Recall r 1 = (2, 4) and r 2 = (6, 1) S = (18,18) for CPU and RAM Dominant resource: RAM for User 1 and CPU for User 2 Equalize dominant shares: 4 x 1 = 6 x 2 Yields x 1 = 3 and x 2 = 2

19 Proportional Fairness (PF or CEEI) Maximize X log x j under the resource constraints Each job has $1 money If resource i is priced at p i then job j will maximize x j by solving: Price paid = x j p r j = $1 Find prices for resources to clear the market Recall r 1 = (2, 4) and r 2 = (6, 1) S = (18,18) for CPU and RAM Set x 1 = 4.09 and x 2 = 1.64 j

20 Fairness: DRF vs PF DRF is EF, SI, IC, but not PO A simple generalization achieves PO [Parkes, Proccacia, Shah, 12] PF is PO, EF, SI but lacks IC A simple modification achieves IC [Cole, Gnatzelis, Goel, 13] and approximately satisfies PO

21 Desiderata 2: Job Latency Completion time C j of job j satisfies: Z Cj t=a j x jt p j Total rate allocated Completion time objective: Minimize X j w j C j Flow time (or Delay): Minimize X j w j (C j a j )

22 Competitive Ratio c competitive max I Cost(A online, I) Cost(OPT offline, I) c For all inputs I

23 Job Latency: DRF vs PF [I-Kulkarni-Munagala, STOC 14, FOCS 15, JACM 18] DRF is Ω(D.5 ) competitive for total completion time Even when all jobs arrive at time 0 DRF does not distinguish between heavy and light jobs! D p D jobs For all j a j =0 p j =1 w j =1 1 job 1 job 1 job Resources in unit supply

24 DRF Allocation x =1/ p D x =1/ p D x =1/ p D D p D jobs 1 job 1 job Completion Time of DRF = O(D p D) OPT schedules singleton jobs in parallel in one step, Then the heavy jobs sequentially OPT = O(D)

25 What does PF do? x = 1 p D(1 + p D) 1 D D p D jobs x = p D 1+ p D 1 1 job 1 job Completion Time of PF O(D) = OPT

26 Our Main Results [I-Kulkarni-Munagala, STOC 14, FOCS 15, JACM 18] PF is O(1) competitive for completion time! Fair! Non-clairvoyant! Holds for any scheduling problem that can be modeled as: Allocate rates {x jt } s.t.{x jt } 2 Packing polytope P - Packing polytope: Convex; if x is in P, and any y x is also in P.

27 Our Main Results [I-Kulkarni-Munagala, STOC 14, FOCS 15, JACM 18] PF is O(1) competitive for total flow time for any monotone PSP problem (with O(1)-speed augmentation) Monotone PSP: All present jobs gets less processing rate under PF when more jobs are added E.g. single source multi-commodity network routing, utilities modelling substitutable resources

28 Conclusion & Open Question Fairness could lead to low overall job latency Our lower bound does not rule out O(1)-speed O(1)-competitive algorithms for general PSP when clairvoyance is allowed. Can one can get O(1)-competitive algorithm for total flow time with (1+eps)-speed for monotone PSP? What is the tight competitive ratio of DRF for total completion time? Is it O(D.5 ) or O(D)?