University Dortmund. Robotics Research Institute Information Technology. Job Scheduling. Uwe Schwiegelshohn. EPIT 2007, June 5 Ordonnancement

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1 University Dortmund Robotics Research Institute Information Technology Job Scheduling Uwe Schwiegelshohn EPIT 2007, June 5 Ordonnancement

2 ontent of the Lecture What is ob scheduling? Single machine problems and results Makespan problems on parallel machines Utilization problems on parallel machines ompletion time problems on parallel machines Exemplary workload problem 2

3 Examples of Job Scheduling Processor scheduling Jobs are executed on a PU in a multitasking operating system. Users submit obs to web servers and receive results after some time. Users submit batch computing obs to a parallel processor. Bandwidth scheduling Users call other persons and need bandwidth for some period of time. Airport gate scheduling Airlines require gates for their flights at an airport. Repair crew scheduling ustomer request the repair of their devices. 3

4 Job Properties Independent obs No known precedence constraints Difference to task scheduling Atomic obs No ob stages Difference to ob shop scheduling Batch obs No deadlines or due dates Difference to deadline scheduling p processing time of ob r release date of ob earliest starting time w m weight of ob importance of the ob size of ob parallelism of the ob 4

5 Machine Environments : single machine Many ob scheduling problems are easy. P m : m parallel identical machines Every ob requires the same processing time on each machine. Use of machine eligibility constraints M if ob can only be executed on a subset of machines Airport gate scheduling: wide and narrow body airplanes Q m : m uniformly related machines The machines have different speeds v i that are valid for all obs. In deterministic scheduling, results for P m and Q m are related. In online scheduling, there are significant differences between P m and Q m. R m : m unrelated machines Each ob has a different processing time on each machine. 5

6 Restrictions and onstraints Release dates r Parallelism m Fixed parallelism: m machines must be available during the whole processing of the ob. Malleable obs: The number of allocated machines can change before or during the processing of the ob. Preemption The processing of a ob can be interrupted and continued on another machine. Gang scheduling: The processing of a ob must be continued on the same machines. Machine eligibility constraints M Breakdown of machines m(t): time dependent availability rarely discussed in the literature 6

7 Obective Functions ompletion time of ob : Owner oriented: Makespan: max =max (,..., n ) completion time of the last ob in the system Utilization U t : Average ratio of busy machines to all machines in the interval (0,t] for some time t. User oriented: Total completion time: Σ Total weighted completion time: Σ w Total weighted waiting time: Σ w ( p r ) = Σ w Σ w (p +r ) Total weighted flow time: Σ w ( r ) = Σ w Σ w r Regular obective functions: non decreasing in,..., n const. const. 7

8 Workload lassification Deterministic scheduling problems All problem parameters are available at time 0. Optimal algorithms, Simple individual approximation algorithms Polynomial time approximation schemes Online scheduling problems Parameters of ob are unknown until r (submission over time). p is unknown (nonclairvoyant scheduling). ompetitive analysis Stochastic scheduling Known distribution of ob parameters Randomized algorithms Workload based scheduling An algorithm is parameterized to achieve a good solution for a given workload. 8

9 Nondelay (Greedy) Schedule No machine is kept idle while a ob is waiting for processing. An optimal schedule need not be nondelay! Example: Σ w Nondelay schedule obs 2 p 3 r 0 w 2 2 Σ w = Optimal schedule 2 Σ w =

10 omplexity Hierarchy Some problems are special cases of other problems: Notation: α β γ (reduces to) α 2 β 2 γ 2 Examples: Σ Σ w P m Σ w P m m Σ w R m Q m Σw r w m prmp M brkdwn P m Σ

11 ontent of the Lecture What is ob scheduling? Single machine problems and results Makespan problems on parallel machines Utilization problems on parallel machines ompletion time problems on parallel machines Exemplary workload problem

12 Σ w Σ w is easy and can be solved by sorting all obs in decreasing Smith order w /p (weighted shortest processing time first (WSPT) rule, Smith, 956). Nondelay schedule Proof by contradiction and localization: If the WSPT rule is violated then it is violated by a pair of neighboring task h and k. S : Σ w =...+ w h (t+p h ) + w k (t + p h + p k ) t h k k h S -S 2 : w k p h w h p k > 0 w k /p k > w h /p h S 2 : Σ w =...+ w k (t+p k ) + w h (t + p k + p h ) 2

13 Other Single Machine Problems Every nondelay schedule has optimal makespan and optimal utilization for any interval starting at time 0. WSPT requires knowledge of the processing times No direct application to nonclairvoyant scheduling prmp Σ is easy. The online nonclairvoyant version (Round Robin) has a competitive factor of 2-2/(n+) (Motwani, Phillips, Torng,994). r,prmp Σ is easy. The online, clairvoyant version is easy. r Σ is strongly NP hard. r,prmp Σ w is strongly NP hard. The WSRPT (remaining processing time) rule is not optimal. 3

14 4 Optimal versus Approximation r,prmp Σ w (-r) and r,prmp Σ w Same optimal solution Larger approximation factor for r,prmp Σ w ( -r ). No constant approximation factor for the total flowtime obective (Kellerer, Tautenhahn, Wöginger, 999) + = ) r (OPT) ( w r w (OPT) w (S) w (OPT) w (S) w = ) r (OPT) ( w ) r (S) ( w = + = ) r (OPT) ( w r w (OPT) w (S) w ) r (OPT) ( w (OPT) w (S) w

15 Approximation Algorithms r Σ Approximation factor e/(e-)=.58 (hekuri, Motwani, Nataraan, Stein, 200) lairvoyant online scheduling: competitive factor 2 (Hoogeveen, Vestens,996) r Σ w Approximation factor.6853 (Goemans, Queyranne, Schulz, Skutella, Wang, 2002) lairvoyant online scheduling: competitive factor 2 (Anderson, Potts, 2004) r,prmp Σ w Approximation factor.3333, Randomized online algorithm with the competitive factor.3333 WSPT online algorithm with competitive factor 2 (all results: Schulz, Skutella, 2002) 5

17 P m and Makespan with m = A scheduling problem for parallel machines consists of 2 steps: Allocation of obs to machines Generating a sequence of the obs on a machine A minimal makespan represents a balanced load on the machines if no single ob dominates the schedule. max (OPT) max max { p }, m p Preemption may improve a schedule even if all obs are released at the same time. max { p }, (OPT) = max max p m Optimal schedules for parallel identical machines are nondelay. 7

18 P m max P m max is strongly NP-hard (Garey, Johnson 979). Approximation algorithm: Longest processing time first (LPT) rule (Graham, 966) Whenever a machine is free, the longest ob among those not yet processed is put on this machine. Tight approximation factor: max max (LPT) (OPT) 4 3 3m The optimal schedule max (OPT) is not necessarily known but a simple lower bound can be used: (OPT) max p m = n 8

19 LPT Proof () If the claim is not true, then there is a counterexample with the smallest number n of obs. The shortest ob n in this counterexample is the last ob to start processing (LPT) and the last ob to finish processing. If n is not the last ob to finish processing then deletion of n does not change max (LPT) while max (OPT) cannot increase. A counter example with n obs Under LPT, ob n starts at time max (LPT)-p n. In time interval [0, max (LPT) p n ], all machines are busy. max (LPT) p n m n = p 9

20 LPT Proof (2) 4 3 max 3m < (LPT) (LPT) (OPT) p ( ) m (OPT) n max = n + + max p n + m max n = p = p n ( m max n p m (OPT) ) + m p ( m) (OPT) max n = p max(opt) < 3p n At most two obs are scheduled on each machine. For such a problem, LPT is optimal. 20

21 A Worst ase Example for LPT obs p parallel machines: P4 max max (OPT) = 2 =7+5 = 6+6 = max (LPT) = 5 = +4=(4/3 -/(3 4))

22 List Scheduling LPT requires knowledge of the processing times. No direct application to nonclairvoyant scheduling Arbitrary nondelay schedule (List Scheduling, Graham, 966) Tight approximation factor: max max (LIST) (OPT) 2 m 6 6 max (LIST)= max (OPT)=6 22

23 Online Transformation Let A be an algorithm for a ob scheduling problem without release dates and with max (A) (OPT) max k Then there is an algorithm A for the corresponding online ob scheduling problem with max max (A') (OPT) 2k (Shmoys, Wein, Williamson, 995) 23

24 Transformation Proof S 0 : Jobs available at time 0=F - =F -2 F 0 = max (A,S 0 ) S i+ : Jobs released in (F i-,f i ] F i = max (A,S i ) such that no ob from S i starts before F i-. Assume that all obs in S i are released at time F i-2 max (OPT) cannot increase while max (A ) remains unchanged. Proof F i-2 + F i F i- k max (A,S i ) = k max (A') F i- F i-2 F i-3 + F i- F i-2 k max (A,S i- ) < k max (A') F i < 2k max (A') 24

25 List Scheduling Extensions The List scheduling bound 2-/m also applies to P m r max (Hall, Shmoys, 989). Online extension of List scheduling to parallel obs: No machine is kept idle while there is at least one ob waiting and there are enough machines idle to start this ob (nondelay). The List scheduling bound 2-/m also applies to P m m max (Feldmann, Sgall, Teng, 994). The List scheduling bound 2-/m also applies to P m m,r max (Naroska, Schwiegelshohn, 2002). 2-/m is a competitive factor for the corresponding online nonclairvoyant scheduling problem. Proof by induction on the number of different release dates 25

26 P m m max Proof The bound holds if during the whole schedule there is no interval with at least m/2 idle machines. max m 2m - (OPT) max m (S) = m 2 The sum of machines used in any two intervals is larger than m unless the obs executed in one interval are a subset of the obs executed in the other interval. m p max m + 2m (S) max (S) max (S) { p } max 2 m p, 2 max m m 26

27 Makespan with Preemptions P m prmp max is easy. Transformation of a nonpreemptive single machine schedule in a preemptive parallel schedule (McNaughton, 959) The single machine schedule is split into at most m schedules of length max (OPT). Each schedule is executed on a different machine. There are at most m- preemptions. P m r, prmp max is easy. Longest remaining processing time algorithm. lairvoyant online scheduling ompetitive factor for allocation as late as possible. ompetitive factor e/(e-)=.58 for allocation of machine slots at submission time (hen, van Vliet, Wöginger, 995) Nonclairvoyant online scheduling: same competitive factor 2-/m as for the nonpreemptive case (Shmoys, Wein Williamson, 995) 27

29 Utilization Utilization U t is closely related to the makespan max if t= max. In online ob scheduling problems, there is no last submitted ob. U t with t being the actual time is better suited than the makespan obective. P m r U t Nonclairvoyant online scheduling: tight competitive factor for any nondelay schedule.3333 (Hussein, Schwiegelshohn, 2006) Proof by induction on the different release dates. 2 2 U 2 (LIST)=0.75 U 2 (OPT)= 29

30 Utilization Proof () Transformation of the ob system Reduction of the release dates time t 2 Interval without idle machines t machines 30

31 Utilization Proof (2) Transformation of the ob system Splitting of obs The system only contains short and long obs. All long obs start at the end of an interval. time Interval without idle machines machines 3

32 Utilization Proof (3) Transformation of the ob system Modification of obs with earlier release dates Nondelay schedule time Optimal schedule machines Interval without idle machines machines 32

33 Utilization Proof (4) If all long obs of a transformed ob system start at their release date, then the utilization is maximal for all t and the equal priority completion time is minimal. time long obs long obs from earlier release dates short obs machines 33

34 Utilization Proof (5) t σ t σ τ r r t r k τ k Nondelay schedule S Optimal schedule t σ t σ τ r r t k r τ r Nondelay schedule S Optimal schedule 34

35 P m r,m U t Parallel obs may cause intermediate idle time even if all obs are released at time 0. Nonclairvoyant online scheduling: ompetitive factor m in the worst case ompetitive factor 2 if the actual time >> max{p } Jobs p +ε +ε +ε +ε 5 r m U 5 (LIST)= ε U 5 (OPT)= 35

36 P m r,m,prmp U t Here, preemption of parallel obs is based on gang scheduling. All allocated machines concurrently start, interrupt, resume, and complete the execution of a parallel ob. There is no migration or change of parallelism. Nonclairvoyant online scheduling: competitive factor 4 (Schwiegelshohn, Yahyapour, 2000) U 3 (A)=7/ U 3 (OPT)=4/5 36

38 P m Σ P m Σ is easy. Shortest processing time (SPT) (onway, Maxwell, Miller, 967) Single machine proof: Σ =n p () + (n-) p (2) + 2 p (n-) + p (n) p () p (2) p (3)... p (n-) p (n) must hold for an optimal schedule. Parallel identical machines proof: Dummy obs with processing time 0 are added until n is a multiple of m. The sum of the completion time has n additive terms with one coefficient each: m coefficients with value n/m m coefficients with value n/m : m coefficients with value If there is one coefficient h>n/m then there must be a coefficient k<n/m. Then we replace h with k+ and obtain a smaller Σ. P m prmp Σ is easy (Shortest remaining processing time). 38

39 P m Σw P m Σw is strongly NP-hard. The WSPT algorithm has a tight approximation factor of.207 (Kawaguchi, Kyan, 986) It is sufficient to consider instances where all obs have the same ratio w /p. Proof by induction on the number of different ratios. J is the set of all obs with the largest ratio in an instance I. The weights of all obs in J are multiplied by a positive factor ε < such that those obs now have the second largest ratio. This produces instance I. The WSPT order is still valid. The WSPT schedule remains unchanged. The optimal schedule may change. 39

40 Different Ratios Induction Proof Σw (WSPT,I ) λ Σw (OPT,I ) (induction assumption) x: contribution of all obs in J to Σw (WSPT,I) y: contribution of all obs not in J to Σw (WSPT,I) x : contribution of all obs in J to Σw (OPT,I) y : contribution of all obs not in J to Σw (OPT,I) x λ x (induction assumption) Σw (WSPT,I)= x+y and Σw (WSPT,I )=ε x+y, Σw (OPT,I)=x +y and Σw (OPT,I ) ε x +y y λ y Σw (WSPT,I) λ Σw (OPT,I) y>λ y λ x y>x λ y x /y >x/y x y-xy >0 x y-xy >ε(x y-xy ) Σw (WSPT,I ) Σw (OPT,I) =(ε x+y)(x +y )>(ε x +y )(x+y) Σw (OPT,I ) Σw (WSPT,I) Σw (WSPT,I) λ Σw (OPT,I) Assumption: w =p holds for all obs. 40

41 WSPT Proof () Transformation of the ob system Splitting of ob into obs and 2. The system only contains short and long obs. All long obs start at the end of busy interval in the list schedule. = = (p p w p 2 + p (S') 2 ) w (S') p (S) = p ( (S') p (S') p 2 ) p 2 (S) p 2 (S') = 2 (S) = time machines w w (S) (OPT) w w w w (S') p (OPT') p (S') (OPT') p 2 p 2 4

42 WSPT Proof (2) Single machine without intermediate idle time w =p holds for all obs. w (S)=w (OPT)= 0.5((p ) 2 +p 2 ) Proof by induction on the number of obs w (S) = = 2 2 ( ) ) 2 2 p + p + p ' ( p ' + p ) ( ) ) ( + ) p 2p ' p p ' p p ' 2 = 42

43 WSPT Proof (3) Equalization of the long obs Assumption of a continuous model (fraction of machines) k long obs with different processing times are transformed into n(k) obs with the same processing time p(k) such that p =n(k) p(k) and p 2 =n(k) (p(k)) 2 hold. p(k)= p 2 / p and n(k)= (p ) 2 / p 2 Then we have k n(k) for reasons of convexity. time machines machines 43

44 WSPT Proof (4) Modification of the ob system Partitioning of the long obs into two groups Equalization of the both groups separately The maximum completion time of the small obs decreases due to the large rectangle. The obs of the small rectangle are rearranged. New equalization of the large rectangle Determination of the size of the large rectangle time machines machines 44

45 Release Dates P m r Σ Approximation factor 2 lairvoyant, randomized online scheduling: competitive factor 2 P m r,prmp Σ Approximation factor 2 lairvoyant, randomized online scheduling: competitive factor 2 P m r Σw Approximation factor 2 lairvoyant, randomized online scheduling: competitive factor 2 P m r,prmp Σw Approximation factor 2 lairvoyant, randomized online scheduling: competitive factor 2 (all results Schulz, Skutella, 2002) 45

46 Parallel Jobs P m m,prmp Σw Use of gang scheduling without any task migration Approximation factor 2.37 (Schwiegelshohn, 2004) P m m,prmp Σ Nonclairvoyant approximation factor 2-2/(n+) if all obs are malleable with linear speedup (Deng, Gu, Brecht, Lu, 2000). P m m Σw Approximation factor 7. (Schwiegelshohn, 2004) Approximation factor 2 if m 0.5m holds for all obs (Turek et al., 994) P m m Σ Approximation factor 2 if the obs are malleable without superlinear speedup (Turek et al., 994) 46

47 Online Problems P m m,r,prmp Σw Nonclairvoyant online scheduling with gang scheduling and w =m p : competitive factor (Schwiegelshohn, Yahyapour, 2000) w =m p guarantees that no ob is preferred over another ob regardless of its resource consumption as all obs have the same (extended) Smith ratio. All obs are started in order of their arrival (FFS). Any ob started after a ob can increase the flow time -r by at most a factor of 2 lairvoyant online scheduling with malleable obs and linear speedup: ompetitive factor 2+ε for a deterministic algorithm ompetitive factor 8.67 for a randomized algorithm (both results hakrabarti et al.,996) 47

49 MPP Problem Machine model Massively parallel processor (MPP): m parallel identical machines Job model Multiple independent users Nonclairvoyant (unknown processing time p ) with estimates Online (submission over time r ) Fixed degree of parallelism m during the whole processing No preemption Obective Machine utilization Average weighted response time (AWRT): p m ( -r ) Based on user groups 49

50 Algorithmic Approach Reordering of the waiting queue Parameters of obs in the waiting queue Actual time Scheduling situations: weekdays daytime (8am 6pm), weekdays nighttime (6pm 8am), weekends Selected sorting criteria Selected obective onsideration of 2 user groups: 0 AWRT + 4 AWRT 2 Parameter training with Evolution Strategies Recorded workloads and simulations Workload scaling for comparison Waiting queue 50

51 Workloads and User Groups User Group R u /R > 8% 2 8 % 2 % 0. % < 0. % User group definition Identifier T KTH LANL SDS 00 SDS 95 SDS 96 Machine SP2 SP2 M-5 SP2 SP2 SP2 Period 06/26/96 05/3/97 09/23/96 08/29/97 04/0/94 09/24/96 04/28/98 04/30/00 2/29/94 2/30/95 Processors (m) /27/95 2/3/96 Jobs (n) Workload scaling 5

52 Sorting riteria ( Job) f f f 2 ( Job) 3 ( Job) = = = Groups i= Groups i= Groups i= wi Ki wi Ki wi Ki + a waittime requestedtime requestedtime + b processors requestedtime + a waittime + b processors + a waittime requestedtime processors f 4 ( Job) = Groups i= w i ( K + a waittime + b requestedtime processors) i Training of parameters w i, K i, a, b with Evolution Strategies 52

53 T Training and T Workload 5,00 0,00 AWRT Improvements in % 5,00 0,00-5,00-0,00 AWRT AWRT 2 AWRT 3 AWRT 4 AWRT 5-5,00-20,00 Method AWRT AWRT 2 AWRT 3 AWRT 4 AWRT 5 UTIL GREEDY s s s s s % EASY s s s s s % 53

54 T Training and All Workloads 20,00 0,00 Obective Improvements in % 0,00-0,00-20,00-30,00-40,00 T KTH LANL SDS 00 SDS 95 SDS 96-50,00 Some workloads are similar (T, LANL). Some workloads are significantly different (T, KTH). 54

55 Results in T Paretofront AWRT 2 in Seconds PF EASY GREEDY T opt GREEDY ALL opt AWRT in Seconds 55

56 Results in SDS 95 Paretofront AWRT 2 in Seconds PF EASY GREEDY T opt GREEDY ALL opt AWRT in Seconds 56

57 onclusion Most deterministic ob scheduling problems are NP hard. Approximation algorithms Polynomial time approximation schemes Simple algorithms omplete problem knowledge is rare in practice. Online algorithms ompetitive analysis Stochastic scheduling Randomized algorithms hallenges Partial information Recorded workloads User estimates Scheduling obectives and constraints 57