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Efficient Scheduling Algorithms for resource management
Denis Trystram
ID-IMAG
CSC, Toulouse, june 23, 2005
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General Context
Recently, there was a rapid and deep evolution of execution supports: super-computers, clusters, grid computing, global computing…
Need of efficient tools for resource management for dealing with these new systems.
This talk will investigate some of the related problems and discuss new solutions
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Parallel computing today.
Different kindsClusters, collection of clusters, grid, global computingSet of temporary unused resourcesAutonomous nodes (P2P)
Our view of grid computing (reasonable trade-off):Set of computing resources under control (no hard
authentication problems, no random addition of computers, etc.)
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Content
• Goal: to illustrate the scheduling problem in grids
• A first report of experiences in Grenoble
• Intra-clusters scheduling
• On-line
• Multi-criteria
• Inter-cluster scheduling. How to deal with more complex actual problems?
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CiGri: a regional grid
It comes from the project CIMENT whose aim was to create new meso-computing facilities (alternative to national computing centers) in the late 90ties.
Our view of grids: collection of clusters. Equipment of 5 clusters in different places around Grenoble, local autonomy, but shared experiences and tools (more than 600 connected machines!).
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CiGri
Starting in november 2002 (national programme ACI).
observatory
PhyNum
icluster
IA64
Hospital
DELL
ID
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Output
Models of applications coming form the real-life.
Synthetic workload generation
Practical tools (OAR + batch scheduling system).
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Brief example: Analysis of job characteristics by logs of Icluster
Icluster is a 225 PCs machine from HP with a rather slow hierarchical interconnection module (5x45machines) – fast Ethernet. Processors are PIII 733Mhz it was replaced in 2003 by a 104 dual-processors cluster.
Objectives of the study:Better understanding of the behaviourBuild a Workload generator
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Data (one year)
11496 submissions of all kind of jobs (interactive ones or not, mono and multi-processors). 85 users.About 10% of these jobs are coming from the system team and were removed from the analysis.
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Evolution du nombre de travaux par jour de la semaine
0100200300400500600700800900
100011001200130014001500160017001800190020002100
Lundi Mardi Mercredi Jeudi Vendredi Samedi Dimanche
Nombre de travaux
MultiBatch
MultiInter
MonoBatch
MonoInter
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Evolution du nombre de travaux par heure du jour
0
200
400
600
800
1000
1200
0h-1h 2h-3h 4h-5h 6h-7h 8h-9h10h-11h12h-13h14h-15h16h-17h18h-19h20h-21h22h-23h
Nombre de travaux
MultiBatch
MultiInter
MonoBatch
MonoInter
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Evolution du cumul de travaux par heure de la semaine
0
20
40
60
80
100
120
140
160
180
200
220
240
260
280
300
320
340
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1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 131 136 141 146 151 156 161 166
Nombre de travaux
Mo I Mo B Mu I Mu B
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Synthetic Generation of workloads
Based on these logs, we derived the probabilistic laws of arrivals.
Percentage of jobs using power of 2 (23) or multiple of ten (17) processors.
Workload characterization using Dowley’s model.
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A new national french initiative:GRID5000
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Target Applications
New execution supports created new applications (data-mining, bio-computing, coupling of codes, interactive, virtual reality, …).
Interactive computations (human in the loop), adaptive algorithms, etc..
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J1
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Users queue
time
job
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J1
users
time
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J1J2
users
time
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J1J2J3…
users
time
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representantJ1J2J3… …
…
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J1J2J3… …
…
……
…
…
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Scheduling: kind of jobs
Execution time
Number of processors
Parallelism overhead due to communications
Profitable part
moldablerigid malleable
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Basics in scheduling
Central scheduling problem
• Parameters: number and types of processors, structure of the application, criterion to optimize.
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Central Scheduling Problem
P | prec, pj | Cmax is NP-hard [Ulmann75]
Thus, we are looking for good heuristics.
Analysis using Competitive ratio r:
maximum over all instances of
The schedule is said -competitive iff
*ωω
ρ ρ ≤)(r
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Formal Definition
The problem of scheduling graph G = (V,E) weighted by function t on m processors:
(with communication delays)
Determine the pair of functions (date,proc) subject to:
•respect of precedence constraints
•Usual objective: to minimize the makespan
),())(,()()(:),( jiciprocitidatejdateEji ++≥∈∀maxC
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L
Trivial remark: allocation is important even while scheduling on identical processors…
If L is large, the problem is very hard (no approximation)
Taking into account heterogeneity
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More complicated models: LogPTentative of designing new computational models closer to the actual parallel systems [Culler et al.]:
4 parameters.
•L latency
•o overhead
•g gap
•P number of processors
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Alternative models: LogP
No overlap.
O + L + O
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Alternative models: LogP
No overlap. g
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Alternative models: LogP
No overlap.
The delay model is a LogP-system where o=g=0
g
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Need to simplify the modelThe game becomes too complicated
No way for finding good approximation algorithms (more and more negative results are available…):
Parallel tasks focus on key optimization parameters… No need of sophisticated analysis at the finest grain!
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Level2 …
…
……
…
…
Level1
Level1
Level1
Level03
processors
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Intra-Cluster Scheduling
Context :
On-line scheduling
independent jobs (applications represented as moldable – non rigid - tasks) are submitted to the scheduler at any time on a queue.
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(strip) Packing problems
The schedule is divided into two successive steps:
1. Allocation problem and
2. Scheduling with preallocation (NP-hard in general [Rayward-Smith 95]).
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Scheduling: on-line vs off-line
On-line: no knowledge about the future
We take the scheduling decision while other jobs arrive
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Scheduling: on-line vs off-line
Off-line: we have a finite set of works
We try to find a good arrangement
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Off-line scheduler
Problem:Schedule a set of independent moldable jobs (clairvoyant).Penalty functions have somehow to be estimated (using complexity analysis or any prediction-measurement method like the one obtained by the log analysis).
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Classical approachusing combinatorial optimization
• Design fast algorithms with performance guaranty
• On-line algorithms
• One or several objective functions
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k-dual Approximation [Shmoys, Hochbaum]
•Let us guess a value of the objective Cmax: .
•Apply an algorithm, if the obtained a guaranty worse than k, then, refine the value of (by dichotomic search).
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Dual approximation
W/m (=1)
Estimated a target using a lower bound
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Canonical allotment
maximal number of processors for executing a task in time lower than 1 unit (normalized).
Jobs are assumed to be monotonic.
1
m
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HINT: analyze the optimal structure
Long Jobs (whose execution times are greater than 1/2) may not use more than m processors
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Thus, we are looking for a schedule in two shelves
3/2
1
m
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2 shelves partitioning
Knapsack problem: minimizing the global surface under the constraint of using less
than m processors in the first shelf.
1
m
1/2
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Dynamic programming
For i = 1..n // # of tasksfor j = 1..m // #proc.
Wi,j = min(– Wi,j-minalloc(i,1) + work(i,minalloc(i,1))– Wi,j + work(i,minalloc(i,1)))
work Wn,m <= work of an optimal solutionbut the half-sized shelf may be overloaded
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2 shelves partitioning
1
m
1/2
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Drop down
1
m
1/2
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1
m
1/2
Insertion of small tasks
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Analysis
•These transformations donot increase the work
•If the 2nd shelf is used more than m, it is always possible to do one of the transformations (using a global surface argument)
•It is always possible to insert the « small » sequential tasks (again by a surface argument)
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Guaranty
•The 2-shelves algorithm has a performance guaranty of 3/2+
•We will use it as a basis for an on-line version (batch scheduling).
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Batch scheduling
Principle: several jobs are treated at once using off-line scheduling.
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Principle of batch
jobs arrival time
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Start batch 1
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Batch chaining
Batch i Batch i+1
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Constructing a batch scheduling
Analysis: there exists a result which gives a guaranty for an execution in batch using the guaranty of the scheduling policy inside the batches.
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Analysis [Shmoys]
previous last batch last batch
Cmax
r(last job)
n
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previous last batch last batch
Cmax
rn
Tk
DkDK-1
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Proposition
*maxmax2 CC ρ≤
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Analysis
Tk is the duration of the last batch
On another hand, and
Thus:
TrC kn+≥*max
ρ
rD nk≤
−1
TTDC kkk++=
−− 11max
*maxmax2 CC ρ≤
*max, CT ii ρ≤∀
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Application
Applied to the previous 3/2-approximation algorithm, we obtain a 3-approximation on-line batch algorithm for Cmax.
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Multi criteria
The Makespan is not always the adequate criterion.
User point of view:Average completion time (weighted or
not)
Other criteria: Stretch, Asymptotic throughput
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How to deal with this problem?
Hierachy: one after the other(Convex) combination of criteriaTransform one criterion in a constraint
Ad hoc algorithms
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A first solution
Construct a feasible schedule from two schedulesof guaranty r for minsum and r’ for makespan with a guaranty (2r,2r’) [Stein et al.].
Instance: 7 jobs (moldable tasks) to be scheduled on 5 processors.
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Schedules s and s’
Schedule s(minsum)
35
41 2
67
Schedule s’(makespan)
7
2
14
635
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New schedule
35
41 2
67
7
2
14
635
r’Cmax
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New schedule
35
41 2
67
7
65
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New schedule
3
41 2
7
65
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New schedule
3
41 2
7
65
2r’Cmax
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New schedule
3
41 2
7
65
2r’Cmax
Similar bound for the first criterion
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Analysis
The best known schedules are:8 [Schwiegelsohn] for minsum and 3/2 [Mounie et al.] for makespan leading to (16;3).
Similarly for the weighted minsum (ratio 8.53).
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Improvement
We can improve this result by determining the Pareto curves (of the best compromises): (1+)/ r and (1+ )r’
Idea: take the first part of schedule s up to r’Cmax
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Pareto curve
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Another way for designing better schedules
We proposed a new solution for a better bound which has not to consider explicitly the schedule for minsum (based on a dynamic framework).
Principle: recursive doubling with smart selection (using a knapsack) inside of each interval.Starting from the previous algorithm for Cmax, we obtain a (6;6) approximation.
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Reservations
On-going work: packing with constraints
q
m
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Inter-Cluster Scheduling
Context
Load-balancing problem
independent jobs (moldable – non rigid - tasks) are submitted to local schedulers at any time.
Each cluster has its own managing internal rule.
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How to manage?
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Need of alternative approaches
Too many parameters…
Game theory (prisonner dilema)
Analogy with economy (auctions)
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To open the discussion
New execution supports are more and more complex, need of multi-criteria results. Ex: quality of service, reliability, etc..Need of robust approaches, able to remain efficient if the instances are disturbed…