# Introducción a Calendarización en Sistemas Paralelas, Grids y Nubes

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1 CICESE Research Center Ensenada, Baja California Mexico Introducción a Calendarización en Sistemas Paralelas, Grids y Nubes Dr. Andrei Tchernykh CICESE Centro de Investigación Científica y de Educación Superior de Ensenada Ensenada, Baja California, México, Escuela de Modelación y Métodos Numéricos CIMAT, Guanajuato, México de junio de A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 1

2 The first problem set of independent tasks identical processors minimize schedule length. Complexity: the problem is not easy to solve since even simple cases such as scheduling on two processors can be proved to be NP-hard. There is no hope of finding a polynomial time algorithm Solution: find an approximation algorithm for the original problem evaluate its worst case mean behavior. One of the most often used general approximation strategies for solving scheduling problems is list scheduling priority list of the tasks is given at each step the first available processor is selected to process the first available task on the list The accuracy of a given list scheduling algorithm depends on the order in which tasks appear on the list A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 2

3 Set of tasks T = {T 1, T 2,..., T n } Set of processors P = {P 1, P 2,..., P m } p j = processing time of task T j 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 3

4 Identical Processors. List Scheduling n W seq = p i i = 1 be the total work of all jobs p max is the maximum processing time of a job. W be the total idle intervals, p ( m 1) idle W C W C seq + seq W m + p idle W idle max is the completion time of the set of tasks. max ( 1) m m W seq p m max + p max W ( m 1) seq C + p, max m m W seq and p max are lower bounds of m performance bound is ρ 2 seq 1 m. seq C opt, it follows that the worst-case 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 4

5 Outline Preliminaries 1.1 Scheduling in Processor and Operating Systems 1.2 Production Scheduling 1.3 Control Systems 1.4 Basic Notions 1.5 The Scheduling Model Deterministic Model Optimization Criteria Scheduling Problem Notation Scheduling Algorithms 1.6 Schedule representation and evaluation 1.7 Three-field notation 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 5

6 Outline Scheduling on Parallel Processors 2. Minimizing Schedule Length Identical Processors Independent tasks List Scheduling LPT Algorithm Preemptions. McNaughton's rule Precedence constraints Graham anomalies 3. Minimizing Due Date Involving Criteria Identical Processors EDF - Earliest Deadline First scheduling LL-Least Laxity scheduling Minimizing number of tardy tasks 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 6

7 Outline 4. Communication Delays Scheduling without Task Duplication Scheduling with Task Duplication 5. Bin Packing Problem Next Fit and First Fit Best Fit (BF) and Worst Fit (WF). One-Dimensional Bin Packing Problem Two-Dimensional Bin Packing Problem 6. Strip Packing Problem Bottom-left algorithm Level-oriented algorithms Split algorithms Shelf algorithms Hybrid algorithms The Slave Algorithm On-line v's Off-line 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 7

8 Outline References 1. J. Blazewicz, K. Ecker, G. Schmidt, J. Weglarz, Scheduling in Computer and Manufacturing Systems, Springer, pp. 495, 2001 ISBN: Handbook of Scheduling: Algorithms, Models, and Performance Analysis. Edited by Joseph Y-T. Leung. Published by CRC Press, Boca Raton, FL, USA, Handbook on Scheduling. From Theory to Applications, by J. Blazewicz, K. Ecker, E. Pesch, G. Schmidt, J. Weglarz, Springer, pp. 647, 2007 ISBN: A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 8

9 Topic 1: Preliminaries Objective Application areas Basic Notions The Scheduling Model 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 9

10 Introduction Application Area: Scheduling in Processor and Operating Systems Production Scheduling Technical and Industrial Processes Control Systems 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 10

11 Application Area: Scheduling in Processor and Operating Systems In operating systems : hundreds of processes waiting to get access to the processor strategy takes various parameters into account, such as priority of the process its parent priority already consumed CPU time assigned resources The scheduler (process dispatcher) is designed to optimize some system performance: optimizing throughput: maximize the number of completed processes per time unit minimizing the makespan maximizing profit for the owner of the machine 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 11

12 Application Area: Scheduling in Processor and Operating Systems Questions regarding the scheduling of activities in computers occur at different levels: Inside processors : sequencing of micro-operations; pipelining scheduling strategies in single processor operating systems : round robin priority based dispatcher algorithms multiprocessor systems, consisting of a CPU, co-processors, and I/O processors: process handling, assignment of activities to the special purpose processors parallel processing on a large number of identical processors as in massive parallelism: o Work distribution, taking into account the network connectivity and communication delays 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 12

13 Application Area: Scheduling in Processor and Operating Systems distributed processing (several computers (workstations, PC's, etc.) are connected in a local area network (LAN), metropolitan area network (MAN), wide area network (WAN), Grids, Clouds real-time operating systems in parallel or distributed systems need careful handling of activities with deadlines Another example of practical interest concerns production systems Typical in this area is the demand for optimal working plans for assembly lines and for flexible manufacturing machines, e.g. in production cells General requirements: - production due dates - resource balancing - maximal production throughput - minimum storage cost 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 13

14 Application Area: Production Scheduling Examples: Control of robot movement has to deal with data, and concerns the real time coordination of moving the arm(s) Assembly lines are of pipeline structure; their optimal design leads to flow shop problems Organizing flexible manufacturing machines leads to problems of optimizing lot sizes under the requirement of optimal throughput while minimizing overhead due to tool change delays and other setup costs Optimal routing of automated guided vehicles (AGV's) leads to questions that again require careful planning and sequencing In a manufacturing environment deterministic scheduling is also known as predictive Its complement is reactive scheduling, which can also be regarded as deterministic scheduling with a shorter planning horizon 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 14

15 Application Area: Technical and Industrial Processes Activities from - production planning - computer aided design - work planning - manufacturing - quality control have to be coordinated The objectives are similar: optimal capacity planning maximal throughput minimum storage cost, etc A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 15

16 Application Area: Control Systems In real-time systems the particular situation dictates conditions different from those before: some processes must be activated periodically with a fixed rate, and others have to meet given deadlines In such systems, meeting the deadlines can be a crucial condition for the correct operation of the environment Examples of application areas are o aircraft control o power plants, heat control, turbine speed control, o frequency and voltage stabilization etc., 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 16

17 Basic Notions 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 17

19 Basic Notions. Introduction Depending of how much is known about the tasks to be processed, we distinguish between three main directions in scheduling theory: Deterministic or static or off-line scheduling assumes that all information required to develop a schedule is known in advance, before the actual processing takes place Especially in production scheduling and in real-time applications the deterministic scheduling discipline plays an important role Stand-alone theory, known as deterministic scheduling theory Characteristically, one has to deal with the development of optimal algorithms Non-deterministic scheduling is less restrictive: only partial information is known for example computer applications where tasks are pieces of software with unknown run-time 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 19

20 Basic Notions. Introduction On-line scheduling: In many situations detailed knowledge of the nature of the tasks is available, but the time at which tasks occur is open If the demand of executing a task arises a decision upon acceptance or rejection is required, and, in case of acceptance, the task start time has to be fixed In this situation schedules cannot be determined off-line, and we then talk about on-line scheduling or dynamic scheduling Non-clairvoyant scheduling: consider problems of scheduling jobs with unspecied execution time requirements Stochastic scheduling: only probabilistic information about parameters is available In this situation probability analysis is typical means to receive information about the system behavior For each type of scheduling one can find justifying applications Here, off-line scheduling (occasionally also on-line scheduling) is considered 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 20

21 Deterministic Scheduling Problems The deterministic scheduling or planning problems arising in different applications have often strong similarities hence essentially the same basic model can be used Common aspects in these applications: processes consist of complex activities to be scheduled modeled by means of tasks or jobs Tasks usually need one of the available machines, maybe even a special machine, and additional resources of limited availability Between tasks there are relations describing the relative order in which the tasks are to be performed order of task execution can be restricted by conditions like precedence constraints Preemption of task execution can be allowed or forbidden 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 21

22 Deterministic Scheduling Problems Timing conditions such as task release times, deadlines or due dates may be given In case of due dates cost functions may define penalties depending on the amount of lateness There may be conditions for time lags between pairs of tasks, such as setup delays In so-called shop problems sequences of tasks, each to be performed on some specified machine, are defined An example is the well-known flow shop or assembly line processing Scheduling problems are characterized not only by the tasks and their specific properties, but also by information about the processing devices Processors or machines for processing the tasks can be identical, can have different speeds (uniform), or their processing capabilities can be unrelated 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 22

23 Deterministic Scheduling Problems The problem is to determine an appropriate schedule, i.e. one that satisfies all conditions imposed on the tasks and processors A schedule essentially defines the start times of the tasks on a specified processor Generally there may exist several possible schedules for a given set of tasks An important condition describes the intended properties of a schedule, as defined by an optimization criterion Common criteria are: - minimization of the makespan of the total task set, - minimization of the mean waiting time of the tasks The optimization criterion allows to choose an appropriate schedule Such schedules are then used as a planning basis for carrying out the various activities Unfortunately, finding optimal schedules is in general a very difficult process Except for simplest cases, these problems turn out to be NP-hard, and hence the time required computing an exact solution is beyond all practical means In this situation, algorithmic approaches for sub-optimal schedules seem to be the only possibility 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 23

24 The Scheduling Model Deterministic Model Optimization Criteria Scheduling Problem and α β γ - Notation Scheduling Algorithms 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 24

25 The Scheduling Model. Deterministic Model Tasks, Processors, etc. Set of tasks T = {T 1, T 2,..., T n } Set of resource types R = {R 1, R 2,..., R s } Set of processors P = {P 1, P 2,..., P m } Examples of processors: CPUs in e.g. a multiprocessor system Computers in a distributed processing environment Production machines in a production environment Processors may be parallel : they are able to perform the same functions dedicated : they are specialized for the execution of certain tasks 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 25

26 The Scheduling Model. Deterministic Model Parallel processors have the same execution capabilities Three types of parallel processors are distinguished o identical : if all processors from set P have equal task processing speeds o uniform : if the processors differ in their speeds, but the speed b i of each processor is constant and does not depend on the tasks in T o unrelated : if the speeds of the processors depend on the particular task unrelated processors are more specialized : on certain tasks, a processor may be faster than on others 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 26

27 The Scheduling Model. Deterministic Model Characterization of a task Tj Vector of processing times p j = [p 1j,..., p m j ], where p ij is the time needed by processor P i to process T j Identical processors : p 1j =... = p m j = p j Uniform processors : p i j = p j / b i, i = 1,..., m p j = standard processing time (usually measured on the slowest processor), b i is the processing speed factor of processor P i Processing times are usually not known a priori in computer systems Instead of exact values of processing times one can take their estimate However, in case of deadlines exact processing times or at least upper bounds are required 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 27

28 The Scheduling Model. Deterministic Model Arrival time (or release or ready time) r j is the time at which task T j is ready for processing if the arrival times are the same for all tasks from T, then r j = 0 is assumed for all tasks Due date d j specifies a time limit by which T j should be completed problems where tasks have due dates are often called "soft" real-time problems. Usually, penalty functions are defined in accordance with due dates Penalty functions G j define penalties in case of due date violations Deadline d j "hard" real time limit, by which T j must be completed Weight (priority) w j... expresses the relative urgency of T j 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 28

29 The Scheduling Model. Deterministic Model Preemption / non-preemption : A scheduling problem is called preemptive if each task may be preempted at any time and its processing is resumed later, perhaps on another processor If preemption of tasks is not allowed the problem is called non-preemptive Conditions among the set of tasks T : precedence constraints T i p T j means that the processing of T i must be completed before T j can be started We say that a precedence relation p is defined on set T mathematically, a precedence relation is a partial order The tasks in T are called dependent if the relation p is non-empty otherwise, the tasks are called independent 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 29

30 The Scheduling Model. Deterministic Model Representation of tasks with precedence constraints: task-on-node graph (Hasse diagram) T 1 T 2 For each T i p T j, an edge is drawn between the corresponding nodes The situation T i p T j and T j p T k is called transitive dependency between T i and T k Transitive dependencies are not explicitly represented T 3 T 6 T 4 T 7 T 5 T 8 T 9 T 10 T 11 T A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 30

31 The Scheduling Model. Deterministic Model task-on-arc graph, activity network. Arcs represent tasks and nodes time events Example 1: T = {T 1,..., T 10 } with precedences as shown by the above Hasse diagram. A corresponding activity network: T 1 T 3 T 10 ' T 10 T 2 T 4 T 6 T 9 T 12 T 5 T 5 ' T 7 ' T 6 T 11 T 8 ' T A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 31

32 task-on-arc graph????? 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 32

33 Error???? 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 33

34 The Scheduling Model. Deterministic Model Task T j is called available at time t if r j t and all its predecessors (with respect to the precedence constraints) have been completed by time t Schedules Schedules or work plans generally inform about the times and on which processors the tasks are executed To demonstrate the principles, the schedules are described for the special case of: - parallel processors - tasks have no deadlines - tasks require no additional resources Release times and precedence constraints may occur 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 34

35 The Scheduling Model. Deterministic Model A schedule S is an assignment of processors to the tasks from T (or an assignment of the tasks to the processors) such that: task T j is processed in the time interval [r j, ) for p j time units, all tasks are completed, at each instant of time, each processor works on at most one task, at each instant of time, each task is processed by at most one processor if tasks T i, T j are in relation T i p T j then the processing of T j is not started before T i has been completed, if T j is no-preemptive then processing of T j is not interrupted; if T j is preemptive then T j may be interrupted only a finite number of times If all tasks are non-preemptive then the schedule is called non-preemptive If all tasks are preemptive, then the schedule is called preemptive 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 35

36 The Scheduling Model. Schedule representation (3) Graphic representation: Gantt chart - this is a two-dimensional diagram The abscissa represents the time axis that usually starts with time 0 at the origin Each processor is represented by a line For a task T j to be processed by P i, a bar of length p(t j ) and that begins at the time marked by s(t j ), is entered in the line corresponding to P i A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 36

37 The Scheduling Model. Schedule representation Example 1: T = {T 1,..., T 12 } with precedences as shown by the Hasse diagram: T 1 T T 3 T 4 T T 6 T 7 T T 9 1 T 10 3 T 11 T 2014 A. Tchernykh. CIMAT 2014 Introducción 12 a calendarización en sistemas paralelas, grids y nubes 37 1

38 The Scheduling Model. Schedule representation Example 2: non-preemptive schedule In the above example, let (2, 2, 8, 2, 3, 2, 4, 4, 2, 1, 3, 1) be the vector of processing times, and assume all release times = 0 Assume furthermore that there are m=3 identical processors (P = {P 1, P 2, P 3 }) available for processing the tasks Gantt chart of a non-preemptive schedule: P 3 T 5 T 8 T 9 T 10 P 2 T 2 T 4 T 6 T 7 T 12 P 1 T 1 T 3 T A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 38

39 The Scheduling Model. Schedule representation P 3 T 5 T 8 T 9 T 10 P 2 T 2 T 4 T 6 T 7 T 12 P 1 T 1 T 3 T A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 39

40 The Scheduling Model. Deterministic Model Given a schedule S, the following can be determined for each task T j : flow time lateness F j := c j r j L j = c j d j tardiness D j = max{c j d j, 0} tardy task U j = 0 if D j = 0 1 else 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 40

41 The Scheduling Model. Deterministic Model. Optimization Criteria Evaluation of schedules Maximum makespan C max = max{c j T j T } Mean flow time F := (1/n) Σ F j Maximum lateness L max = max{l j T j T } Mean tardiness D := (1/n) Σ D j 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 41

42 The Scheduling Model. Deterministic Model. Optimization Criteria Given a set of tasks and a processor environment there are generally many possible schedules Evaluating schedules: distinguish between good and bad schedules This leads to different optimization criteria Minimizing the maximum makespan C max C max criterion: C max -optimal schedules have minimum makespan the total time to execute all tasks is minimal Minimizing schedule length is important from the viewpoint of the owner of a set of processors (machines): This leads to both, the maximization of the processor utilization factor (within schedule length C max ), and the minimization of the maximum in-process time of the scheduled set of tasks 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 42

43 The Scheduling Model. Deterministic Model. Optimization Criteria Deadline related criteria If deadlines are specified for (some of) the tasks we are interested in a schedule in which all tasks complete before their deadlines expire Question: does there exist a schedule that fulfills all the given conditions? Such a schedule is called valid (feasible) Here we are faced in principle with a decision problem If, however, a valid schedule exits, we would of course like to get it explicitly If a valid schedule exists we may wish to find a schedule that has certain additional properties, such as minimum makespan or minimum mean flow Hence in deadline related problems we often additionally impose one of the other criteria 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 43

44 The Scheduling Model. Deterministic Model. Optimization Criteria Example 3 Gantt chart of a preemptive schedule: P 3 T 5 T 8 T 6 T 11 T 3 P 2 T 2 T 4 T 7 T 9 T 10 T 12 P 1 T 1 T 3 T 6 T 3 T 9 T C max = 11.3 (1) In the schedule of example the flow time of tasks F(T 1 )=2,F(T 2 ) = 2,F(T 3 )=???, etc A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 44

45 The Scheduling Model. Deterministic Model. Optimization Criteria Example 4: non-preemptive schedule with due dates For the task set as specified before, let in addition due dates be given by the vector (8, 2, 16, 4, 4, 8, 8, 8, 10, 8, 10, 11). In the schedule below, task T 10 with due date 8 violates its due date by two time units. P 3 T 5 T 8 T 9 T 10 T 12 P 2 T 2 T 4 T 6 T 3 P 1 T 1 T 7 T A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 45

46 The Scheduling Model. Deterministic Model. Optimization Criteria (2) In the schedule of example task T 10 has lateness???; for all other tasks L j is less equal???. The tardiness of T 10 =????, and it is??? for all other tasks; hence U 10 =???, and U j =??? for all other tasks (3) In the same schedule task T 1 has earliness??, T 2 has earliness??, etc A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 46

47 Examples (1) In the schedule of example 3 the flow time of tasks T 1 and T 2 is 2, that of T 3 is 11.3, etc. (2) In the schedule of example 4 task T 10 has lateness 2; for all other tasks L j is less than or equal 0. The tardiness of T 10, and it is 0 for all other tasks; hence U 10 = 1, and U j = 0 for all other tasks (3) In the same schedule task T 1 has earliness 6, T 2 has earliness 0, etc A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 47

48 Example: Consider the task set as in Example 1, with processing times and due dates as specified in the respective Examples 2 and 3. m=3, processing times (2, 2, 8, 2, 3, 2, 4, 4, 2, 1, 3, 1), due dates (8, 2, 16, 4, 4, 8, 8, 8, 10, 8, 10, 11). T 1 T 2 T 3 T 4 T 5 T 6 T 7 T 8 T 9 T 10 T 11 T A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 48

49 P 3 T 5 T 8 T 9 T 10 P 2 T 2 T 4 T 6 T 7 T 12 P 1 T 1 T 3 T A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 49

50 Topic 2 Scheduling on Parallel Processors 2.1 Minimizing Schedule Length Identical Processors Uniform Processors 2.2 Minimizing Mean Flow Time Identical Processors Uniform Processors 2.3 Minimizing Due Date Involving Criteria Identical Processors Uniform Processors 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 50

51 Independent tasks 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 51

52 Identical Processors P Cmax The first problem considered is P Cmax where set of independent tasks identical processors minimize schedule length. Complexity analysis: the problem is not easy to solve since even simple cases such as scheduling on two processors can be proved to be NP-hard. There is no hope of finding a polynomial time algorithm Solution: find an approximation algorithm for the original problem and evaluate its worst case as well as its mean behavior. One of the most often used general approximation strategies for solving scheduling problems is list scheduling priority list of the tasks is given at each step the first available processor is selected to process the first available task on the list The accuracy of a given list scheduling algorithm depends on the order in which tasks appear on the list A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 52

53 Identical Processors. List Scheduling n W seq = p i i = 1 be the total work of all jobs p max is the maximum processing time of a job. W be the total idle intervals, p ( m 1) idle W C W C seq + seq W m + p idle W idle max is the completion time of the set of tasks. max ( 1) m m W seq p m max + p max W ( m 1) seq C + p, max m m W seq and p max are lower bounds of m performance bound is ρ 2 seq 1 m. seq C opt, it follows that the worst-case 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 53

54 Identical Processors. LPT Algorithm for P Cmax Approximation algorithm for P C max : One of the simplest algorithms is the LPT algorithm in which the tasks are arranged in order of non-increasing p j. Algorithm LPT for P C max. begin Order tasks such that p 1... p n ; for i = 1 to m do s i := 0; -- processors P i are assumed to be idle from time s i = 0 on j := 1; repeat s k := min{ s i }; Assign task T j to processor P k at time s k ; -- the first non-assigned task from the list is scheduled on the first processor that becomes free s k := s k + p j ; j := j + 1; until j = n; -- all tasks have been scheduled end; 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 54

55 Identical Processors. LPT Algorithm for P Cmax Theorem If the LPT algorithm is used to solve problem P Cmax, then RLPT = m. an example showing that this bound can be achieved. Let n = 2m + 1, p = [2m 1, 2m 1, 2m 2, 2m 2,...,m + 1, m + 1, m, m, m]. For m = 3, Next figure shows two schedules, an optimal one and an LPT schedule A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 55

56 Identical Processors. LPT Algorithm for P Cmax Example: m = 3 identical processors; n = 2m + 1, p = [2m 1, 2m 1, 2m 2, 2m 2,..., m + 1, m + 1, m, m, m]. Time complexity of this algorithm is O(nlogn) the most complex activity is to sort the set of tasks. For m = 3, p = [5, 5, 4, 4, 3, 3, 3]. P 1 T 1 T 3 P 1 T 1 T 5 T 7 P 2 T 2 T 4 P 2 T 2 T 6 P 3 T5 T6 T t P 3 T 3 T t (a) an optimal schedule, (b) LPT schedule A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 56

57 Identical Processors. LPT Algorithm for P Cmax Example: n = (m 1)m + 1, p = [1, 1,...,1, 1, m], p is empty, L = (T n, T 1, T 2,...,T n 1 ), L' = (T 1, T 1,...,T n ). The corresponding schedules for m = 4 P 1 T 13 P 1 T 1 T5 T 9 T 13 P 2 T1 T 4 T7 T 10 P 2 T 2 T6 T 10 P 3 T2 T5 T8 T 11 P 3 T 3 T7 T 11 P 4 T T 3 6 T 9 T 12 P 4 T 4 T8 T t t (a) an optimal schedule, (b) an approximate schedule 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 57

58 Preemptions 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 58

59 Identical Processors, P pmtn Cmax Problem P pmtn Cmax relax some constraints imposed on problem P Cmax and allow preemptions of tasks. It appears that problem P pmtn Cmax can be solved very efficiently. It is easy to see that the length of a preemptive schedule cannot be smaller than the maximum of two values: the maximum processing time of a task and the mean processing requirement on a processor: The following algorithm given by McNaughton (1959) constructs a schedule whose length is equal to C * max. C max * = max{max {p j j }, 1 m j=1 Σ n p j } A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 59

60 Identical Processors, P pmtn Cmaxю McNaughton's rule Algorithm McNaughton's rule for P pmtn C max begin C * max := max{σ n pj /m, max{p j=1 j j = 1,...,n}}; -- min schedule length t := 0; i := 1; j := 1; repeat if t + p j C * max then begin Assign task T j to processor P i, starting at time t; t := t + p j ; j := j + 1; -- assignment of the next task continues at time t + p j end else begin Starting at time t, assign task T j for C * max - t units to P i ; -- task T j is preempted at time C * max, -- assignment of T j continues on the next processor at time 0 p j := p j - (C * max - t); t := 0; i := i + 1; end; 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 60

61 until j = n ; -- all tasks have been scheduled end; Remarks: The algorithm is optimal. Its time complexity is O(n) Question of practical applicability: preemptions are not free of cost (delays) two kinds of preemption costs have to be considered: time and finance. Time delays are not crucial if the delay caused by a single preemption is small compared to the time the task continuously spends on the processor Financial costs connected with preemptions, on the other hand, reduce the total benefit gained by preemptive task execution; but again, if the profit gained is large compared to the losses caused by the preemptions the schedule will be more useful and acceptable A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 61

62 Precedence constraints 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 62

63 Identical Processors, P prec C max Given: task set T with vector of processing times p precedence constraints p priority list L m identical processors Let C max be the length of the list schedule 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 63

64 Identical Processors, P prec C max, Graham anomalies The above parameters can be changed: vector of processing times p' p (component-wise), relaxed precedence constraints p' p, priority list L' and another number of processors m' Let the new value of schedule length be C ' max. List scheduling algorithms have unexpected behavior: 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 64

65 Identical Processors, P prec C max, Graham anomalies the schedule length for problem P prec C max if: may increase the number of processors increases, task processing times decrease, precedence constraints are weakened, or the priority list changes 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 65

66 Identical Processors, P prec C max, Graham anomalies (a) T 1 /3 T 2 /4 T 3 /2 T /13 7 T /2 8 T 4 /4 T 5 /4 T 6 /2 (b) P 1 T 1 T 3 T 4 T 5 T 6 T 8 P 2 T T (a) A task set, m = 2, L = (T 1, T 2, T 3, T 4, T 5, T 6, T 7, T 8 ), (b) an optimal schedule t 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 66

67 Identical Processors, P prec C max, Graham anomalies P 1 P 2 T1 T3 T4 T6 T2 T8 T5 T A new list L' = (T 1, T 2, T 3, T 4, T 5, T 6, T 8, T 7 ). t P 1 T1 T 3 T4 T6 T8 P 2 T 2 T 5 T t Processing times decreased; p' j = p j 1, j = 1, 2,..., n A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 67

68 Identical Processors, P prec C max, Graham anomalies P 1 P 2 T1 T 5 T2 T6 T7 P 3 T 3 T 4 T Number of processors increased, m = 3 t 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 68

69 Identical Processors, P prec C max, Graham anomalies (a ) T /3 1 T /4 2 T /2 3 T 7 /13 T 8 /2 T /4 4 T /4 5 T /2 6 (b ) P 1 T 1 T 3 T 5 T 7 P 2 T 2 T 4 T 6 T t Figure 4-6 (a) Precedence constraints weakened, (b) resulting list schedule A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 69

70 Identical Processors, P prec C max, Graham anomalies These list scheduling anomalies have been discovered by Graham [Gra66], who has also evaluated the maximum change in schedule length that may be induced by varying one or more problem parameters. o Let the processing times of the tasks be given by vector p, o let T be scheduled on m processors using list L, and o let the obtained value of schedule length be equal to Cmax. On the other hand, let the above parameters be changed: o a vector of processing times p' p (for all the components), o relaxed precedence constraints ', o priority list L' and the number of processors m'. o Let the new value of schedule length be C ' max A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 70

71 Identical Processors, P prec C max, Graham anomalies Corollary (Graham 1966) For an arbitrary list scheduling algorithm LS for P C max we have R LS 2 1 m if m' = m. P 1 T 13 P 1 T 1 T5 T 9 T 13 P 2 T1 T4 T7 T10 P 2 T 2 T6 T 10 P 3 T2 T5 T8 T 11 P 3 T 3 T7 T 11 P 4 T3 T6 T 9 T12 t P Schedules for Corollary (a) an optimal schedule, (b) an approximate schedule. T 4 T8 T A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 71

72 Topic 3 Scheduling on Parallel Processors 2.1 Minimizing Schedule Length Identical Processors Uniform and Unrelated Processors 2.2 Minimizing Mean Flow Time Identical Processors Uniform and Unrelated Processors 3 Minimizing Due Date Involving Criteria Identical Processors Uniform and Unrelated Processors 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 72

73 Model Arrival time (or release or ready time) r j is the time at which task T j is ready for processing if the arrival times are the same for all tasks from T, then r j = 0 is assumed for all tasks Due date d j specifies a time limit by which T j should be completed problems where tasks have due dates are often called "soft" real-time problems. Usually, penalty functions are defined in accordance with due dates Penalty functions G j define penalties in case of due date violations Deadline d j "hard" real time limit, by which T j must be completed Weight (priority) w j... expresses the relative urgency of T j 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 73

74 Identical Processors. Deadline Criteria P r j, d ~ j If deadlines are given: check if a feasible schedule exists (decision problem) Single processor problem P1 p j = 1, d j can be solved in polynomial time EDF algorithm is optimal More than one processor: most problems are known to be NP-complete The problems P p j = 1, d j and P prec, p j {1, 2}, d j are NP-complete Algorithmic approaches: exhaustive search heuristic algorithms approximation algorithms 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 74

75 Identical Processors. Deadline Criteria P r j, d ~ j Scheduling strategies: A strategy is called "feasible", if the algorithm generates schedules where all tasks observe their deadlines (assuming this is actually possible) three interesting deadline scheduling strategies: EDF Earliest Deadline First scheduling LL Least Laxity scheduling 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 75

76 Identical Processors. Deadline Criteria P r j, d ~ j Earliest Deadline First Scheduling Policy means that the task that has the earliest deadline (task that has to be processed first) is to be scheduled next. EDF scheduler views task deadlines as more important than task priorities. Experiments have shown that the earliest deadline first policy is the most fair scheduling algorithm A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 76

79 Identical Processors. Deadline Criteria P r j, d ~ j Example: Comparison of strategies Set of independent tasks: T = {T 1, T 2,..., T 6 } Tasks: (deadline, total execution time, arrival time): T 1 = (5, 4, 0), T 2 = (6, 3, 0), T 3 = (7, 4, 0), T 4 = (12, 9, 2), T 5 = (13, 8, 4), T 6 = (15, 12, 2) Execution on three identical processors: EDF-schedule (no preemptions): total execution time is 16 least laxity schedule (with preemptions): 8 preemptions, total execution time is 15 optimal schedule with 3 preemptions, total execution time = 15 Execution on a single, three times faster processor: possible with no preemptions; total execution time is 40/3 Hence: a larger number of processors is not necessarily advantageous 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 79

80 Identical Processors. Deadline Criteria P pmtn, r j, d ~ j Feasibility testing of problem P pmtn, r j, d ~ j is done by applying a network flow approach (Horn 1974) Given an instance of P pmtn, r j, d ~ j, let e 0 < e 1 <...< e k, k 2n 1 be the ordered sequence of release times and deadlines together (e i stands for r j or d ~ j) (time intervals) Construct a network with source, sink and two sets of nodes (Figure): the first set (nodes w i ) corresponds to time intervals in a schedule; node w i corresponds to interval [e i 1, e i ], i = 1, 2,..., k the second set corresponds to the tasks 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 80

81 Identical Processors. Deadline Criteria P pmtn, r j, d ~ j c =m(e e ) c 2=m(e 2 e 1) w 1 w 2 c = e e 1 0 T 1 T 2 b=p c=p 2 2 b=p c=p 1 1 c =m(e e ) k k k-1 c=e e k 1 k w k T n b=p c=p n n 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 81

82 Identical Processors. Deadline Criteria P pmtn, r j, d ~ j Flow conditions: The capacity of an arc joining the source to node w i is m(e i e i 1 ) o this corresponds to the total processing capacity of m processors in this interval If task T j is allowed to be processed in interval [e i 1, e j ] then w i is joined to T j by an arc of capacity e i e i 1 Node T j is joined to the sink of the network by an arc with lower and upper capacity equal to p j Finding a feasible flow pattern corresponds to constructing a feasible schedule; this test can be made in O(n 3 ) time the schedule is constructed on the basis of the flow values on arcs between interval and task nodes A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 82

83 Example. n = 5, m = 2, p = [5, 2, 3, 3, 1], r = [2, 0, 1, 0, 2], and d = [8, 2, 4, 5, 8]. S c=2 c=2 c=4 [0,1] [1,2] c=1 c=1 [2,4] c=2 c=1 c=1 c=2 c=1 b=2 c=2 b=5 c=5 1 T 3 2 T 1 T 2 b=3 c=3 S c=2 c=1 c=2 b=3 c=3 c=6 [4,5] [5,8] (a) corresponding network c=3 c=1 c=1 c=3 c=2 T 4 T 5 b=1 c= A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 83

84 [0,1] T [1,2] S 1 S 2 4 [2,4] 1 2 T 2 T [4,5] 3 1 T [5,8] T5 (b) feasible flow pattern 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 84

85 (c) optimal schedule P 1 T 2 T 2 T 3 T 1 T 1 P 2 T 4 T 3 T 4 T 1 T 4 T t 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 85

86 Scheduling on Parallel Processors Communication Delays and Multiprocessor Tasks Introductory Remarks Scheduling Multiprocessor Tasks o Parallel Processors o Refinement Scheduling Scheduling Uniprocessor Tasks with Communication Delays o Scheduling without Task Duplication o Scheduling with Task Duplication o Considering Processor Network Structure Scheduling Divisible Tasks 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 86

87 Scheduling Uniprocessor Tasks with Communication Delays The following simple example serves as an introduction to the problems. Let there be given three tasks with precedences as shown in Figure (a). T 1 T 2 T 3 (a) Precedence graph The computational results of task T1 are needed by both successor tasks, T2 and T3. We assume unit processing times. For task execution there are two identical processors, connected by a communication link. To transmit the results of computation T1 along the link takes 1.5 units of time A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 87

88 Scheduling Uniprocessor Tasks with Communication Delays P 2 P 1 T 3 T 1 T 2 (b) Schedule without consideration of communication delays The schedule in Figure (b) shows a schedule where communication delays are not considered. P 2 P 1 T 1 T 3 T 2 (c) Schedule considering communication from T1 to T3 The schedule (c) is obtained from (b) by introducing a communication delay between T1 and T A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 88

89 Scheduling Uniprocessor Tasks with Communication Delays Schedule (d) demonstrates that there are situations where a second processor does not help to gain a shorter schedule. P 2 P 1 T 1 T 2 T 3 (d) Optimal schedule without task duplication The fourth schedule, (e), demonstrates another possibility: if task T1 is processed on both processors, an even shorter schedule is obtained. The latter case is usually referred to as task duplication. P 2 T 1 P 1 T 1 T 3 T 2 (e) Optimal schedule with task duplication 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 89

90 Scheduling Uniprocessor Tasks with Communication Delays Communication delays are the same for all tasks so-called uniform delay scheduling. Other approaches distinguish between coarse grain and fine grain parallelism: high computation-communication ratio can be expected in coarse grain parallelism. As pointed out before, task duplication often leads to shorter schedules; this is in particular the case if the communication times are large compared to the processing times A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 90

91 Bin Packing Problem 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 91

92 Outline 1. Introduction Metaphorically, there never seem to be enough bins for all one needs to store. Mathematics comes to the rescue with the bin packing problem and its relatives. The bin packing problem raises the following question: given a finite collection of n weights w1, w2, w3,..., wn, and a collection of identical bins with capacity C (which exceeds the largest of the weights), what is the minimum number k of bins into which the weights can be placed without exceeding the bin capacity C? 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 92

93 Outline How few bins are needed to store a collection of items? This problem, known as the 1-dimensional bin packing problem, is one of many mathematical packing problems which are of both theoretical and applied interest. It is important to keep in mind that "weights" are to be thought of as indivisible objects rather than something like oil or water. For oil one can imagine part of a weight being put into one container and any left over being put into another container. However, in the problem being considered here we are not allowed to have part of a weight in one container and part in another. One way to visualize the situation is as a collection of rectangles which have height equal to the capacity C and a fixed width, whose exact size does not matter. When an item is put into the bin it either falls to the bottom or is stopped at a height determined by the weights that are already in the bins A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 93

94 Outline The diagram below shows a bin of capacity 10 where three identical weights of size 2 have been placed in the bin, leaving 4 units of empty space, which are shown in blue A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 94

95 Outline By contrast with the situation above, the bin below has been packed with weights of size 2, 2, 2 and 4 in a way that no room is left over A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 95

96 Basic ideas The bin packing problem asks for the minimum number k of identical bins of capacity C needed to store a finite collection of weights w1, w2, w3,..., wn so that no bin has weights stored in it whose sum exceeds the bin's capacity. Traditionally capacity C is chosen to be 1 and weights are real numbers which lie between 0 and 1, for convenience of exposition, C is a positive integer and the weights are positive integers which are less than the capacity. Example 1: Suppose we have bins of size 10. How few of them are required to store weights of size 3, 6, 2, 1, 5, 7, 2, 4, 1, 9? 2014 A. Tchernykh. CIMAT 2014 Introducción a calendarización en sistemas paralelas, grids y nubes 96

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