Aperiodic Task Scheduling
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1 Aperiodic Task Scheduling Radek Pelánek
2 Preemptive Scheduling: The Problem 1 processor arbitrary arrival times of tasks preemption performance measure: maximum lateness no resources, no precedence constraints
3 Example 1 Solve the example (manually). 2 Try to find out a scheduling algorithm.
4 Earliest Deadline First Earliest Deadline First Algorithm EDF At any instant execute the task with the earliest absolute deadline among all the ready tasks.
5 Earliest Deadline First Example EDF Schedule
6 Earliest Deadline First Optimality of EDF Theorem (Horn) Given a set of n independent tasks with arbitrary arrival times, the EDF algorithm is optimal with respect to minimazing the maximum lateness.
7 Earliest Deadline First Proof of Horn s Theorem Basic idea of the proof: let σ be optimal schedule; we transform it into EDF schedule σ EDF without increasing maximum lateness schedule is divided into time slices of 1 unit transformation: interchange 1 appropriate time slice
8 Earliest Deadline First Proof of Horn s Theorem (cont.)
9 Earliest Deadline First Guarantee Test is the set of tasks schedulable? sort tasks by increasing deadlines synchronous activation (a i = 0) schedulability guaranteed by conditions: i : Σ i k=1c k d i asynchronous activation: dynamical version of the previous test
10 Earliest Deadline First EDF is Fine... EDF: is optimal works on-line easy to implement simple guarantee test
11 Earliest Deadline First... but Not a Silver Bullet EDF is not optimal with more than 1 processor Try to find a specific set of tasks: the set is schedulable on 2 processor EDF schedule misses some deadline
12 Earliest Deadline First EDF with More Processors 2 Processors J 1 J 2 J 3 a i C i d i Schedulable, but EDF schedule misses deadline for J 3.
13 Least Slack Time Least Slack Time Algorithm LST At any instant execute the task with the least slack time (that is d i C i ) among all the ready tasks. LST is also optimal.
14 Least Slack Time Example Find a set of task such that EDF and LST produce different schedules.
15 Non-preemptive Scheduling: The Problem 1 processor arbitrary arrival times of tasks preemption not allowed performance measure: maximum lateness no resources, no precedence constraints
16 EDF Non-optimality of EDF EDF is not optimal for non-preemptive scheduling. Find a set of task such that EDF does not produce optimal schedule.
17 EDF Non-optimality of EDF
18 EDF EDF and Non-idle Schedules non-idle algorithm - does not permit the processor to be idle when there are active jobs restriction to non-idle algorithms EDF is optimal
19 Brute Force Optimal Scheduling no on-line algorithm can generate optimal schedule (example above: time 0, stay idle or start J 1?) off-line: non-preemptive scheduling is NP-complete branch-and-bound (backtracking) algorithms, worst case complexity O(n n!)
20 Brute Force Search Tree
21 Brute Force Reminder: Backtracking, n Queen Problem
22 Brute Force Pruning branch is abandoned when: the addition of any node to the current path causes a missed deadline a feasible schedule is found at the current path size of the search tree is exponential in the worst case significant pruning in the average case
23 Brute Force Pruning: Example
24 Brute Force Example J 1 J 2 J 3 J 4 a i C i d i Draw the search tree (with pruning).
25 Scheduling with Precedence Constraints generally NP-complete we consider two special cases, polynomial algorithms remark on heuristical approach
26 Latest Deadline First Precedence Constraints: Problem 1 1 processor precedence constraints synchronous activation ( i : a i = 0) (preemption does not matter) performance measure: maximum lateness
27 Latest Deadline First Example 1 solve the example (manually) 2 construct the EDF schedule 3 try to find out an optimal scheduling algorithm
28 Latest Deadline First Latest Deadline First Algorithm one possible solution: latest deadline first (LDF) note: different interpretations EDF algorithm = earliest deadline job is scheduled to run first LDF algorithm = latest deadline job is put into schedule first
29 Latest Deadline First Latest Deadline First Algorithm LDF among tasks without successors select the task with the latest deadline remove this task from the precedence graph and put it into a stack repeat until all tasks are in the stack the stack represents the order in which tasks should be scheduled LDF is optimal.
30 Latest Deadline First LDF: example
31 Modified EDF Precedence Constraints: Problem 2 1 processor precedence constraints arbitrary arrival times of tasks preemption performance measure: maximum lateness
32 Modified EDF Example Precedence constraints: A C; A D; B D, B E; E D; C F ; D F A B C D E F a C d
33 Modified EDF Basic Idea 1 transform set J of dependent tasks into set J of independent tasks 2 apply EDF to set J transformation done by modification of arrival times and deadline times
34 Modified EDF Modification of Arrival Times If J Y J X then: s X a X J X cannot start earlier than its activation time s X a Y + C Y J X cannot start earliear than the minimum finishing time of J Y
35 Modified EDF Modification of Arrival Times (cont.) new arrival time: a X = max(a X, max(a Y + C Y, J Y J X )) modified arrival time must be computed in the correct order (given by precedence constraints)
36 Modified EDF Modification of Deadlines If J X J Y then: f X d X J X must finish within its deadline f X d Y C Y J X must finish not later than the maximum starting time of J Y
37 Modified EDF Modification of Deadlines (cont.) new deadline: d X = min(d X, min(d Y C Y, J X J Y )) modified deadline times must be computed in the correct order (given by precedence constraints)
38 Modified EDF Optimality Theorem There exists feasible schedule for the modified task set J under EDF if and only if the original task set is schedulable.
39 Modified EDF Example Precedence constraints: A C; A D; B D, B E; E D; C F ; D F A B C D E F a C d a d
40 Modified EDF Example precedence constraints: A C, B C, C E, D F, B D, C F, D G all tasks: a i = 0, D i = 25, computation times: 2, 3, 3, 5, 1, 2, 5 compute modified arrival times and deadlines compute schedule according to EDF
41 Heuristical algorithms Heuristical search more complicated problems (e.g., non-preemptive, precedence constraints) NP-complete brute-force search (with pruning) heuristical search only some schedules are considered no guarantee of optimality note: general computer science approach
42 Heuristical algorithms Heuristical search constructs partial schedules (like brute-force search) considers only one (or few) tasks for extension of a schedule selection based on heuristic function H
43 Heuristical algorithms Heuristic function H(i) = a i H(i) = C i H(i) = d i first come first serve shortest job first earliest deadline first more complicated parameters (taking into account precedence relations, resources,...) weighted combinations of different parameters
44 Overview of Problems and Algorithms independent precedence sync. act. EDF, O(n log n) LDF, O(n 2 ) preemptive async. act. EDF, LST, O(n 2 ) modified EDF, O(n 2 ) nonpreemptive async. act. tree search, O(n n!) heuristic search
45 Summary aperiodic task scheduling (arbitrary arrival times) 1 processor, no resources precedence constraints: yes/no preemption: yes/no performance measure: maximum lateness algorithms: earliest deadline first (EDF), least slack time (LST), branch and bound, latest deadline first (LDF), transformations
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