Today s topic: So far, we have talked about. Question. Task models

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1 Overall Stucture of Real Time Systems So far, we have talked about Task Task n Programming Languages to implement the Tasks Run-TIme/Operating Systems to run the Tasks RTOS/Run-Time System Hardware Question Overall Stucture of Real Time Systems How to schedule the Tasks such that given timing constraints are satisfied? Task Scheduler Task n RTOS/Run-Time System Hardware Task models Today s topic: Non periodic/periodic (three parameters) : arrving time : computing time : deadline (relative deadline) REL TIME SHEULING (BSIS)

2 onstraints on task sets Timing constraints: deadline for each task, Relative to arriving time or absolute deadline Other constraints Precedence constraints Precedence graphs imposed e.g by input/output relation Resource constraints: mutual exclusion Resource access protocols Scheduling Problems Given a set of tasks (ready queue). heck if the set is schedulable. If yes, construct a schedule to meet all deadlines. If yes, construct an optimal schedule e.g. minimizing response times 7 8 Tasks with the same arrival time ssume a list of tasks (,, )(,, )...(,n,n) that arrive at the same time i.e. How to find a feasible schedule? (OBS: there may be many feasible schedules) Earlist ue ate first (E) [Jackson 9] E: order tasks with nondecreasing deadlines. Simple form of EF (earlist deadline first) Example: (,)(,)(,) Schedule: (,)(,)(,) FT: E is optimal If EF cann t find a feasible schedule for a task set, then no other algorithm can, i.e. The task set is non schedulable. 9 E: Schedulability test If +...+k <=k for all k<=n for the schedule with nondescreasing ordering of deadlines, then the task set is schedulable Response time for task i, Ri =+...+i Prove that E is optimal? E: Examples (, )(,)(,) is schedulable: Feasible schedule: (,)(,)(,) Note that (,)(,)(,) is also feasible (,)(,)(,) is schedulable The feasible schedule: (,)(,)(,) Why not shortest task first? (,)(,)(,) is not schedulable (,)(,)(,) is not feasible: + >!

3 E: optimality ssume that Ri is the finishing time (relative to the release time) of task i. Note that R means response time. Let Li = Ri-i (the lateness for task i) E: Exercises Prove: E is optimal in finding a feasible schedule Program the schedulability test for E FT: E is optimal with respect to minimizing the maximum lateness Lmax= MXi(Li) (the general form of optimality of E) Note that even a task set is non schedulable, E may minimize the maximal lateness (minimizes loss for soft tasks?) Tasks with different arrival times ssume a list of tasks S= (,, )(,, )...(n,n,n) Preemptive EF [Horn 97]: Whenever new tasks arrive, sort the ready queue according to earlist deadlines first at the moment Run the first task of the queue if it is non empty Preemptive EF: Schedulability test t time i, if the list ordered according to EF (,, )(,, )...( i, i, i) satisfies k <= k for all k=,...i, then S is schedulable at time i If S is schedulable at all i s, S is schedulable FT: Preemptive EF is optimal [ertouzos 97] in finding feasible schedules. Preemptive EF: Example onsider (,, )(,,)(,,8) eadlines are relative to arrival times t, (,) t, (,)(,) t, (,8)(,9) Preemptive EF: Response time calculation omplicated But possible 7 8

4 Preemptive EF: Exercises Write a program to calculate the response times for (non)preemptive EF Preemptive EF: Optimality ssume that Ri is the finishing time (relative to the release time) of task i. Note that R means response time. Let Li = Ri-i (the lateness for task i) FT: preemptive EF is optimal with respect to minimizing the maximum lateness Lmax= MXi(Li) (the general form of optimality of preemptive EF) 9 Non preemptive EF (on-line version) On-line non preemptive EF: example lternative : Run a task until it s finished and then sort the queue according to EF +The algorithm may be run on-line, easy to implement, less overhead (no more context switch than necessay) - However it is not optimal, it may not find the feasible schedule even it exists e.g (,,)(,,)(,7,): the second task misses its deadline. Note that the feasible schedule: (,,)(,,)(,7,) T 7 T ssume that is absolute deadline On-line EF Schedule Feasible Schedule PU idling 7 Missing the deadline! On-line non preemptive EF: Optimal? If we only consider non-idle algorithms (PU waiting only no task to run), is EF is optimal? Unfortunately no! Off-line Non preemptive EF (complete search) lternative : the decision should be made according to all the parameters in the whole list of tasks onsider the example: (,,)(,,)(,7,) Example T= (,, ) T= (,,) = (,,) Run T,,T: the rd task will miss its deadline Run T,,T: it is a feasible schedule

5 Off-line Non preemptive EF (NP-hard) Practical methods: Bratley s algorithm Unfortunately, to find a feasible non-preemptive schedule for task set with different arrival times is not easy The worst case is to test all possible combinations of n tasks (NP-hard, difficult for large n) Search until a non-schedulable situation occur, then backtrack [Bratley s algorithm] simple and easy to implement but may not find a schedule if n is too big (worst case) Example (Bratley s alg.) Heuristic methods: Spring algorithm T 7 T T 7 T T T 8 T T T Similar to Bratley s alg. But Use heuristic function H to guide the search until a feasible schedule is found, otherwise backtrack: add a new node in the search tree if the node has smallest value according to H e.g H(task i) = i, i, i etc [Spring alg.] However it may be difficult to find the right H 7 T 7 8 Example Heuristics H(Ti) = i FIFO H(Ti) = i SJF H(Ti) = i EF H(Ti) = i +w*i EF+SJF... EF: + and Simple (+) Preemptive EF, Optimal (+) No need for computing times (+) On-line and off-line (+) Preemptive schedule easy to find (+) But preemptive EF is difficult to implement efficiently (-) Must use a list of timers, one per task Nonpreemptive (feasible) schedule difficult to find (-) But minimal context switch (+) nd the only way to schedule non preemtive tasks 9

6 Other scheduling algorithms lassical ones HPF (priorities = degrees of importance of tasks) Weighted Round Robin LRT (Latest Release Time or reverse EF) LST (Least Slack Time first) Latest Release Time (reversed EF) Release time = arrival time Idea: no advantage to completing any hard task sooner than necessary. We may want to postpone the execution of hard tasks e.g to improve response times for soft tasks. LRT: Schedule tasks from the latest deadline to earliest deadline. Treat deadlines as release times and arrival times as eadlines. The latest eadline first FT: LRT is optimal in finding feasible schedule (for preemptive tasks) LRT: Example LRT: + and - T T T 8 7 T It needs rrival times (-) It got to be an off-line algorithm (-) Only for preemptive tasks (-) It could optimize Response times for soft tasks (+) (=absolute deadline) 7 Reverse time: we get the schedule: T(9,)T(,)(,7)T(7,8)T(8,) OBS: from to 9, soft tasks may be running! Least slack time first (LST) LST: Example Let Si = i-i (the Slack time for task i) Si is the maximal (tolerable) time that task i can be delayed Idea: there is no point to complete a task earlier than its deadline. Other (soft) tasks may be executed first Slack stealing LST: order the queue with nondecreasing slack times T T T 8 9 T (=absolute deadline) S=-9-= at S=--= at S=-9-= at 9 FT: preemptive LST is optimal in finding feasible schedules 7 9 omment: a task should run until a Slack reaches (to avoid context switch) nd if more than one -slack: nonschedulable

7 LST: + and It needs omputing times (-) Only for preemptive tasks (-) Not easy to implement! (-) But it can run on-line (+) and it may improve response times? Independent tasks OBS! we have assumed that tasks are independent! meaning that we can compute them in arbitrary orderings only if the orderings (schedules) are feasible ll algorithms we have studied so far are applicable only to independent tasks 7 8 Summary: scheduling independent tasks Task types lgorithms For Independent tasks Same arrival times E,Jackson O(n log n), optimal Preepmtive ifferent arrival times EF, Horn 7 O(n**), Optimal LST, LRT optimal Non preemptive ifferent arrival times Tree search Bratley 7 O(n n!), optimal Spring, Stankovic et al 87 O(n**), Heuristic ependent tasks In practice, tasks are dependent. We often have conditions or constraints e.g. must be computed before B B must be computed before and Such conditions are called precedence constraints which can be represented as irected cyclic Graphs (G) known as Precedence graphs Such graphs are also known as Task Graph 9 ependent tasks: Examples Input/output relation Some task is waiting for output of the others, data flow diagrams T T sampling T T T7 output Synchronization Some task must be finished before the others e.g. It is holding a shared resource Other dependence relations (e.g priority-orderings?) Precedence graph: Example must be computed before B B must be computed before and B

8 Precedence graph: Examples N/OR-precedence graphs E F N-node, all incomming edges must be finished first OR-node: some of the incomming edges must be finished B B B E Not a precedence graph! onjunct and isjunct join: We will only consider conjunct join! Scheduling under Timing and Precedence constraints Feasible schedules should meet Timing constraints: deadlines and also Precedence constraints: Precedence graphs Overlapping area of blue and red is what we need Precedence constraints restrict the search area (Guiding!) ependent tasks with the same arrival times ssume a list of tasks: (,,)(,,)...(,n,n) In addition to the deadlines...n, the tasks are also constrained by a G Solution: Latest eadline First (LF), Lawler 97 Schedules satisfying Precedence constraints ll possible schedules (feasible/infeasible) Schedules Satisfying Timing constraints FT: LF is optimal (in finding feasible schedules) Latest eadline First (LF) LF: Example It constructs a schedule from tail to head using a queue:. Pick up a task from the current G, that Has the latest deadline and oes not precede any other tasks (a leaf!). Remove the selected task from the G and put it to the queue Repeat the two steps until the G contains no more tasks. Then the queue is a potentilly feasible schedule. The last task selected should be run first. T T T T T T T T Note that this is similar to LRT 7 8

9 9 LF: Example T T T T T T T T LF: T LF: Example T T T T T T T LF: T LF: Example T T T T T T T LF: T,T LF: Example T T T T T T LF: T,T LF: Example T T T T T T LF: T,T, LF: Example T T T T T T LF: T,T,,

10 LF: Example LF: Example T T T T T T T T T LF: T,T,,,T LF: T,T,,,T,T LF: Example Earlest eadline First (EF) T T T T It is a variant of LF, but start with the root of the G:. Pick up a task with earlest deadline among all nodes that have no fathers (the roots). Remove the selected task from the G and put it to the queue Repeat the two steps until the G contains no more tasks. Then the queue is a feasible schedule. Unfortunately, EF is not optimal (see the following example) LF: T,T,,,T,T Feasible Schedule 7 8 LF: Example LF: Example T T T T T T T T T T T T T T T T EF: T 9

11 EF: Example LF: Example T T T T T T T T T T T T T T EF: T EF: T, LF: Example LF: Example T T T T T T T T T T EF: T,,T EF: T,,T,,T,T LF: Example ependent tasks with different arrival times T T T T will miss its eadline: ssume a list of tasks: S = (,,)(,,)...(,n,n) In addition to the deadlines...n, the tasks are also constrained by a G Solution: The omplete Search guided by the G The Bratley s algorithm The Spring algorithm LF: T,T,,,T,T Feasible EF: T,,T,,T,T Infeasible

12 Better algorithms? ssume a list of tasks: S = (,,)(,,)...(,n,n) In addition to the deadlines...n, the tasks are also constrained by a G Idea: Transform the task set S (constrained by the G) to an Independent task set S* such that S is schedulable under G iff S* is schedulable Idea: how to transform S to S*? Idea: If Ti ->Tj is in the G i.e. Ti must be executed before Tj, we replace the arrival time for Tj and deadline for Ti with j* = max(j, i+i) Tj can not be computed before the completion of Ti i*=min(i,j-j) Ti should be finished early enough to meet the deadline for Tj 7 8 lgorithm (EF*): transform S to S* Let arrival times and deadlines be absolute times Step : Transform the arrival times from roots to leafs For all initial (root) nodes Ti, let i* = i REPET: Pick up a node Tj whose fathers arrival times have been modified. If no such node, stop. Otherwise: Let j* =max(j, max{i*+i: Ti->Tj}) Step : Transform the deadlines from leafs to roots For all terminal (leafs) nodes Tj, let j* = j REPET: Pick up a node Ti all whose sons deadlines have been modified. If no such node, stop. Otherwise: Let i* =min(i, min{j*-j: Ti->Tj}) Step : use EF to schedule S*=(*,,*)...(n*.n,n*) EF*: optimality FT: S is schedulable under a G iff S* is schedulable EF* is optimal in finding a feasible schedule 9 7 Example EF*: Example() T T T T T T T T T T T T T T T T Step : Modifying the arrival times (top-down) 7 7

13 EF*: Example() EF*: Example() T T T T T T T T T T T () T () () * T T * () T () T () Step : Modifying the arrival times (top-down) Step : Modifying the deadlines (bottom-up) 7 7 EF*: Example() EF*: Example() S* T T T T T () S* T T T T T () * T () () * T () () * () T () T () * () T () T () Step : Modifying the deadlines (bottom-up) Step : now we don t need the G any more! 7 7 EF*: Example() EF*: Example() S* S* T T T T T T T T * * * * Step : schedule S* using EF Finally we have a schedule: T,T,,,T,T 77 78

14 Summary: scheduling aperiodic tasks Task types Same arrival times Preepmtive ifferent arrival times Non preemptive ifferent arrival times lgorithms for Independent tasks lgorithms for ependent tasks E,Jackson O(n log n), optimal LF, Lawler 7 O(n**) Optimal EF, Horn 7 O(n**), Optimal LST, optimal LRT, optimal EF* hetto et al 9 O(n**) optimal Tree search Bratley 7 O(n n!), optimal Spring, Stankovic et al 87 O(n**) Heuristic Spring s above 79

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