EECS 571 Principles of Real-Time Embedded Systems. Lecture Note #7: More on Uniprocessor Scheduling

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1 EECS 571 Principles of Real-Time Embedded Systems Lecture Note #7: More on Uniprocessor Scheduling Kang G. Shin EECS Department University of Michigan

2 Precedence and Exclusion Constraints Thus far, we assumed tasks are independent and always preemptible If a set of tasks, T = {T 1, T 2,..., T n } with D 1 D 2 D n, that form a task graph are assigned to the same processor, then 1. Schedule T n in [D n e n, D n ] 2. While not all tasks have been scheduled do Schedule T k, k = max{m : m A} as late as possible end do where A = set of as-yet-unscheduled tasks all of whose successors have already been scheduled. 3. Move tasks forward to the extent possible, keeping their execution order as specified in Step 2.

3 An Example Task graph with precedence constraints and all T i s released at time 0 T = (e, d ) = (3,6) T = (3,7) 2 3 T = (2,20) T = (5,21) 4 T = (6,27) 5 T = (6,28) 6 Upon completion of Step 2 T1 T2 T3 T4 T5 T After moving tasks in Step 3 T1 T2 T3 T4 T5 T

4 Scheduling with AND/OR Constraints AND task AND-JOIN; OR task OR-JOIN Minpath Algorithm (Fig. 3.21) reduces AND/OR scheduling to a standard problem while A = {all OR tasks} = do 1. Choose T i A none of whose precedents are an OR task 2. Find k s.t. L(k) L(j) j PD i = {immediate predecessors of T i } { ei if T L(i) = i has no precedents e i + max{l(k) : T k PD i } otherwise 3. Remove all edges terminating in T i except the one from T k 4. Relable T i as an AND task end do

5 MINPATH Example 5 T 1 8 T 3 1 T 4 6 T 2 2 T 5 1 T 7 4 T 6 2 T 8 1. Pick T 7 2. k = 4 Keep edge (4,7) 3. A = {T 6, T 8 } 4. Pick T 6 5. k = 7 Keep edge (7,6) and remove edge (5,6) 6. A = {T 8 }, pick T 8 7. k = 7 keep edge (7,8), and remove (2,8) & (6,8). 5 T 1 8 T 3 1 T 4 6 T 2 2 T 5 1 T 7 4 T 6 2 T 8

6 Generic Task Model T = {T 1, T 2,..., T n } T i = (e i, d i, r i ) Three relationships: T i precedes, or excludes, or preempts T j A task is eligible if it has been released and if all its predecessors completed execution Modified release time r i = { ri if no task precedes T i max[r i, r j + e j T j precedes T i ] otherwise A schedule is valid if o Processor is not left idle when one or more tasks ready to run o Precedence, exclusion, and preemption constraints are all met throughout the schedule

7 Algorithm for generating a valid schedule t := 0 while ( unfinished tasks) do if ( i s.t. t = r i or t = f i) then - select highest-priority eligible task with shortest deadline - if more than 1 eligible task w/ same deadline then break ties end if t := t + 1 end while

8 Example for generating a valid initial schedule T 1 T 2 T 3 T 4 r i e i D i Constraint: T 1 EXCLUDES T 2 Question: generate a valid schedule. Time points of interest: {0,1,10,11,13,14}. T 2 T 1 T T T Busy Period Busy Period After reordering in the first busy period 0 1 T 1 T

9 General Sched Alg Z(i) = set of tasks containing T i and those scheduled before T i within the same busy period, e.g., Z(1) = {1,2}, Z(2) = {2}, Z(3) = {3,4}, Z(4) = {4}. Lateness of T i = f i D i Lateness of a schedule = max task lateness in the schedule Two sets of tasks of interest: G 1 (i): set of tasks that cannot be preempted by T i, but that, if moved in the schedule to execute after T i, may reduce the max lateness. G 2 (i): set of tasks that, if preempted by T i, may reduce the max lateness. Compute a lower bound of the lateness of the valid initial schedule Apply sched alg in Figure 3.27 that uses the valid initial schedule as the root of the search tree and all of the above

10 Primary and Alternative Tasks Critical tasks are scheduled based on their worst-case execution time average execution time Low utilization Reclaim for less critical tasks the time left unused by critical tasks a primary ( full-quality ) & alternative ( bare-bones ) of each critical task. Completing either version is acceptable How to choose runtime limit of a primary? Example: primary alternative Worst-case runtime 12 5 Avg runtime 7 4 Period Primary Alternative Runtime limit of the primary

11 Interesting Special Case Set of periodic tasks with periods {P m,2p m,2 2 P m,...,2 i P m }. Level-i task if its period = 2 i P m ; l i = runtime limit of primary version π i of T i. Scheduling alg 1. Schedule all level-0 tasks (called schedule S 0 ) over an interval P m, ensuring all their alternatives are scheduled, then schedule max # of primaries that will fit in the remaining time Note: The alternative is never scheduled to run before the corresponding primary. 2. Concatenate two S 0 schedules to form one schedule of length 2P m. For all level-1 tasks (a) schedule alternatives; drop some level-0 primaries (first those with longest runtime limit) if insufficient space to fit all alternatives (b) see if any level-1 primaries can be scheduled in ascending order of their runtime limits (c) concatenate 2 copies of the resultant schedule, If a primary completes successfully, its corresponding alternative is not needed, thus reclaiming the time allotted to it.

12 Example T 1 T 2 T 3 T 4 T 5 runtime limit l(i) WC alternative exec time α(i) period P(i) A(2) Pr(1) A(1) A(3) Pr(1) A(1) A(3) Pr(1) A(1) A(2) A(2) A(3) A(4) A(2) Pr(1) A(1) A(3) Pr(5) A(2) A(3) A(1)

13 Increased Reward w/ Increased Service (IRIS) Tasks Examples: computation of π, Newton s alg, chess-playing,... R(x) = 0 if x < m r(x) if m x o + m r(o + m) if x > o + m Optimization problem: schedule tasks so that the reward is maximized subject to the requirements that mandatory portions of all tasks are completed NP-complete when there is no restriction on release times, deadlines, and reward functions Consider special cases

14 Identical Linear Reward Functions Reward: R i (x) = 0 if x < m i x m i if m i x o i + m i o i if x > o i + m i Optimization alg: Task set, T = {T 1,..., T n } M = {M 1,..., M n }; O = {O 1,..., O n }. 1. Run EDF on T to generate schedule S t. If feasible, done stop 2. Run EDF on M to generate schedule S m ; if infeasible thent is infeasible stop. Let a 0 = first task begins execution in S m a i = i-th instant in S m when the scheduled task changes, or processor becomes idle. τ j = task in S m that executes in [a j, a j+1 ] L t (j)(l m (j)) = total execution time given, after a j, to τ j in S t (S m ) 3. For j = k 1 to 0 (where k = # of a i s) if L m (j) > L t (j) then modify S t by assigning L m (j) L t (j) in [a j, a j+1 ] to τ j reducing time assigned to other tasks in [a j, a j+1 ] by L m (j) L t (j) and update L t (1),..., L t (j) accordingly.

15 Example T i m i o i r i D i T 4 misses deadline S t0 T1 T2 T3 T 4 T S m M M M 3 M a =0, a =1, a =2, a =5, a =11, k= S t1 T1 T 4 T 2 T3 T

16 Other IRIS Reward Functions Non-identical linear reward functions R i (x) = 0 if x < m i w i (x m i ) if m i x o i + m i w i o i if x > o i + m i w 1 w 2 w n : nondecreasing order of weights 0/1 reward functions (no partial reward) R i (x) = { 0 if x < mi + o i 1 if x o i + m i Non-identical concave reward functions R i (x) = { fi (x) if 0 x < o i f i (o i ) if x o i