Slack and Lateness D i i t a i s i f i d i R R i slack = d - PDF document

Slack and Lateness D i  i t a i s i f i d i R R i slack = d slack i = d i - f i f D i lateness L i = f i - d i  i t a i s i d i f i R i Task model Tasks and jobs  Sequence of instructions that in the absence of A task running several times on different input other activities is continuously executed by the data generates a sequence of instances (jobs): processor until completion. Task  i activation time C i Job 1 Job 2 Job 3 start time  i,1  i,2  i,3 t a i s i f i computation Response time R i C i  i time t a i,1 a i,k a i,k+1 finishing time Real-Time Task Activation modes  It is a task characterized by a timing constraint on  Time driven its response time, called deadline: The task is automatically activated by the relative deadline D i operating system at predefined time instants.   i t  Event driven a i s i f i d i The task is activated at an event arrival or by absolute deadline response time R i ( d i = a i + D i ) explicitly invocating a system call. A real-time task  i is said to be feasible if it completes within its absolute deadline, that is, if f i  d i , o equivalently, if R i  D i 1

Types of tasks Assumptions  Aperiodic  Implicit deadlines Activated by events. Task activation times are  i D i = T i unknown and unbounded.  Sporadic  Constrained deadlines Activated by events Activated by events. Task activation times are Task activation times are  i unknown and bounded: consecutive activations are D i ≤ T i separated by a minimum interarrival time.  Arbitrary deadlines  Periodic Activated by a timer. Task activation times are Deadlines can be less than, known and bounded: Consecutive jobs are greater than, or equal to periods separated by a constant interval (period). Analysis under fixed priority Aperiodic/Sporadic task interrupt Let  = {  1 , …,  n } be a set of n periodic tasks. computation time C i Implicit deadlines job  ik     n C i C i C i … … Utilization-based test     i    only only U U n 2 / 2 / 1 1 n n 1 1 [Liu & Layland, 1973] i sufficient t  a i,1 a i,k a i,k+1 i 1  Aperiodic: (C i , D i ) a i,k+1 > a i,k n   Hyperbolic Bound    only U 1 2 a i,k+1  a i,k + T i [Bini - Buttazzo 2 , 2001] i (C i , D i , T i ) sufficient  Sporadic:  i 1 minimum interarrival time 11 Analysis under fixed priority Periodic task input Constrained deadlines C i computation time C i U i = T i Response Time Analysis necessary  i R i  D i output and [Audsley et al., 1993] timer ( period T i ) utilization factor sufficient sync job  ik j T i T i ik Iterative solution: C i  i i  ( 0 )  R C a i,1 =  i a i,k a i,k+1 t i k iterate while  k 1  ( s  1 ) task phase i 1 R  ( s )  ( s  1 ) R R ( s )   R C i C a i,k =  i + (k  1) T i i i i i k T  i (  i , C i , T i , D i )  k 1 k d i,k = a i,k + D i 12 2

Analysis under EDF Workload Analysis under FP Constrained deadlines Arbitrary deadlines necessary necessary Processor Demand  t  D dbf ( t )  t Workload Analysis and  i  t  A i W i ( t )  t and [Baruah et al., 1990] sufficient [Lehoczky et al., 1989] sufficient  dbf ( t ) is called demand bound function and denotes the  W i ( t ) is called workload in (0, t ] at priority level P i and computation time of tasks with deadlines ≤ t computation time of tasks with deadlines ≤ t denotes the computation time requested in (0 t ] by tasks denotes the computation time requested in (0, t ] by tasks with priority higher than or equal to P i n t + T i  D i Σ dbf ( t ) = C k T i i –1 t Σ i = 1 C k W i ( t ) = C i + T k  D is the set of points where the test has to be performed k =1 D = { d k | d k  L b } L b = max { D max , min (H, L * ) }  A i is the set of points where the test has to be performed, equal to the activation times ≤ D i , including D i    n  ( T D ) U i i i H = lcm(T 1 , … , T n ) L * i 1  1 U 13 16 EDF example Workload Analysis under FP  1 1 task C i T i D i T i - D i U i 1 1 1 25 A task  i is feasible iff:     U  1 1 4 2 2 1/4 4 2 7 28  2 5  t ≤ D i : W i (t) ≤ t  2 3 6 5 1 1/2 H = 4  3  7 = 84  3 2 14 9 5 1/7 W 2 (t) 18 2 1 5 16      n  ( T D ) U 14 12 28 4 2 7 i i i *     L i 1 16 12  3 1 U 7 3 10 28 8 6 L b = max ( 9, min (84,16) ) = 16 D max = 9 4 2 0 D = { d k | d k  L b } = {2, 5, 6, 9, 10, 11, 14} 0 2 4 6 8 10 12 14 16 18 20 14 17 EDF example Workload Analysis under FP  1 1  1 1 A task  i is feasible iff:  2 3  2 7  t ≤ D i : W i (t) ≤ t  3 2 W 2 (t) 9 14 23 18 dbf(t) ( ) 16 16 14 14 12 12 10 10 8 8 6 6 4 4 2 2 0 0 0 2 4 6 8 10 12 14 16 18 20 22 24 0 2 4 6 8 10 12 14 16 18 20 L b 15 18 3

Workload Analysis under FP Workload Analysis under FP  1  1 3 1 A task  i is feasible iff:  2 2  t ≤ D i : W i (t) ≤ t  2 3 miss  3 2 W 2 (t) 9 14 23 18 W 3 (t) 3 ( ) 16 16 14 14 12 12 10 10 8 8 6 6 4 4 2 2 0 0 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 A 3 = {4, 6, 8, 9} 19 22 Workload Analysis under FP Workload Analysis under FP Theorem [Bini-Buttazzo, 2002] AND OR  1 1  2 3 A task set is feasible under fixed priorities iff: miss W i ( t ) ≤ t  3 2 t  P i–1 (D i ) i = 1...n 9 14 23 W 3 (t) 3 ( ) 16 14 Where P i (t) is defined by: 12 10 8 P 0 (t) = { t } In an unfeasible schedule: 6  t ≤ D i : W i (t) ≤ t t 4 P i (t) = P i-1 U P i-1 (t) T i 2 T i 0 0 2 4 6 8 10 12 14 16 18 20 20 23 Workload Analysis under FP Example  1 Theorem [Lehoczky-Sha-Ding, 1989] 3 6 9 12 15 18 21  2 A task set is feasible under fixed priorities iff: 8 16 24  3  i = 1… n  t ≤ D i : W i ( t ) ≤ t 20  1 P (t) = { t } P 0 (t) = { t } Problem Problem t How many points need to be tested? P i (t) = P i-1 T i U P i-1 (t) T i  1 : P 0 (D 1 ) = P 0 (3) = {3} When checking  i feasibility, we need to verify W ( t ) ≤ t for D i and for all release times r hk ≤ D i of jobs  hk with priority P h ≥ P i ,  2 : P 1 (D 2 ) = P 0 (6) U P 0 (8) = {6, 8} that is, for all t in A i :  3 : P 2 (D 3 ) = P 1 (16) U P 1 (20) T i A i = r hk r hk = kT h , h = 1 ... i , k = 1 ... U D i = [ P 0 (15) U P 0 (16)] U [ P 0 (18) U P 0 (20)] T h = {15, 16, 18, 20} 21 24 4

Example Example  1  1 3 6 9 12 15 18 21  2  2 8 16 24  3  3 20  2 P (t) = { t } P 0 (t) = { t } N N N N N N Y Y t P i (t) = P i-1 T i U P i-1 (t)  1 T i  1 : P 0 (D 1 ) = P 0 (3) = {3}  2  2 : P 1 (D 2 ) = P 0 (6) U P 0 (8) = {6, 8}  3  3 : P 2 (D 3 ) = P 1 (16) U P 1 (20) = [ P 0 (15) U P 0 (16)] U [ P 0 (18) U P 0 (20)] N N N Y = {15, 16, 18, 20} 25 28 Example Example  1  1 3 6 9 12 15 18 21  2  2 8 16 24  3  3 20  3 P 0 (t) = { t } P (t) = { t } N N N N N N Y Y t P i (t) = P i-1 T i U P i-1 (t)  1 T i  1 : P 0 (D 1 ) = P 0 (3) = {3}  2  2 : P 1 (D 2 ) = P 0 (6) U P 0 (8) = {6, 8}  3  3 : P 2 (D 3 ) = P 1 (16) U P 1 (20) = [ P 0 (15) U P 0 (16)] U [ P 0 (18) U P 0 (20)] N N N Y = {15, 16, 18, 20} 26 29 Workload Analysis under FP Workload Analysis under FP  1  1  2  2 t 1 t 2  3 W i (t) ≤ t i ( ) t  P i–1 (D i ) i = 1...n P 2 (D 3 ) i = 1 C 1 ≤ T 1 P i-1 (D i ) is the minimum set of points t 1 C 2 + C 1 ≤ t 1 for checking the feasibility of  i . C 2 + 2C 1 ≤ 2 T 1 T 1 i = 2 t 2 C 2 + 3C 1 ≤ T 2 C 2 + C 1 ≤ t 2 T 1 27 30 5

Workload Analysis under FP C 2 C 2 + 3C 1 ≤ T 2 T 2 FT 1 C 2 + 2C 1 ≤ 2T 1 C 1 T 2 – FT 1 T 2 T 1 F+1 31 Workload Analysis under FP C 2 C 2 + (F+1)C 1 ≤ T 2 T 2 OR FT 1 C 2 + FC 1 ≤ FT 1 C 1 T 2 – FT 1 T 2 T 1 F+1 32 Analysis summary Under EDF (Processor Demand Criterion):  t  D dbf ( t )  t n t + T i  D i Σ dbf ( t ) = C k T T i i = 1 Under Fixed Priorities (Workload Analysis):      i 1 ,..., n t A : W i ( t ) t i t i –1 Σ W i ( t ) = C i + C k T k k =1 6