Design of Real-Time Software

Transcription

1 Design of Real-Time Software Analysis of hard real-time tasks under fixed-priority pre-emptive scheduling Reinder J. Bril Technische Universiteit Eindhoven Department of Mathematics and Computer Science System Architecture and Networking Group P.O. Box 513, 5600 MB Eindhoven, The Netherlands January

2 Overview Context Schedulability conditions Basic response time analysis Jitter analysis for periodic tasks Resource sharing Practical factors Concluding remarks References 2

3 Context Overview Basic scheduling model (Recap) Motivation for FPPS Schedulability conditions Basic response time analysis Jitter analysis for periodic tasks Resource sharing Practical factors Concluding remarks References 3

4 A basic scheduling model Model (of a system in general): Abstraction (of that system) leaving out details irrelevant to a given set of criteria preserving the properties of interest Scheduling model (for Real-Time SW) explicitly addresses relevant issues in real-time systems...but must be mapped eventually onto an execution environment OS, hardware, run-time system,... 4

5 Event: Reinder J. Bril, A basic scheduling model indicates a state change requiring a timely response, i.e. neither too early nor too late; external (e.g. at RT(C)S boundary), internal (e.g. a task triggering another), or timed. Task: actions in response of event. Processor: executes a single task at the time. Schedule: assignment of tasks to processor; set on n tasks 1 n ; : R {0, 1, n}, where (t) = 0 means idle. 5

6 Schedule: example Schedule (t) of three independent periodic tasks 1, 2, 3, where 1 has highest priority and 3 lowest priority. 6

7 Events: implicit Tasks: Initial basic assumptions Reinder J. Bril, released strictly periodically, elastically, or sporadically; independent; no self-suspension. Processor: only one Scheduling algorithm: fixed-priority pre-emptive scheduling (FPPS), i.e. processor is used to execute the highest priority task that has work pending; non-idling; tasks have unique priorities; Overhead of scheduling and context switching is ignored. 7

8 Basic model for a task Task i : sequence of jobs ik with k Z. Basic timing notions (taken from R) for ik activation time a ik ; start time s ik ; finalization time f ik ; Derived timing notions response (or active) interval [a ik, f ik ); response time R ik = f ik a ik ; phasing i = a i,0 ; Reinder J. Bril, r.j.bril@tue.nl 8

9 Basic model for a task continued (Best-case and worst-case) characteristics: deadline: R ik [BD i, WD i ], where BD i R + {0}, WD i R + ; computation time C ik [BC i, WC i ], where BC i, WC i R + ; inter-arrival time (or period): strictly periodic task (i.e. without activation jitter): period T i R + ; elastic task: best-case period BT i and worst-case period WT i, where WT i < BT i ; sporadic task: worst-case period WT i (best-case period BT i ) deadlines and inter-arrival times: 0 BD i WD i WT i Notation: we will also use WT i and BT i for periodic tasks. 9

10 Relative deadlines: Deadlines best-case BD i (typically assumed to be zero); worst-case WD i ; Absolute deadlines: best-case bd ik = a ik + BD i ; worst-case wd ik = a ik + WD i ; Reinder J. Bril, r.j.bril@tue.nl 10

11 Overview of basic assumptions Single processor; Set on n tasks 1 n : (released strictly periodically: a ik = i + kt i ); (independent: no resource sharing and no precedence relations); arbitrary phasing; `ready to run upon activation; no self-suspension; a job does not start before previous job completed (f i,k-1 s ik ); hard deadlines (i.e. BD i R ik WD i ) and WD i WT i. Scheduling: (FPPS and unique priorities); (instantaneous pre-emption and non-idling); overhead of context switching and task scheduling is ignored; Notational convenience: tasks are given in order of decreasing priority, i.e. 1 has highest priority and n has lowest priority. 11

12 De-facto standard Motivation for FPPS Supported by commercial RTOS Rate monotonic analysis (RMA): Reinder J. Bril, Adopted by leading companies and institutions [Obenza 94]: Boeing, Honeywell, IBM, McDonnell Douglas, NASA, ; IBM research, CMU/SEI. Usage: From simple control applications to large defense and aero-space applications. Documentation [Klein et al 93]: Practitioner s Handbook by CMU/SEI (KAP). 12

13 Overview Context Schedulability conditions Exact, necessary, and sufficient conditions; Recapitulation of [Liu and Layland 73]. Basic response time analysis Jitter analysis for periodic tasks Resource sharing Practical factors Concluding remarks References 13

14 Schedulability conditions Requirement Req: all jobs of all tasks of must meet their deadline constraints, i.e. BD i R ik WD i for all i and all k. Derived notions for task i worst-case response time WR i def WR i supr, k where is the phasing of the set ; critical instant: a (hypothetical) instant that leads to WR i ; best-case response time BR i def, k BR i inf R ( ) optimal instant: a (hypothetical) instant that leads to BR i. ik ik ( ) 14

15 Schedulability conditions Re-phrased requirement Req: BD i BR i WR i WD i for all i and all k; note: a best-case part: BReq = BD i BR i, and a worst-case part: WReq = WR i WD i. Types of conditions: Exact condition EC: EC Req Sufficient condition SC: SC Req Necessary condition NC: NC Req Specializations for best-case and worst-case, e.g. best-case sufficient condition BSC: BSC BReq Examples: BC i BD i : best-case sufficient condition; WC i WD i : worst-case necessary condition. 15

16 Recapitulation of [Liu and Layland 73] Assumptions (additional): fixed computation time and inter-arrival time: i.e. C i = BC i = WC i and T i = BT i = WT i ; best-case deadlines ignored: i.e. assume BD i = 0; worst-case deadlines equal to periods: i.e. WD i = T i. Utilization U i = C i / T i : utilization factor of task i ; U = U i : (processor) utilization factor. Necessary condition: U 1. Th. 1: A critical instant occurs upon a simultaneous release of a task with all its higher priority tasks. 16

17 Critical Instant Reinder J. Bril, Task 1 Task 2 Task 2 is preempted by a single activation of the higher priority task 1. C 2 + C 1 Task 1 Task 2 The interference increases when the activation of task 1 is advanced. C 2 + 2C time 17

18 Recapitulation of [Liu and Layland 73] Th. 2: rate monotonic priority assignment (RMA) is an optimal fixed priority assignment, i.e. if a set of tasks can be scheduled based on a fixed priority assignment, then can be scheduled based on RMA. Note: Th. 2 only holds for arbitrary phasing; see [Goossens et al 97]. Sufficient condition: U n(2 1/n 1) also referred to as `Liu and Layland bound LL(n); adding tasks increases U and decreases `RHS ; converges to ln(2) ( 0.69) for n ; for the proof, see [Devillers et al 00]. 18

19 Alternative utilization bound Sufficient condition `Hyperbolic bound HB(n) [Liu 00]* HB( n) : i1 improves LL(n), i.e. LL(n) HB(n); the bound is `tight ; Note: n ( U i 1) 2 adding tasks increases `LHS. * independently conceived by [Bini et al 01]. 19

20 Exercises Reinder J. Bril, Consider the exercises C.1, C.2, and C.3 of the module Implementing the real-time task model. Determine the utilization of the task-sets. Determine whether or not the necessary condition U 1 holds for the task-sets. Determine whether or not the sufficient conditions for the task-sets holds: LL-bound; Hyperbolic bound. 20

21 An example (leading) Task set consisting of 3 tasks: Task T C U Notes: RM priority assignment and D i = T i (RMS); Necessary condition: U 1 + U 2 + U 3 = , hence could be schedulable; Sufficient condition: LL(n): U n (2 1/n 1): U 1 + U 2 = 0.88 > `LL(2) 0.83, therefore U 1 + U 2 +U 3 > `LL(3), hence another test required; HB(n): (U 1 + 1)(U 2 + 1) 2.05 > 2, therefore `HB(3) > 2, hence another test required. 21

22 Overview Context Schedulability conditions Basic response time analysis Worst-case response time analysis Best-case response time analysis Jitter analysis for periodic tasks Resource sharing Practical factors Concluding remarks References 22

23 Basic response time analysis Assumptions ( reset ) deadlines at most equal to periods: WD i WT i ; worst-case and best-case characteristics; arbitrary (but fixed and unique) priority assignments. Worst-case response time analysis (Critical instant); Techniques: timeline + calculation Best-case response time analysis Introduction Optimal instant; Techniques: timeline + calculation. 23

24 Worst-case response time analysis Critical instant for task i : i assumes its WR i. A critical instant for task i : Task i is released simultaneously with all tasks with a higher priority. The highest amount of pre-emption of a task is found after a simultaneous release of higher priority tasks. General for a set of tasks. Note: A rather than the, because there may be more instants for which i assumes its WR i. 24

25 WCRT Techniques Time line: Task 1 Task 2 Task time WR 1 = 3 WR 2 = 17 WR 3 = 56 25

26 WCRT Techniques Calculation: Recursive equation for task i : x x WC i WC ji WTj WR i is the smallest positive solution Assume a task j with a higher priority than i ; x/wt j denotes the maximum number of preemptions of task i in an interval [0, x) by task j ; x/wt j WC j denotes the maximal preemption time of task i in an interval [0, x) by task j. Intuition: LHS: amount of time available (or provided) in [0, x); RHS: max. amount of time requested in [0, x) by i and j < i j. j 26

27 WCRT Techniques Calculation: Iterative procedure: WR WR (0) i ( k 1) i WC Stopped when either: i WC i ji WC ji the same value is found for two successive iterations; or the deadline WD i is exceeded (hence, not schedulable). All intermediate values are at most equal to WR i ; Terminates when j<i U j < 1. See [Harter 84], [Joseph et al 86] or [Audsley et al 91]. j WR WT ( k ) i j WC j 27

28 Example for task 3 : WCRT Techniques WR 3 (0) = C 3 + j < 3 C j = = 19 Reinder J. Bril, r.j.bril@tue.nl WR 3 (1) = C 3 + j < 3 WR 3 (0) /T j C j = / /1911 = = 22 WR 3 (2) = / /1911 = = 36 WR 3 (3) = / /1911 = = 39 WR 3 (4) = / /1911 = = 50 WR 3 (5) = / /1911 = = 53 WR 3 (6) = / /1911 = = 56 WR 3 (7) = / /1911 = = 56 Because WR 3 (6) = WR 3 (7) = 56 D 3 = T 3, WR 3 =

29 WCRT Techniques Calculation (visualization): Task Task Task WR (0) = 19 WR (6) 3 = WR (7) 3 = = 56 WR (1) 3 = = 22 WR 3 (2) = = time 29

30 WCRT Exercises Reinder J. Bril, Consider the exercises C.1, C.2, and C.3 of the module Implementing the real-time task model. Determine the worst-case response times of the tasks under FPPS. 30

31 WCRT Exercise* Task set consisting of 3 tasks: Task WT = WD WC WU Reinder J. Bril, r.j.bril@tue.nl Determine `worst-case schedulability of. Draw a time line with a critical instant for. Explain, using the time line, why 25 and 27 are also solutions for the recursive equation for WR 3. 31

32 Best-case response time analysis See [Redell et al 02] or [Bril et al 04]. Motivation Best-case and worst-case notions are duals: Best-case: earliest, shortest, or minimal; Worst-case: latest, longest, or maximal; Best-case deadline: response no earlier than BD May simplify design and reduce cost: No need to delay (i.e. buffer) output artificially; Desirable for analysis in distributed systems: but also applies to single-processor systems; examples: airbag, engine control. 32

33 BCRT Optimal instant Reinder J. Bril, Optimal instant for task i : i assumes its BR i. An optimal (or favourable) instant: Job ik ends simultaneously with the release of all tasks with a higher priority, and ik s release time is equal to its start time, i.e. a ik = s ik. The lowest amount of pre-emption of a task is found before a simultaneous release of higher priority tasks. Specific for each task! Note: An rather than the, because there may be more instants for which i assumes its BR i. 33

34 BCRT Techniques Time line: optimal instant for 2 Task 1 Task 2 T 2 BR 2 = 14 f 2,k time 34

35 BCRT Techniques Time line: optimal instant for 3 Task 1 Task 2 Task 3 T 3 BR 3 = 22 f 3,k time 35

36 BCRT Techniques Calculation: Recursive equation for task i : BR i is the largest positive solution Assume a task j with a higher priority than i ; x/bt j 1 denotes the minimal number of preemptions of task i in an interval (0, x) by task j ; (x/bt j 1)BC j denotes the minimal preemption time of task i in an interval (0, x) by task j. Intuition: x x BC i BC ji BT 1 j LHS: amount of time available (or provided) in (0, x); RHS: min. amount of time requested in (0, x) by i and j < i j. j 36

37 Calculation: Iterative procedure: BR BR (0) i ( k 1) i Stopped when: BCRT Techniques WR i BC i ji BR BT ( k ) i the same value is found for two successive iterations; or the deadline BD i is exceeded (hence, not schedulable). All intermediate values are at least equal to BR i. j 1BC j 37

38 Example for task 3 : Assume BD i = 0. BR 3 (0) = WR 3 = 56 BCRT Techniques Reinder J. Bril, r.j.bril@tue.nl BR 3 (1) = C 3 + j < 3 BR 3 (0) /T j 1C j = / / = = 42 BR 3 (2) = / / = = 39 BR 3 (3) = / / = = 36 BR 3 (4) = / / = = 25 BR 3 (5) = / / = = 22 BR 3 (6) = / / = = 22 Because BR 3 (5) = BR 3 (6) = 22 BD 3 = 0, BR 3 =

39 BCRT Techniques Calculation (visualization): Task Task Task BR (0) 3 = WR 3 = = 56 BR BR(1) (5) 3 3 = = 5 BR (6) = = = 22 f 3,k time BR 3 (2) = = 39 39

40 BCRT Exercises Reinder J. Bril, Consider the exercises C.1, C.2, and C.3 of the module Implementing the real-time task model. Determine the best-case response times of the tasks under FPPS. Can all tasks assume their best-case response times for a single phasing? 40

41 BCRT Exercise* Task set consisting of 3 tasks: Task BT BC = BD BU Determine the best-case response times BR i for all three tasks of. Draw a time line with an optimal instant for 3. Explain, using the time line, why 3 is also a solution for the recursive equation for BR 3. 41

42 Overview Context Schedulability conditions Basic response time analysis Jitter analysis for periodic tasks Distributed systems analysis Response jitter Activation jitter Finalization and activation jitter Resource sharing Practical factors Concluding remarks References 42

43 Distributed systems analysis Finalization of a task f on one processor activates a task a on another processor: Periodic tasks: Finalization jitter FJ: variation in finalization times; Activation jitter AJ: variation in activation times (e.g. output of one task triggers a next task); Example: multimedia in a networked environment. Elastic tasks: Minimal inter-finalization time of f determines WT a of task a. Maximal inter-finalization time of f determines BT a of task a. Sporadic tasks: Minimal inter-finalization time of f determines WT a of task a. 43

44 Distributed systems analysis Example: f CPU-1 a, k CPU-2 f triggers a FJ f causes AJ a, which influences response times of both a and k Goal jitter analysis: Determine schedulability in the context of jitter, hence, determine response (or finalization) times. Note: strictly spoken, the variations in the delay of the messages on the bus should be taken into account as well 44

45 Advanced exercises Reinder J. Bril, Proof that the type of a is determined by f, e.g. a is an elastic task if f is an elastic task. For both an elastic and a sporadic task f, express WT a in terms of WT f, WR f, and BR f. For an elastic task, express BT a in terms of BT f, WR f, and BR f. How about BT a for a sporadic task? 45

46 Jitter analysis for periodic tasks Types of (absolute) jitter (recap): Response jitter RJ i : variations in response times; Activation (or release) jitter AJ i : variation in release times (e.g. output of one task triggers a next task); Finalization (or end) jitter FJ i : variation in end times; 46

47 Response jitter Response jitter RJ i of a task i : RJ i def sup( R, k, l ik ( ) R il ( )) A bound on response jitter RJ i WR i BR i because WR i and BR i are not necessarily assumed for the same phasing 47

48 Example (leading): WR 2 = 17, BR 2 = 14; Response jitter Reinder J. Bril, r.j.bril@tue.nl WR 2 and BR 2 both assumed for a single phasing, hence, FJ 2 = WR 2 BR 2 = 3, and the bound for RJ 2 is therefore tight. Task 1 Task time 48

49 Activation jitter of periodic tasks Assumptions: revised assumption: WD i WT i AJ i. lifted restrictive assumption: sup( a a ( k l) T ) k, l ik il Worst-case response times: See [Audsley et al 93] or [Tindell et al 94]; Critical instant revisited: Task i is released simultaneously with all tasks with a higher priority and all tasks with a higher priority experience a maximal release delay at that simultaneous release, and a minimal release delay at subsequent releases. Hence, a maximal pre-emption of i occurs. i AJ i 49

50 Activation jitter of periodic tasks New example: Task T C AJ T 1 = 9 AJ 1 = Task 1 Notes: WR 2 is independent of AJ 2 ; WR 2 increases 3 due to AJ 1. Task 2 20 T 2 = time 50

51 Activation jitter of periodic tasks Worst-case response times: Recursive equation for task i : x WC i ji x AJ WTj j WC j Where AJ j is the activation jitter of j. WR i is the smallest positive solution of the equation. Iterative procedure: Similar to the case without jitter. Note: equation also holds for elastic and sporadic tasks! Because AJ j = 0 for those tasks. 51

52 Activation jitter of periodic tasks Best-case response times: Optimal instant revisited: Job ik ends simultaneously with the release of all tasks with a higher priority, and ik s release time is equal to its start time, and all tasks with a higher priority experience a maximal release delay at that simultaneous release, and a minimal release delay at previous releases. Hence, a minimal pre-emption of i occurs. 52

53 Activation jitter of periodic tasks Same (new) example: Task T C AJ T 1 AJ Task 1 T 2 Notes: Task 2 BR 2 is independent of AJ 2 ; BR 2 does not change here. 14 f 2 time 53

54 Activation jitter of periodic tasks Best-case response times: Recursive equation for task i : x j BCi 1 ji BT j Where AJ j is the activation jitter of j, and w + = max{w,0}. BR i is the largest positive solution of the equation. Iterative procedure: x AJ Similar to the case without jitter. Note: equation also holds for elastic and sporadic tasks! Because AJ j = 0 for those tasks. BC j 54

55 Finalization and activation jitter Finalization jitter FJ i of a periodic task i : FJ i def sup(, k, l f ik ( ) f il ( ) ( k l) T i ) A bound on finalization jitter: FJ i AJ i WR i BR i because WR i and BR i are not necessarily assumed for the same phasing. Exercise: derive this latter bound 55

56 Response, finalization and activation jitter Example: T 1 AJ 1 RJ 2 = WR 2 BR 2 = = 6 FJ 2 = AJ 2 + WR 2 BR 2 = = 13 Task 1 T 2 AJ 2 RJ 1 = WR 1 BR 1 = 0 FJ 1 = AJ 1 = Task time BR 2 FJ 2 T 2 56

57 Advanced exercise Reinder J. Bril, Define worst-case processor utilization WU in the context of periodic tasks with activation jitter. Can you describe a (worst-case) necessary condition using WU? What about the analysis? This lecture considered elastic, periodic, and sporadic tasks. What is the effect of modeling a periodic task with jitter as an elastic task? a periodic task with jitter as a sporadic task? 57

58 Overview Context Schedulability conditions Basic response time analysis Jitter analysis for periodic tasks Resource sharing Practical factors Concluding remarks References 58

59 Response-time analysis Worst-case response time analysis: Blocking time B i : Longest time i can be blocked by a task with a lower priority. Depends on the resource access protocol. Includes blocking due to non-interrupt-able code of system-calls. Recursive equation for task i : x AJ x Bi WC i ji WTj WR i is the smallest positive solution Best-case response time analysis: Yet unknown. j WC j 59

60 Overview Context Schedulability conditions Basic response time analysis Jitter analysis for periodic tasks Resource sharing Practical factors From event to task activation Context switches Interrupts Concluding remarks References 60

61 From external event to task activation Steps: External event occurs; Detection by sensor; Interrupt generated by sensor may take some time before the bus is free Interrupt arrival at CPU may take some time before the interrupt is handled Immediate interrupt service may be pre-empted by higher priority interrupts Activation of the scheduler may be delayed to the next clock-tick Activation of the task hence, both a delay and potential jitter between the arrival of the external event and activation of the task! 61

62 Context switches Reinder J. Bril, Question: how many running jobs can a job pre-empt? Answer: at most 1! context switch i j Let CS denote the context-switch time of the system, i.e. max time the system spends on a context switch; optionally including time of the scheduler to service the event interrupt that triggered the context switch <dummy> 62

63 Context switches Extending the analysis: Replace C j by C j + 2CS; Replace C i by C i + 2CS; Questions: Reinder J. Bril, r.j.bril@tue.nl Can these extensions be applied for the necessary condition, sufficient condition, and response-time analysis? Motivate your answer. Can you ignore the context switch out-of a task for the response time analysis of that task, i.e. use C i + CS rather than C i + 2CS? Motivate your answer. 63

64 External interrupts The immediate interrupt service Let will pre-empt a running task ; even when the sporadic task handling the interrupt has a lower priority than. T k : the minimum inter-arrival time of the interrupt corresponding with sporadic task k ; s : the set of sporadic tasks; IH k : the cost of handling that interrupt. Extension of the recursive equation k s x T k IH k

65 Clock interrupt Simple (basic) approach: similar to external interrupts, i.e. extension of the recursive equation Refinement: x IH Tclk distinguish between clk a fixed cost to serve the clock interrupt; additional costs to move tasks from the waiting queue to the ready queue Question: How to model the additional costs?

66 Exercises Measure the overhead of the tick interrupt handler. 66

67 Overview Context Schedulability conditions Basic response time analysis Jitter analysis for periodic tasks Resource sharing Practical factors Concluding remarks References 67

68 Concluding remarks Many of the restrictive assumptions have been lifted, e.g. other dependencies between tasks e.g. precedence relations; tasks with varying priorities [Harbour et al 91]; arbitrary deadlines [Klein et al 93]; tasks with offsets; tasks that suspend themselves; limited-pre-emptive scheduling [Regher 02], [Bril et al 07]. Unlike worst-case response time analysis, best-case response time analysis has not been addressed yet for any of these lifted assumptions! Further elaboration falls outside the scope of this lecture, however. 68

69 Concluding remarks Be aware: The analysis presented has the explicitly stated assumptions as preconditions! For many situations, only looking at the first job upon a critical instant is not sufficient, even when deadlines are at most equal to periods. Instead, the response times of all jobs in a so-called busy (or level-i active) period have to be examined! Examples: tasks with varying priorities [Harbour et al 91]. preemption thresholds [Regehr 02]; Controller Area Network (CAN), see [Davis et al 07]; fixed-priority scheduling with deferred preemption (FPDS), see [Bril et al 07]. 69

70 References - I [Audsley et al 91] N.C. Audsley and A. Burns and M.F. Richardson and A.J. Wellings, Hard Real-Time Scheduling: The Deadline Monotonic Approach, In: Proc. 8 th IEEE Workshop on Real-Time Operating Systems and Software (RTOSS), pp , May [Audsley et al 93] N.C. Audsley and A. Burns and M.F. Richardson and K. Tindell and A.J. Wellings, Applying New Scheduling Theory to Static Priority Pre-emptive Scheduling, Software Engineering Journal, 8(5): , [Bini et al 01] E. Bini and G. Buttazzo and G. Buttazzo, A hyperbolic bound for the rate monotonic algorithm, In: Proc. 13 th Euromicro Conference on Real-Time Systems (ECRTS), pp , June [Bril et al 04] R.J. Bril, E.F.M. Steffens, and W.F.J. Verhaegh, Bestcase response times and jitter analysis of real-time tasks, Journal of Scheduling, 7(2): , [Bril et al 07] R.J. Bril, J.J. Lukkien, and W.F.J. Verhaegh, Worst-case response time analysis of real-time tasks under fixed-priority scheduling with deferred preemption revisited, In: Proc. 19th Euromicro Conference on Real-Time Systems (ECRTS 07), pp , July

71 References - II [Davis et al 07] R.I. Davis and A. Burns and R.J. Bril and J.J. Lukkien, Controller Area Network (CAN) Schedulability Analysis: Refuted, Revisited and Revised", Real-Time Systems, ISSN (online), January 2007, ISSN (print), 35(3): , April [Devillers et al 00] R. Devillers and J. Goossens, Liu and Layland's schedulability test revisited, Information Processing Letters, 73(5-6): , March [Goossens et al 97] J. Goossens and R. Devillers, The non-optimality of the monotonic priority assignments for hard real-time offset free systems, Real-Time Systems, 13(2): , September [Harbour et al 91] M. Gonzalez Harbour, M.H. Klein, J.P. Lehoczky Fixed priority scheduling of periodic tasks with varying execution priority. In: Proc. 12 th IEEE Real-Time Systems Symposium, pp , [Harter 84] P. Harter, Response times in level-structured systems, Department of Computer Science, University of Colorado, USA, Tech. Rep. CU-CS , [Joseph et al 86] M. Joseph and P. Pandya, Finding Response Times in a Real-Time System, The Computer Journal, 29(5): ,

72 References - III [Klein et al 93] M.H. Klein, T. Ralya, B. Pollak, R. Obenza, and M. González Harbour, A Practitioner s Handbook for Real-Time Analysis: Guide to Rate Monotonic Analysis for Real-Time Systems, Kluwer Academic Publishers (KAP), [Liu and Layland 73] C.L. Liu and J.W. Layland, Scheduling Algorithms for Multiprogramming in a Real-Time Environment, Journal of the ACM, 20(1): 46-61, January [Liu 00] J.W.S. Liu, Real-Time Systems, Prentice Hall, [Obenza 94] R. Obenza, Guaranteeing real-time performance using RMA, Embedded Systems Programming, pp , [Redell et al 02] O. Redell and M. Sanfridson, Exact best-case response time analysis of fixed priority scheduled tasks, In: Proc. 14 th Euromicro Conference on Real-Time Systems (ECRTS), pp , June [Regehr 02] J. Regehr, Scheduling tasks with mixed pre-emption relations for robustness to timing faults, In: Proc. 23 rd IEEE Real-Time Systems Symposium, pp , [Tindell et al 94] K.W. Tindell and A. Burns and A.J. Wellings, An Extendible Approach for Analyzing Fixed Priority Hard Real-Time Tasks, Real-Time Systems, 6(2): ,