Periodic scheduling 05/06/

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1 Periodic scheduling T T or eriodic scheduling, the best that we can do is to design an algorithm which will always find a schedule if one exists. A scheduler is defined to be otimal iff it will find a schedule if one exists. P. Marwedel, Univ. Dortmund, Informatik, /06/7 - -

2 Periodic scheduling Let i be the eriod of task T i, c i be the execution time of T i, d i be the deadline interval, that is, the time between a job of T i becoming available and the time after which the same job T i has to finish execution. l i be the laxity or slack, defined as l i = d i - c i P. Marwedel, Univ. Dortmund, Informatik,

3 Accumulated utilization Accumulated utilization: n i ci i Necessary condition for schedulability (with m=number of rocessors): m P. Marwedel, Univ. Dortmund, Informatik,

4 Indeendent tasks: Rate monotonic (RM) scheduling Most well-known technique for scheduling indeendent eriodic tasks [Liu, 973]. Assumtions: All tasks that have hard deadlines are eriodic. All tasks are indeendent. d i = i, for all tasks. c i is constant and is known for all tasks. The time required for context switching is negligible. or a single rocessor and for n tasks, the following equation holds for the accumulated utilization µ: n c / i n n( ) i i P. Marwedel, Univ. Dortmund, Informatik,

5 Rate monotonic (RM) scheduling - The olicy - RM olicy: The riority of a task is a monotonically decreasing function of its eriod. At any time, a highest riority task among all those that are ready for execution is allocated. Theorem: If all RM assumtions are met, schedulability is guaranteed. P. Marwedel, Univ. Dortmund, Informatik,

6 Maximum utilization for guaranteed schedulability Maximum utilization as a function of the number of tasks: n i lim( n( n c i i / n n( ) / n ) ln() P. Marwedel, Univ. Dortmund, Informatik,

7 Examle of RM-generated schedule T reemts T and T3. T and T3 do not reemt each other. P. Marwedel, Univ. Dortmund, Informatik,

8 Case of failing RM scheduling Task : eriod 5, execution time Task : eriod 7, execution time 4 µ=/5+4/7=34/ ( / -) 0.88 Missed deadline Missing comutations scheduled in the next eriod P. Marwedel, Univ. Dortmund, Informatik,

9 More in-deth: Proof of RM otimality Definition: A critical instant of a task is the time at which the release of a task will roduce the largest resonse time. Lemma: or any task, the critical instant occurs if that task is simultaneously released with all higher riority tasks. Proof: Let T={T,,T n }: eriodic tasks with i: i i +. Source: G. Buttazzo, Hard Real-time Comuting Systems, Kluwer, 00 P. Marwedel, Univ. Dortmund, Informatik,

10 Critical instances () Resonse time of T n is delayed by tasks T i of higher riority: T n T i c n +c i t Delay may increase if T i starts earlier T n T i c n +3c i t Maximum delay achieved if T n and T i start simultaneously. P. Marwedel, Univ. Dortmund, Informatik,

11 Critical instances () Reeating the argument for all i =, n-: The worst case resonse time of a task occurs when it is released simultaneously with all higher-riority tasks. q.e.d. Schedulability is checked at the critical instants. If all tasks of a task set are schedulable at their critical instants, they are schedulable at all release times. P. Marwedel, Univ. Dortmund, Informatik,

12 Proof of the RM theorem Let T={T, T } with <. Assume RM is not used rio(t ) is highest: T c T c t Schedule is feasible if c +c () Define = / : # of eriods of T fully contained in T P. Marwedel, Univ. Dortmund, Informatik,

13 Case : c Assume RM is used rio(t ) is highest: Case *: c (c small enough to be finished before nd instance of T ) T T Schedulable if (+) c + c () t * Tyos in [Buttazzo 00]: < and mixed u] P. Marwedel, Univ. Dortmund, Informatik,

14 Proof of the RM theorem (3) Not RM: schedule is feasible if c +c () RM: schedulable if (+) c + c () rom (): c +c Since : c +c c +c Adding c : (+)c +c +c Since c : (+)c +c +c Hence: if () holds, () holds as well or case : Given tasks T and T with <, then if the schedule is feasible by an arbitrary (but fixed) riority assignment, it is also feasible by RM. P. Marwedel, Univ. Dortmund, Informatik,

15 Case : c > Case : c > (c large enough not to finish before nd instance of T ) T T t Schedulable if c + c (3) c +c () Multilying () by yields c + c Since : c + c c + c Same statement as for case. P. Marwedel, Univ. Dortmund, Informatik,

16 Calculation of the least uer utilization bound Let T={T, T } with <. Proof rocedure: comute least uer bound U lu as follows Assign riorities according to RM Comute uer bound U u by setting comutation times to fully utilize rocessor Minimize uer bound with resect to other task arameters As before: = / c adjusted to fully utilize rocessor. P. Marwedel, Univ. Dortmund, Informatik,

17 Case : c T T t Largest ossible value of c is c = c (+) Corresonding uer bound is U ub c c c c ( ) c c ( ) c ( ) { } is <0 U ub monotonically decreasing in c Minimum occurs for c = P. Marwedel, Univ. Dortmund, Informatik,

18 - 8 - P. Marwedel, Univ. Dortmund, Informatik, 006 Universität Dortmund Universität Dortmund Case : c Largest ossible value of c is c = ( -c ) Corresonding uer bound is: T T t c c c c c c c U ub ) ( { } is 0 U ub monotonically increasing in c (indeendent of c if {}=0) Minimum occurs for c =, as before.

19 - 9 - P. Marwedel, Univ. Dortmund, Informatik, 006 Universität Dortmund Universität Dortmund Utilization as a function of G= / - or minimum value of c : c U ub ; Let G G G G G G G G G G G G G U ub ) ( / / Since 0 G< : G(-G) 0 U ub increasing in Minimum of U ub for min(): = G G U ub

20 Proving the RM theorem for n= end G Uub G Using derivative to find minimum of Uub : duub G( G) ( G ) G G dg ( G) ( G) 0 G ; G Considering only G, since 0 ; G : U lub ( ( ) ) 4 ( ) ( ) 0.83 This roves the RM theorem for the secial case of n= P. Marwedel, Univ. Dortmund, Informatik,

21 Proerties of RM scheduling rom the roof, it is obvious that no idle caacity is needed if =. In general: not required if the eriod of all tasks is a multile of the eriod of the highest riority task, that is, schedulability is then also guaranteed if µ. RM scheduling is based on static riorities. This allows RM scheduling to be used in standard OS, such as Windows NT. A huge number of variations of RM scheduling exists. In the context of RM scheduling, many formal roofs exist. P. Marwedel, Univ. Dortmund, Informatik,

22 ED ED can also be alied to eriodic scheduling. ED otimal for every eriod otimal for eriodic scheduling ED must be able to schedule the examle in which RMS failed. P. Marwedel, Univ. Dortmund, Informatik,

23 Comarison ED/RMS RMS: ED: T not reemted, due to its earlier deadline. P. Marwedel, Univ. Dortmund, Informatik,

24 ED: Proerties ED requires dynamic riorities ED cannot be used with a standard oerating system just roviding static riorities. P. Marwedel, Univ. Dortmund, Informatik,

25 Deendent tasks The roblem of deciding whether or not a schedule exists for a set of deendent tasks and a given deadline is NP-comlete in general [Garey/Johnson]. Strategies:. Add resources, so that scheduling becomes easier 3. Slit roblem into static and dynamic art so that only a minimum of decisions need to be taken at run-time. P. Marwedel, Univ. Dortmund, Informatik,

26 Soradic tasks If soradic tasks were connected to interruts, the execution time of other tasks would become very unredictable. Introduction of a soradic task server, eriodically checking for ready soradic tasks; Soradic tasks are essentially turned into eriodic tasks. P. Marwedel, Univ. Dortmund, Informatik,

27 Resource access rotocols Critical sections: sections of code at which exclusive access to some resource must be guaranteed. Can be guaranteed with semahores S. P(S) V(S) Task Task Exclusive access to resource guarded by S P(S) V(S) P(S) checks semahore to see if resource is available and if yes, sets S to used. Uninterrutable oerations! If no, calling task has to wait. V(S): sets S to unused and starts sleeing task (if any). P. Marwedel, Univ. Dortmund, Informatik,

28 Priority inversion Priority T assumed to be > than riority of T. If T requests exclusive access first (at t 0 ), T has to wait until T releases the resource (time t 3 ), thus inverting the riority: In this examle: duration of inversion bounded by length of critical section of T. P. Marwedel, Univ. Dortmund, Informatik,

29 Duration of riority inversion with > tasks can exceed the length of any critical section Priority of T > riority of T > riority of T3. T reemts T3: T can revent T3 from releasing the resource. P. Marwedel, Univ. Dortmund, Informatik,

30 The MARS Pathfinder roblem () But a few days into the mission, not long after Pathfinder started gathering meteorological data, the sacecraft began exeriencing total system resets, each resulting in losses of data. The ress reorted these failures in terms such as "software glitches" and "the comuter was trying to do too many things at once". P. Marwedel, Univ. Dortmund, Informatik,

31 The MARS Pathfinder roblem () VxWorks rovides reemtive riority scheduling of threads. Tasks on the Pathfinder sacecraft were executed as threads with riorities that were assigned in the usual manner reflecting the relative urgency of these tasks. Pathfinder contained an "information bus", which you can think of as a shared memory area used for assing information between different comonents of the sacecraft. A bus management task ran frequently with high riority to move certain kinds of data in and out of the information bus. Access to the bus was synchronized with mutual exclusion locks (mutexes). P. Marwedel, Univ. Dortmund, Informatik,

32 The MARS Pathfinder roblem (3) The meteorological data gathering task ran as an infrequent, low riority thread, When ublishing its data, it would acquire a mutex, do writes to the bus, and release the mutex... The sacecraft also contained a communications task that ran with medium riority. High riority: retrieval of data from shared memory Medium riority: communications task Low riority: thread collecting meteorological data P. Marwedel, Univ. Dortmund, Informatik,

33 The MARS Pathfinder roblem (4) Most of the time this combination worked fine. However, very infrequently it was ossible for an interrut to occur that caused the (medium riority) communications task to be scheduled during the short interval while the (high riority) information bus thread was blocked waiting for the (low riority) meteorological data thread. In this case, the long-running communications task, having higher riority than the meteorological task, would revent it from running, consequently reventing the blocked information bus task from running. After some time had assed, a watchdog timer would go off, notice that the data bus task had not been executed for some time, conclude that something had gone drastically wrong, and initiate a total system reset. This scenario is a classic case of riority inversion. P. Marwedel, Univ. Dortmund, Informatik,

34 Coing with riority inversion: the riority inheritance rotocol Tasks are scheduled according to their active riorities. Tasks with the same riorities are scheduled CS. If task T executes P(S) & exclusive access granted to T: T will become blocked. If riority(t) < riority(t): T inherits the riority of T. T resumes. Rule: tasks inherit the highest riority of tasks blocked by it. When T executes V(S), its riority is decreased to the highest riority of the tasks blocked by it. If no other task blocked by T: riority(t):= original value. Highest riority task so far blocked on S is resumed. Transitive: if T blocks T and T blocks T0, then T inherits the riority of T0. P. Marwedel, Univ. Dortmund, Informatik,

35 Examle How would riority inheritance affect our examle with 3 tasks? T3 inherits the riority of T and T3 resumes. V(S) P. Marwedel, Univ. Dortmund, Informatik,

36 Priority inversion on Mars Priority inheritance also solved the Mars Pathfinder roblem: the VxWorks oerating system used in the athfinder imlements a flag for the calls to mutex rimitives. This flag allows riority inheritance to be set to on. When the software was shied, it was set to off. The roblem on Mars was corrected by using the debugging facilities of VxWorks to change the flag to on, while the Pathfinder was already on the Mars [Jones, 997]. P. Marwedel, Univ. Dortmund, Informatik,

37 Remarks on riority inheritance rotocol Possible large number of tasks with high riority. Possible deadlocks. Ongoing debate about roblems with the rotocol: Victor Yodaiken: Against Priority Inheritance, htt:// inds alication in ADA: During rendez-vous, task riority is set to the maximum. More sohisticated rotocol: riority ceiling rotocol. P. Marwedel, Univ. Dortmund, Informatik,

38 Summary Periodic scheduling Rate monotonic scheduling Proof of the utilization bound for n=. ED Deendent and soradic tasks (briefly) Resource access rotocols Priority inversion The Mars athfinder examle Priority inheritance The Mars athfinder examle P. Marwedel, Univ. Dortmund, Informatik,