Clock-driven (or Static) Scheduling (Baker and Shaw and Chapter 5 ...

Classification of Scheduling Algorithms

Schedule TableOur scheduler will schedule periodic jobs using a staticschedule that is computed offline and stored in a table T.» Each entry (t k , T(t k )) gives a decision time t k , when schedulingdecision is made, and name of task T(t k ).» T(t k ) = T i , if T i is to be scheduled at time t k , andI, if no periodic task is scheduled at time t k» For most of this chapter, we assume the table is given.» Later, we consider one algorithm for producing the table.• Note: This algorithm need not be highly efficient.» We will schedule aperiodic jobs (if any are ready) in intervalsnot used by periodic jobs.

Static, Timer-driven Scheduling

ExampleConsider a system of four tasks,T 1 = (4, 1), T 2 = (5, 1.8), T 3 = (20, 1), T 4 = (20, 2)Consider the following static schedule:The first few table entries would be: (0, T 1 ), (1, T 3 ),(2, T 2 ), (3.8, I), (4, T 1 ), … (19.8, I)

FramesLet us refine this notion of scheduling…To keep the table small, we divide the time line intoframes and make scheduling decisions only at frameboundaries.» Each job is executed as a procedure call that must fit within aframe.» The phase of each periodic task is a nonnegative integermultiple of the frame size.» Multiple jobs may be executed in a frame, but the table is onlyexamined at frame boundaries (the number of “columns” in thetable = the number of frames per hyperperiod).» In addition to making scheduling decisions, the scheduler alsochecks for various error conditions, like task overruns, at thebeginning of each frame.We let f denote the frame size.

Frame Size ConstraintsWe want frames to be sufficiently long so that every jobcan execute within a frame nonpreemptively. So,f ≥ max 1≤i≤n (e i ).To keep table small, f should divide H. Thus, for at leastone task T i ,⎣p i /f⎦ −p i /f = 0.Let F = H/f. (Note: F is an integer.) Each interval oflength H is called a major cycle. Each interval oflength f is called a minor cycle.There are F minor cycles per major cycle.

Frame Constraints (Cont.)We want the frame size to be sufficiently small so that between therelease time and deadline of every job, there is at least one frame.• A job released “inside” a frame is not noticed by the scheduleruntil the next frame boundary.• Moreover, if a job has a deadline “inside” frame k + 1, itessentially must complete execution by the end of frame k.Thus, 2f − gcd(p i , f) ≤ D i .

ExampleConsider a system of four tasks, T 1 = (4, 1), T 2 = (5, 1.8),T 3 = (20, 1), T 4 = (20, 2)By first constraint, f ≥ 2.Hyperperiod is 20, so by second constraint, possible choicesfor f are 2, 4, 5, 10, and 20.Only f = 2 satisfies the third constraint. The following is apossible cyclic schedule:

Job SlicesWhat do we do if the frame size constraints cannot be met?Example: Consider T = {(4, 1), (5, 2, 7), (20, 5)}. By first constraint,f ≥ 5, but by third constraint, f ≤ 4!Solution: “Slice” the task (20, 5) into subtasks, (20, 1), (20, 3), and(20, 1). Then, f = 4 works. Here’s a schedule:

Summary of Design DecisionsThree design decisions:» choosing a frame size,» partitioning jobs into slices, and» placing slices in frames.In general, these decisions cannot be madeindependently.We will look at an algorithm for making thesedecisions later.

Pseudo-code for Cyclic Executive

Improving Response Times ofAperiodic JobsIntuitively, it makes sense to give hard real-timejobs higher priority than aperiodic jobs.However, this may lengthen the response time ofan aperiodic job.Note that there is no point in completing a hardreal-time job early, as long as it finishes by itsdeadline.

Slack StealingLet the total amount of time allocated to allthe slices scheduled in frame k be x k .Definition: The slack available at thebeginning of frame k is f − x k .Change to scheduler:» If the aperiodic job queue is nonempty, letaperiodic jobs execute in each frame wheneverthere is nonzero slack.

Example

Implementing Slack StealingUse a pre-computed “initial slack” table.» Initial slack depends only on static quantities.Use an interval timer to keep track of availableslack.» Set the timer when an aperiodic job begins to run. If itgoes off, must start executing periodic jobs.» Problem: Most OSs do not provide sub-millisecondgranularity interval timers (as we shall see).» So, to use slack stealing, temporal parameters must beon the order of 100s of msecs. or secs.

Scheduling Sporadic JobsSporadic jobs arrive at arbitrary times.They have hard deadlines.Implies we cannot hope to schedule every sporadic job.When a sporadic job arrives, the scheduler performs anacceptance test to see if the job can be completed by itsdeadline.We must ensure that a new sporadic job does not causea previously-accepted sporadic job to miss its deadline.We assume sporadic jobs are prioritized on an earliestdeadline-first(EDF) basis.

Acceptance TestLet σ (i, k) be the initial total slack in frames i through k, where 1 ≤ i≤ k ≤ F. (This quantity only depends on periodic jobs.)Suppose we are doing an acceptance test at frame t for a newlyarrivedsporadic job S with deadline d and execution cost e.Suppose d occurs within frame l + 1, i.e., S must complete by the endof frame l.Compute the current total slack in frames t through l usingσ c (t, l) = σ (t, l) −∑ d k≤d (e k − ξ k )The sum is over previously-accepted sporadic jobs with equal orearlier deadlines. ξ k is the amount of time already spent executing S kbefore frame t.

Acceptance Test (Continued)We’ll specify the rest of the test “algorithmically”…To summarize, the scheduler must maintain the following data:• pre-computed initial slack table σ(i, k);• ξ k values to use at the beginning of the current frame t;• the current slack σ k of every accepted sporadic job S k ;• a queue of accepted sporadic jobs in a nondecreasing order oftheir deadlines.

Executing Sporadic TasksAccepted sporadic jobs are executed like aperiodic jobsin the original algorithm (without slack stealing).» Remember, when meeting a deadline is the main concern, thereis no need to complete a job early.» One difference: The aperiodic job queue is in FIFO order,while the sporadic job queue is in EDF order.Aperiodic jobs only execute when the sporadic job queueis empty.» As before, slack stealing could be used when executingaperiodic jobs (in which case, some aperiodic jobs couldexecute when the sporadic job queue is not empty).

Example

S 1 is released at time 3.» Must be scheduled in frames 2, 3, and 4.» Acceptance test – at the beginning of frame 2:• Slack time (4) is less than execution time – job is rejected.S 2 is released at time 5.» Must be scheduled in frames 3 through 7.» Acceptance test – at the beginning of frame 3:• Slack time is 5.5 – job is accepted.• First part (2 units) executes in current frame.S 3 is released at time 11.» Must be scheduled in frames 4 and 5.» S 3 executes ahead of S 2» Acceptance test – at the beginning of frame 4:• Slack time is 2 (is enough both for S 3and the part of S 2) – job is accepted• First part (1 unit) S 3executes in current frame, followed by the second part of S 2S 4 is released at time 14.» Acceptance test – at the beginning of frame 5:• Slack time is 4.5 (accounted for slack committed by S 2and S 3) – job is rejected• Remaining portion of S 3completes in current frame, followed by the part of S 2• Remaining portions of S 2execute in the next two frames

Practical ConsiderationsHandling frame overruns.» Main Issue: Should offending job be completed or aborted?Mode changes.» During a mode change, the running set of tasks is replaced by anew set of tasks (i.e., the table is changed).» Can implement mode change by having an aperiodic orsporadic mode-change job. (If sporadic, what if it fails theacceptance test???)General Workloads and Multiprocessors.» Like uniprocessors, but table probably takes longer to precompute.

Pseudo-code for Mode Changertask MODE CHANGER (oldMode, newMode):fetch the deleteList of periodic tasks to be deleted;mark each periodic task in the deleteList;inform the cyclic executive that a mode change has commenced;fetch the newTaskList of periodic tasks to be executed in newMode;allocate memory space for each task in newTaskList and create each of these task;fetch the newSchedule;perform acceptance test on each sporadic job in the system according to thenewSchedule,if every sporadic job in system can complete on time according to the newSchedule,inform the cyclic executive to use the newSchedule;else,compute the latestCompletionTime of all sporadic jobs in system;inform the cyclic executive to use the newSchedule atmax (latestCompletionTime, thresholdTime);

Multi-rate Control SystemsMore complicated control systems have multiple sensors andactuators and must support control loops of different rates.Example 2: Helicopter flight controller.Note: Having only harmonic rates simplifies the system.

Network Flow Algorithm (INF) forComputing Static SchedulesInitialization: Compute all frame sizes in accordance with the secondtwo frame-size constraints:⎣p i /f⎦−p i /f = 02f − gcd(p i , f) ≤ D iAt this point, we ignore the first constraint, f ≥ max 1≤i≤n (e i ). Recallthis is the constraint that can force us to “slice” a task into subtasks.Iterative Algorithm: For each possible frame size f, we compute anetwork flow graph and run a max-flow algorithm. If the flow thusfound has a certain value, then we have a schedule.

Flow GraphDenote all jobs in the major cycle of F frames asJ 1 , J 2 , …, J N .Vertices:» N job vertices, denoted J 1 , J 2 , …, J N.» F frame vertices, denoted 1, 2, …, F.» source and sink.Edges:»(J i , j) with capacity f if J i can be scheduled in frame j.»(source, J i ) with capacity e i .»(f, sink) with capacity f.

Illustration of Flow Graph

Finding a ScheduleThe maximum attainable flow value is clearly∑ i=1,…,N e i . This corresponds to the exact amount ofcomputation to be scheduled in the major cycle.If a max flow is found with value ∑ i=1,…,N e i , thenwe have a schedule.If a job is scheduled across multiple frames, thenwe must slice it into corresponding subjobs.

Example

ExampleExample: Consider T = {(4, 1), (5, 2, 7), (20, 5)}. By second andthird constraints, f = 2 and 4!For the first iteration of the Network Flow Algorithm frame size is 4.Maximum flow of the graph is 18, which is the total execution timeof all jobs in a hyperperiod.Flow of edges represent a schedule of the tasks:

Example

Non-independent TasksTasks with precedence constraints are no problem.» We can enforce precedence constraint like “J i precedes J k ” bysimply making sure J i ’s release is at or before J k ’s release, andJ i ’s deadline is at or before J k ’s deadline.» If slices of J i and J k are scheduled in the wrong order, we canjust swap them.Critical sections pose a greater challenge.» We can try to “massage” the flow-network schedule into onewhere nonpreemption constraints are respected.» Unfortunately, there is no known efficient, optimal algorithm fordoing this (the problem is actually NP-hard).

Pros and Cons of Cyclic ExecutivesMain Advantage: CEs are very simple ⎯ youjust need a table.» For example, additional mechanisms for concurrencycontrol and synchronization are not needed. In fact,there’s really no notion of a “process” here ⎯ justprocedure calls.» Can validate, test, and certify with very highconfidence.» Certain anomalies will not occur.» For these reasons, cyclic executives are thepredominant approach in many safety-criticalapplications (like airplanes).

Aside: Scheduling Anomalies(Section 4.8.1 of Liu)Here’s an example: On a multiprocessor, decreasing a job’sexecution cost can increase some job’s response time.Example: Suppose we have one job queue, preemption, but nomigration.

Pros and Cons (Continued)Disadvantages of cyclic executives:» Very brittle: Any change, no matter how trivial,requires a new table be computed!» Release times of all jobs must be fixed, i.e., “realworld”sporadic tasks are difficult to support.» Temporal parameters essentially must be multiples of f.» F could be huge!» All combinations of periodic tasks that may executetogether must a priori be analyzed.» From a software engineering standpoint, “slicing” oneprocedure into several could be error-prone.

ReadingsT.P.Baker and A.Shaw. “The Cyclic Executive Model and Ada.” In Proc. ofIEEE Real-Time Systems Symposium, pp. 120-129, Madrid, Spain, 1988.Chapter 5 of Liu. Clock-Driven Scheduling.Section 4.8.1 of Liu. Anomalous Behavior of Priority-Driven Systems.T.H. Cormen, C.E. Leiserson, R.L. Rivest, C. Stein. Introduction toAlgorithms, Second Edition. McGraw-Hill&MIT Press, 2001/2.ISBN: 0-07-013151-1 (0262032937)(www.introductiontoalgorithms.com)Reference in Liu’s book: Goldberg, A. V. “Processor efficientimplementation of a maximum flow algorithm”, Information ProcessingLetter, vol.38, pp.179–185, May 1991.(http://www.stormingmedia.us/cgibin/96/9643/A964323.php?pageNo=5&categoryID=53&masterCategoryID=24)The same author: Andrew V. Goldberg. “Recent Developments in MaximumFlow Algorithms” (Invited Lecture). SWAT 1998: 1-10(http://link.springer.de/link/service/series/0558/bibs/1432/14320001.htm)