Washington WASHINGTON UNIVERSITY IN ST LOUIS Real-Time: Periodic Tasks Fred Kuhns Applied Research...

WashingtonWASHINGTON UNIVERSITY IN ST LOUIS

Real-Time Periodic Tasks

Fred KuhnsApplied Research Laboratory

Computer ScienceWashington University

Fred Kuhns (041023) CSE522ndash Advanced Operating Systems 2

Priority-Driven Schedulers

bull Assumptions some of which will be relaxed at the end of this sectionndash Independent tasksndash No sporadic or aperiodic tasksndash Jobs ready when releasedndash Jobs may be preempted at any timendash Jobs never suspend themselvesndash Scheduling decisions made at job release and on completionndash Interrelease times may vary use the minimum for periodndash Uni-Processor environmentndash System overhead (including context switch) is negligible

compared to job execution times and periodsndash Unlimited priority levels


What is a priority-driven scheduler

bull Event-driven work-conserving schedulersndash scheduling decision made when event occurs for example a

job is released or completesndash Resources allocated to the highest priority job and only left

idle if no jobs are waiting (ready)

bull Jobs assigned explicit or implicit prioritiesndash consequently FIFO and Round Robin may be considered

priority driven Priorities are defined by the queue ordering and may be varied when job executes

bull Algorithm defined by priorities used and a set of rules (preemption priority changes etc)


Concepts

bull Algorithms that consider a jobs urgency generally perform better than those that do notndash as the deadline for a job approaches its urgency should

increase Contrast EDF with FIFO the FIFO implicit priority is a jobs position in a queue

bull Static versus Dynamic assignment of workloadsndash for now we only consider the case where a set of tasks are

statically assigned to a processor

bull Fixed versus Dynamic priority assignment ndash Fixed priority RM and DMndash Dynamic Task Fixed Job EDFndash Dynamic Job LST (Least Slack First)


Concepts Schedulable Utilization

bull A scheduling algorithm can feasibly schedule any set of periodic tasks on a processor if the total utilization of the tasks is equal to or less than the schedulable utilization of the algorithmldquo

bull Schedulable utilization is necessarily lt= 1

bull While dynamic priority algorithms have better average performance they are less predictable during overload In fact their worst case behavior is more difficult to predictndash consider how RM with fixed priorities will perform during

overload the highest priority tasks will generally complete on time which the lowest will not Contrast this with EDF which uses dynamic priorities


Example Dynamic Algorithms

bull Earliest Deadline Firstndash Assigns priorities to jobs according to absolute deadline The

sooner the deadline the higher the priorityndash Priority assigned when job released For example job arrives

and is placed in a queue using EDF algorithmndash Dynamic task priority static job priority

bull Least Slack Time First (LST)ndash Job with the least amount of slack time is scheduled nextndash Dynamic task priority Dynamic job priority


Schedulable Utilizations of EDF

bull A system of independent preemptable tasks with relative deadlines equal to their respective periods can be feasibly scheduled on one processor if and only if the total utilization is equal to or less than 1

bull Schedulability of the EDF algorithm ndash if Dk pk then density = Δ = U utilization test

is both necessary and sufficient ndash if Dk lt pk for some k then the test is only a

sufficient condition

n

k kk

k

pD

eDensity

1

1)min(


Schedulable Utilization Case 1

ric ric+pi

1

i

k

n

helliphellip

hellip

Jic misses a deadline

T1

Ti

Tk

Tn

We know that if the utulization is gt 1 then the tasks are not schedulable But if the tasks are not schedulable does that imply the utilization is gt 1Case 1 The Interval I is the current period of task Ti All jobs released before ric

complete by time ric + pi Solve for the total required time and compare to the total utilization Show that if a deadline is missed then the utilization is gt 1

I

ikk

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k

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ii ep

t

p

ett

)(

t



t

ric ric+pi

1

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helliphellip

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T1

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Tk

Tn

I

Case 2Let t-1 be the last time a job runs which has a deadline after (ric+pi) Define Trsquo to be the set of tasks with jobs released before ric and with deadlines after t Then it suffices to only consider the jobs in tasks T-Trsquo Proceed as before using t-1 as the origin

TiTTj

j

j

i

ici ep

t

p

ertttI

1

)(

t-1

rsquo1

rsquoi


Fixed Priority Algorithms

bull Rate Monotonic (RM)ndash Priorities based on Task period smaller periods have higher priority

ndash if pi lt pk (rate of Ti gt rate of Tk) then i gt k

bull Deadline Monotonic (DM)ndash Priority based on task deadline smaller relative deadline higher

priority

ndash if Di lt Dk then i gt k

bull No fixed priority algorithm is optimalbull Under special cases can achieve a utilization of 1

ndash a set of tasks are simply periodic if for every pair of tasks Ti and Tk with piltpk pk=mpi m an integer (They are harmonic)

ndash for simply periodic independent preemptable tasks with Di ge pi a set of tasks is schedulable on a uniprocessor system iff U 1

bull Among fixed-priority algorithms DM is optimalndash if a set of tasks can be scheduled using fixed priorities then DM will

produce a feasible schedule

ndash if Dk = xmiddotpk for all k and some constant x then RM is identical to DM


bull Sufficient utilization bounds for RM let Dk = pk

bull Utilization checkndash This is a sufficient condition that is it is possible for a set of tasks to

fail this test but still be schedulable

bull Time-demand analysis (Dk lt= pk)ndash assumes worst case task interactions evaluates computational

requirements at task critical instantsbull Simulation can be used

ndash it is sufficient to perform simulation over largest period when all tasks are in phase

Schedulability of Fixed Priority Algorithms

500for

150for 112

32for 11

)1(

)(1

1

1

n

n

RM n

n

nU


Critical Instantbull Central concept for fixed priority systemsbull A critical instant of a task is the instant in time

when a job (in Ti) has its maximum response time

bull If a task can be scheduled in its critical instant then it will not miss a deadline

bull Occurs when a job in Ti is released concurrently with all higher priority jobs ndash T = T1 T2 Tn ordered by priority (i lt j i lt j)

ndash Ti = T1 T2 Ti-1 All higher priority jobs

ndash ric = rkl for some l for every k in 1 2 i-1

bull At critical instantndash Lower priority task is preempted by higher priority tasks

delaying completionndash Maximum interference when all phases are equal


advancing release time increases interference

ei + 3ej


ei + 4ej

Critical Instant


Time Demand Analysis

bull Computes total demand for processor time by a job released at a critical instant of the task and all higher priority tasks

bull If this worst case response time is less than or equal to the jobs deadline then it is schedulable

w(t)

t

wi(t) = ei +k = 1

i - 1tpk

ek

for 0 lt t pi


Critical Instant for Fixed Priority Systems

Wk

1

i

k

n

helliphellip

hellip

T1

Ti

Tk

Tn

I

Response time for job Jk1 is wk1 Perform a time demand analysis Let W be the required total processor time then at time W = t = wk1 + k

kjj

j

jkkk e

p

weW

)( 1


Critical Instant Time demand analysis

t

W

kjkj

j

jkk e

p

tew

1

Decreasing the phase k shifts the graph up thus increasing the response time Likewise decreasing the phases of higher priority jobs (j) will also increase the jobrsquos response time Therefore the maximum response time of a job occurs when all jobs are in phase

wk1 = t

j

k


Practical Considerations

bull Preemptive versus Non-Preemptivebull Self Suspensionbull Context Switchesbull Limited Priority Levelsbull Tick schedulingbull Variable priority


Blocking and Nonpreemptivity bull Higher priority job is blocked by a lower priority job during a

nonpreemptive interval resulting in a priority inversionbull Must adjust schedulability tests

ndash let qi = maximum nonpreemptable execution time

ndash qi lt= ei

ndash let bi(nb) = max(qk) i+1 le k le n

bull Fixed priority systems change time demand analysis as follows

bull Change utilization equation as follows

)p(D t ep

tbetw iik

i

k kiii min0for )(

1

1

)(iUp

bU X

i

ii


Nonpreemptivity

bull EDF In a system where jobs are scheduled on the EDF basis a job Jk with relative deadline Dk can block a job Ji with relative deadline Di only if Dk is larger then Di

bull In other words for a job to be blocked we need rkltri and dkgtdi

bull For deadline driven systems

1n ievey for 1)min()min(1

n

k ii

i

kk

k

pD

b

pD

e


Blocking Time Continued

bull Self-suspension or self-blockndash We can treat as another blocking factor bi(ss)

ndash If higher priority job self-suspends then its computation time may be deferred until the feasible interval for some other job

bull If a job Ji can self-suspend at most Ki times then the total possible blocking time is given by (K+1 represents the K self-suspensions and scheduling on release)

1

1i ) of SSmaxmin(T of SSmax )(

i

kkki Tessb

)()1()( npbKssbb iiii


Putting it together with system overhead

bull Context Switch assume fixed job prioritiesndash let CS = context switch time time to ldquoplacerdquo or ldquoremoverdquo

a job tofrom the processor So 2CS is the total context switch overhead for a given job

ndash we can account for this by increasing a jobs execution time by 2CS or 2(K+1)CS if self-suspend K times

bull Updating our tests (must be true for every i)

1

1

1

)( priority Fixed

1)min()min(

Deadline

)1(2)()1()(

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Limited Priorities and Tick schedulingbull Limited priorities

ndash N task priorities S system priorities and N = S then must provide a mapping

bull When N gt S then nondistinct priorities result (different task priorities map to same system priority)

bull worst case a job is delayed by other jobs with the same priority

ndash Uniform mappingndash Constant Ratio mapping and grid ratio

bull Tick Schedulingndash tick period = p0

ndash the tick task has period p and an execution time that is a function of the queue lengths

ndash A job may wait on scheduling queue when runnable


Priority-Driven Schedulers

bull Assumptions some of which will be relaxed at the end of this sectionndash Independent tasksndash No sporadic or aperiodic tasksndash Jobs ready when releasedndash Jobs may be preempted at any timendash Jobs never suspend themselvesndash Scheduling decisions made at job release and on completionndash Interrelease times may vary use the minimum for periodndash Uni-Processor environmentndash System overhead (including context switch) is negligible

compared to job execution times and periodsndash Unlimited priority levels










Concepts
























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500for

150for 112

32for 11

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RM n

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ei + 3ej


ei + 4ej

Critical Instant





w(t)

t

wi(t) = ei +k = 1

i - 1tpk

ek

for 0 lt t pi



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1

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k

n

helliphellip

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ndash qi lt= ei




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Nonpreemptivity





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k ii

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1


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Concepts
























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500for

150for 112

32for 11

)1(

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1

1

n

n

RM n

n

nU












ei + 3ej


ei + 4ej

Critical Instant





w(t)

t

wi(t) = ei +k = 1

i - 1tpk

ek

for 0 lt t pi



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1

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helliphellip

hellip

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1

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Nonpreemptivity





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k ii

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500for

150for 112

32for 11

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1

1

n

n

RM n

n

nU












ei + 3ej


ei + 4ej

Critical Instant





w(t)

t

wi(t) = ei +k = 1

i - 1tpk

ek

for 0 lt t pi



Wk

1

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k

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ndash qi lt= ei




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1

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ii


Nonpreemptivity





n

k ii

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1


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1

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500for

150for 112

32for 11

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1

1

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n

RM n

n

nU












ei + 3ej


ei + 4ej

Critical Instant





w(t)

t

wi(t) = ei +k = 1

i - 1tpk

ek

for 0 lt t pi



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1

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ndash qi lt= ei




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Nonpreemptivity





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1

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500for

150for 112

32for 11

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1

1

n

n

RM n

n

nU












ei + 3ej


ei + 4ej

Critical Instant





w(t)

t

wi(t) = ei +k = 1

i - 1tpk

ek

for 0 lt t pi



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1

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k

n

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1

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Nonpreemptivity





n

k ii

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1


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1

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)1(2)()1()(

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tbetw

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500for

150for 112

32for 11

)1(

)(1

1

1

n

n

RM n

n

nU












ei + 3ej


ei + 4ej

Critical Instant





w(t)

t

wi(t) = ei +k = 1

i - 1tpk

ek

for 0 lt t pi



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1

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k

n

helliphellip

hellip

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ndash qi lt= ei




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1

1

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ii


Nonpreemptivity





n

k ii

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kk

k

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b

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1


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kkki Tessb








1

1

1

)( priority Fixed

1)min()min(

Deadline

)1(2)()1()(

i

kk

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tbetw

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500for

150for 112

32for 11

)1(

)(1

1

1

n

n

RM n

n

nU












ei + 3ej


ei + 4ej

Critical Instant





w(t)

t

wi(t) = ei +k = 1

i - 1tpk

ek

for 0 lt t pi



Wk

1

i

k

n

helliphellip

hellip

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Nonpreemptivity





n

k ii

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1


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1

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ei + 3ej


ei + 4ej

Critical Instant





w(t)

t

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ek

for 0 lt t pi



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1

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1

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Nonpreemptivity





n

k ii

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1


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1

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for 0 lt t pi



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Nonpreemptivity





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k ii

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1


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1

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1)min()min(

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Nonpreemptivity





n

k ii

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1


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1

1

1

)( priority Fixed

1)min()min(

Deadline

)1(2)()1()(

i

kk

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bU X

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ii


Nonpreemptivity





n

k ii

i

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b

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1


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1

1

1

)( priority Fixed

1)min()min(

Deadline

)1(2)()1()(

i

kk

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tbetw

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ndash qi lt= ei




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Washington WASHINGTON UNIVERSITY IN ST LOUIS Real-Time: Periodic Tasks Fred Kuhns Applied Research...

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Transcript of Washington WASHINGTON UNIVERSITY IN ST LOUIS Real-Time: Periodic Tasks Fred Kuhns Applied Research...