Fine-Tuning MAC-Level Protocols for Optimized Real-Time Quality-of-Service

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Real-Time Quality-of-Service

Mathieu Grenier, Nicolas Navet

To cite this version:

Mathieu Grenier, Nicolas Navet. Fine-Tuning MAC-Level Protocols for Optimized Real-Time

Quality-of-Service. [Research Report] RR-6247, INRIA. 2007, pp.22. �inria-00158172v3�

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Thème COM

Fine-Tuning MAC-Level Protocols for Optimized

Real-Time Quality-of-Service

Mathieu Grenier — Nicolas Navet

N° 6247

June 2007

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Unité de recherche INRIA Lorraine

LORIA, Technopôle de Nancy-Brabois, Campus scientifique,

615, rue du Jardin Botanique, BP 101, 54602 Villers-Lès-Nancy (France)

Téléphone : +33 3 83 59 30 00 — Télécopie : +33 3 83 27 83 19

Quality-of-Servi e Mathieu Grenier

∗

, Ni olas Navet

∗

ThèmeCOM Systèmes ommuni ants

ProjetTRIO

Rapportdere her he n°6247June200721pages

Abstra t: Indistributedreal-timesystems,meetingthereal-time onstraintsismandatorybutthesatisfa tion

of other appli ation-dependent riteria is most generally required as well. In parti ular, Networked Control

Systems (NCS) areknownto besensitiveto ommuni ationdelays su h asframe response time jitters. Well

knownMediumA essControl(MAC)algorithmssu hNon-PreemptiveDeadlineMonotoni (NP-DM)or

Non-PreemptiveEarliestDeadline First(NP-EDF)aree ientin termsofbandwidthusagebuttheymayperform

poorly regarding other appli ation dependent performan e riteria. This paper highlights a lass of on-line

s hedulingpoli iestargetedats hedulingframesattheMAClevel,andprovidesas hedulabilityanalysisthat

isvalidforallpoli ieswithinthe onsidered lass. Asitwillbeshown,thesealgorithmsareeasilyimplementable

on COTS omponents(e.g., ControllerArea Network ontrollers)andoergoodtrade-osbetweenfeasibility

and thesatisfa tionofotherappli ation-dependent riteriasu hastheresponsetimejitter.

Key-words: Real-timesystems, ontrol systems, ommuni ationsystems,s heduling,proto ols,eld buses.

∗

LORIA - TRIO Group, Campus S ientique - BP 239, 54506 Vandoeuvre-lès-Nan y Cedex, Fran e. Email:

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Résumé: Danslessystèmesdistribuéstempsréel,lerespe tdes ontraintestempsréelestsine quanon mais

la satisfa tion d'autres ritèresest également le plus souvent requis. En parti ulier, les "systèmes ontrlés

en réseau" (Networked Control System - NCS) sont onnus pour leur sensibilité aux délais de

ommuni a-tion omme la gigue sur les temps de réponse. Les proto oles les plus ourants pour l'a ès au médium de

ommuni ation(Medium A ess Control - MAC)tel queDeadline Monotoni (NP-DM) ouEarliest Deadline

Firstnon-préemptif(NP-EDF) sont e a esen termesd'utilisation debandepassantemais peuventavoirun

omportementmédio reau regarddes autres ritèresdeperforman esdépendantsdel'appli ation. Ce papier

présenteune lassedepolitiquesd'ordonnan ementenlignepourl'ordonnan ementdestramesauniveauMAC,

et fournit uneanalysed'ordonnan ementappli ableàtouteslespolitiquesappartenantà ette lasse. Comme

nous lemontrerons, es algorithmessontfa ilementimplémentables surdes omposantsdu ommer e, omme

des ontrleurs de ommuni ationCAN, et orent un bon ompromis entre faisabilité et la satisfa tion des

ritèrespropresàl'appli ation ommelagiguesurlestempsderéponse.

Mots- lés : Systèmestempsréel,systèmesde ontrle,systèmesde ommuni ation,ordonnan ement,

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1 Introdu tion

Context of paper Indistributed real-time systems,therespe t oftiming onstraintisabasi requirement

but other appli ation-dependent riteriabesidesfeasibility areofinterest. Forinstan e,in NetworkedControl

Systems (NCS), where the ontrol system is distributed through a network, message ex hanges indu e

non- onstant delays in the ontrol-loop, whi h ae ts the performan e and even the stability of the ontrolled

system [1, 2,3℄. Theframe s hedulingalgorithm at theMedium A essControl (MAC) hasalargeinuen e

onthesedelaysandthus,should be hosena ordingto theappli ation'sperforman erequirements.

Problem denition TheproblemistodeviseMAClevels hedulingalgorithmsthataree ientintermsof

bandwidthusageas well asforthesatisfa tionofappli ation dependent riteria. Moreover,asfaraspossible,

these algorithmsshould beimplementable onCOTShardwareandindu e smalloverhead. Inthefollowing,we

will onsiderthat theMACs hedulingalgorithm isimplementedontopof aprioritybus,su hasCAN [4,5℄,

VAN [6℄orJ1850[7℄. Indeed,prioritybuseshaveprovento beee tiveinareal-time ontext, andarewidely

availableandused.

Related work Many studies have been devoted to nd s heduling solutions that are well suited to the

appli ation's requirements,bothforthes hedulingoftasksandthes hedulingofmessages.

1. S hedulingoftasks:

In[8℄, amodiedversionof theConstantBandwidth Server(CBS), initially proposed in [9℄, is used to

eliminate jitters. Data inputand data output o urat xed points in time and ontrol tasks runin a

CBS,whi hisanabstra tionofadedi atedCPUoeringa hosenfra tionoftheoriginalCPU,toensure

that ontroltasksnishbeforetheoutputofthedata.

Tobetterttothepro essingrequirementsofa ontrolsystem,newtaskmodelshavebeen on eived. In

[10,11℄,theelasti taskmodelisproposedtohandleoverruns: taskadapttheirperiodatruntimeinsu h

awayastokeepthesystemsunderloaded. In[12,13℄,itisproposedthat ontroltasksaresubdividedin

three dierentparts: sampling, omputation, a tuation. Samplingand a tuationsub-tasksareassigned

ahigh priorityinordertoredu e thejitters.

Anothersolutionistoadjust theparametersofthetaskstoa hievethedesiredgoals. In[14℄,the

worst- ase end-of-exe ution jitter is minimized by hoosing appropriate deadlines. In [15℄, initial osetsand

prioritiesareadjustedto redu ejitterbyminimizing preemption.

Improvements analsobebroughtby well hoosingthe parametersof thes hedulingpoli ies. In[16℄, a

priorityallo ations hemeis proposed toredu e theaverageresponsetime while,in [17℄, theproblemof

hoosings hedulingpoli iesandprioritiesonaPosix1003.1b ompliantoperatingsystem(OS)ista kled.

Finally, anotherwayisto reatenews hedulingpoli ies. In[18℄,thes heduler issynthetized asatimed

automatafrom thePetrinetmodelingthesystemandtheproperties expe tedfrom thesystem. In[19℄,

alsostartingfromaPetrinetmodelofthesystem,anoptimals hedulingsequen eisfoundbyexamining

themarking graphof the Petrinet. In dire t link with this study is thework published in [20℄, where

on-linetasks hedulingpoli iesare on eivedforoptimizingappli ation-dependentperforman es riteria.

2. Messages heduling:

E ients hedulings hemeshavebeenproposedin[21℄and[22℄toensurefairnessandbandwidthisolation

betweenstreamsofmessagessentonaCANbus,respe tivelywitha entralized(i.e.,amasternodehas

full ontroloverthes heduling)and de entralizeds heme.

Other studies proposed solutions for implementing Non-Preemptive Earliest Deadline First (NP-EDF)

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CAN in [23, 24, 25℄ with solutions where deadlines areen oded on alogarithmi time s ale, whi h has

been shown to minimize the priority inversions due to the limited number of priority bits. In [26℄, a

NP-EDF implementation isproposed ontop ofFTT-CAN while [27℄ addresses thesameproblem when

theappli ationmakesuseofthestandardizedCANappli ationlayersCAL CiA[28℄andCANOpen[29℄.

Anothers heme,publishedin [30℄,allowstoredu e theresponsetimejitterdueto bit-stungby

intro-du ingsomerestri tionsontheprioritiesthat anbeallo atedtothestreamofframesonaCANnetwork.

Toa hievethesamegoal,theauthorsin[31,32℄proposeageneti algorithm toadjustthe initialosets

ofmessagesin ordertoredu etransmissionjitter(samegoalas[15℄fortheunipro essor ase).

Overviewoftheapproa h Inthisstudy,theproblemaddressedistoe ientlygranta esstothenetwork

at the MAC level. The performan e riteria are the s hedulability of the systems and the satisfa tion of

appli ation dependant riteria, su h astheframe response time jitter whi h is ru ial for Networked Control

Systems. Theproposalis omprisedofthreedistin tsteps:

1. dene thepropertiesthat agood MAC-level real-times hedulingpoli ies must possess and identify a

sub- lass,parti ularlypromising intermsofperforman es,

2. deriveas hedulabilityanalysisforthissub- lass lassofs hedulingpoli iesandaddressthe

implementa-tionissues,

3. explore the sear h spa e and nd out themost e ient poli ies. Two ases an be distinguished. The

asewhere the poli y has to be e ient onaverage (it anbeused with dierenttra streams whose

hara teristi sareapriorinotknown). And,the asewherethepoli yshouldbenetunedforaparti ular

appli ation.

Organisation of the paper Inse tion 2, the omputational model is stated. In se tion3, the properties

requiredfrom agood s hedulingalgorithm inthereal-time ontextareexihibited. Amongthis setof good

poli ies, thesub- lass onsidered in thefollowingispresented. Inse tion 4,theimplementationissueson top

oftheCANprioritybusaredis ussed. Se tion5isdevotedtothes hedulabilityanalysis. Finally,experiments

showingtheee tiveness oftheproposalaresummarizedinse tion6.

2 Network model

This studydealswiththes hedulingofframesattheMediumA essControllevel;in parti ularitisassumed

thatsegmentationtakespla eintheupperlayersofthe ommuni ationsta k. Thea esstothebusisgranted

by an algorithm (e.g., prioritybased, token bus,et .) that is termedthe s heduling poli y in therest of the

paper.

2.1 Tra stream model

A set

F

(C

k

, D

k

, T

A

k,n

. For the sake of larity, sporadi streamsand jittersinthe availabilitydates arenot onsideredherebut anbetakenintoa ountas

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Con rete and non- on rete tra streams A on rete tra stream

(f

i

, A

i,1

)

isastreamforwhi h the release time of its rst frame

A

i,1

is known before run-time while the releasetimes of non- on rete streams are unknown. A set of

n

non- on rete tra streams is denoted by

Ω = {f

1 , f

2 , ..., f

n

}

, where

f

k

is the streamwithindex

k

,whileasetof

n

on retetra streamsis

ω

= {(f

1 , A

1,1

), (f

2 , A

2,1

)...(f

n

, A

n,1

)

},where

A

k,1

is the initial osetof stream

f

k

(i.e. the releasetimeof the rstinstan e). Without restri tionson the initial osets,thereis aninnitenumberofmappingfrom thesetof non- on retetra streams

Ω

to theset of on retetra streams

ω

.

Feasibility and optimality A on reteset of tra streams

ω

issaid feasible (or s hedulable) ifnoframe ofthesystemisre eivedafteritsabsolutedeadline

D

k,n

= A

k,n

+ D

k

. Anon- on retesetoftra streams

Ω

is feasible,ifallthe on rete sets

ω

,whi h anbegeneratedfrom

Ω

, arefeasible.

A s heduling poli y is optimal with respe t to a ertain riterion (e.g. feasibility, average response time)

within its lass if noother poli y of the lass performsbetter withrespe t to the riterion. In thefollowing,

optimal isusedto meanoptimal withrespe t tofeasibility. As hedulingpoli yis non- on reteoptimal (with

respe ttofeasibility)ifitsu essfullys hedules allthenon- on rete setsthat ares hedulablewithapoli yof

the lass. Asshownin[34℄,Non-PreemptiveEarliestDeadlineFirst(NP-EDF -smallertheabsolutedeadline,

greaterthepriority)isnon- on reteoptimalwithinthe lassofnon-idlingpoli y,where non-idlingmeansthat

thenetworkisnotidlewhenthereareframesawaitingfortransmission.

2.2 Dening poli ies through priority fun tion.

Priority fun tions is a onvenient way of formally dening in a non-ambiguous manner s heduling poli ies,

whi h toourbest knowledge, was introdu edfortherst timein [35℄. Thepriorityfun tion

Γ

k,n

(t)

indi ates thepriorityofaframe

f

k,n

attime

t

. Thenetworkisallo ated,atea htime,a ordingtotheHighest Priority First (HPF)paradigm.

Fun tion

Γ

k,n

(t)

takesitsvaluefromatotallyorderedset

P

,whi his hosenin[35℄tobethesetof multi-dimensional

IR

-valuedve tors

P = {(p

1 , ..., p

n

) ∈ R

n

_{| n ∈ N}}

providedwith alexi ographi al order. Between

twove tors, oordinates are ompared onebyonestartingfrom the left,the rstdierent oordinate de ides

thepriorityorderwiththe onventionthesmallerthenumeri alvalue,thehigherthepriority. Forinstan e,

Γ

i,j

(t) = (3, 4, 5)

and

Γ

k,n

(t) = (3, 4, 6)

impliesthat

f

i,j

hasahigherprioritythan

f

k,n

attime

t

. Priorityve tors of dierent sizes anbe ompared with thesamerule asaboveand the onventionthat amissing oordinate

is thelowestnumeri alvalue(e.g.

Γ

i,j

(t) = (3, 4, 5)

and

Γ

k,n

(t) = (3, 4)

meansthat

f

k,n

hasahigherpriority than

f

i,j

attime

t

). Finally, twove torsareequalitheyhavethesamesizeand ifthe omponentsareequal one by one. As it will be explained in se tion 4, on apriority bussu h as CAN, the priority ve tor an be

mappedtothepriority odedintheIdentiereld oftheframes. Thisallowstotakeadvantageofthenative

arbitrationme hanismoftheprioritybustoperformthearbitrationa ordingtothes hedulingpoli ydened

bythepriorityfun tion

Γ

k,n

(t)

.

Mostreal-times hedulingpoli ies anbedenedeasilyusingpriorityfun tions. Forinstan e,thepriorityof

aframe

f

k,n

under NP-EDFandNon-PreemptiveDeadlineMonotoni (NP-DM-smallertherelativedeadline, greaterthepriority),prioritybe omes:

Γ

N P−EDF

_k,n

(t) =

(

(A

k,n

+ D

k

, k, n)

if

t < B

k,n

(−∞, k, n)

if

t ≥ B

k,n

,

Γ

N P−DM

k,n

(t) =

(

(D

k

, k, n)

if

t < B

k,n

(−∞, k, n)

if

t ≥ B

k,n

where

B

k,n

is thetransmissionbeginof

f

k,n

(the last two oordinates

k, n

areneeded to ensurede idability, seedenition2). The

−∞

valueoftherst omponentofthepriorityve torafterthestartoftransmission(i.e.

t

≥ B

k,n

)ensuresnon-preemptiveness. A lassofpoli iesofparti ularinterest,towhi hNP-EDFandNP-DM belong,isthePriority PromotionatExe utionBeginning poli ies.

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Denition 1 [35℄A s heduling poli y

A

isaPriority Promotion atExe utionBeginning(PPEB)poli yif the priority of aframemay hange onlyatthe startof transmission

B

k,n

ofthe frame:

Γ

P P EB

k,n

(t) =

(

₋

_→

P

k,n

if

t < B

k,n

−

→

Q

k,n

if

t

≥ B

k,n

Besidesprovidingnon-ambiguousdenition ofthes hedulingpoli y, priorityfun tions enableusto

distin-guish lassesofs hedulingpoli iesandtoderivegeneri resultsthatarevalidforallthepoli ieswithina ertain

lass. Thenextse tionpresentsthesub- lassofnon-preemptives hedulingpoli iesthat willbestudiedin the

rest ofthepaper.

3 Study Domain

An arbitrarypriorityfun tion doesnotne essarilydene as hedulingofinterestfor real-time omputing nor

apoli y that anbeimplemented inpra ti e. Inthisse tion, thepropertiesrequirefrom agood s heduling

poli yispre ised,wheregood meansapoli y that anbeimplementedinpra ti eandthatissuitedto

real-time omputing. Then,amongthesetofallgoodpoli ies,theparti ular lassofs hedulingpoli ies onsidered

in thisstudyisdened.

3.1 Good s heduling poli ies

A good poli y mustmeeta ertainnumberof riteria,whi hareneeded forthepoli yto beimplementedin

areal-time ontext.

De idablepoli ies Poli iesmustbede idable: atanytime

t

,thereisexa tlyoneframeofmaximalpriority among the set ofa tiveframes(i.e. frames whi h ontend forthe hannel). This on ept ofde idabilitywas

introdu edin [35℄fortheunipro essor ase.

Denition 2 [35℄Apriority fun tionisde idable i,atea htime

t

su hthat workispending,thereisexa tly one framewiththe highest priority amongthe setof pending frames.

Forinstan e, thelast two omponents of

−

→

P

N P−EDF

k,n

= (A

k,n

+ D

k

, k, n)

ensure de idability in ase of equal deadlines. Notethat thispropertyisalsorequiredforanimplementationonaprioritybus.

Temporalinvariant poli ies Inthisstudy,forthesakeofpredi tabilityofthesystem,theonlys heduling

poli ies of interestare su h that the relative priority between twoframes doesnot depend on the numeri al

value ofthe lo al lo ks. Therelativepriority mustremain thesameif thearrivalof allframes isshiftedto

theleftortheright. Thepoli yisthusindependentofthevalueofthelo al lo ksatstart-uptime.

Su h poli ies are said shift temporal invariant (STI) poli ies. NP-EDF is a STI poli y sin e the priority

betweentwoframesonlydependsontheosetbetweenarrivaldatesandonrelativedeadlines. Onthe ontrary,

apoli y dened by

−

→

P

k,n

= (C

k

· A

k,n

, k, n)

is notSTI;just onsider

f

i,1

and

f

j,1

with

A

i,1

= 0

,

C

i

= 10

and

A

j,1

= 1

with

C

j

= 1

andthesametwoframesex eptthatthearrivaldatesareshiftedtotherightbyoneunit oftime.

Denition 3 Lettwo on retetra streamsetsbe

ω

= {(f

1 , A

1,1

), (f

2 , A

2,1

)...(f

n

, A

n,1

)

}and

ω

′

= {(f

1 , A

1,1

+

Φ), (f

2 , A

2,1

+ Φ)...(f

n

, A

n,1

+ Φ)

}where

ω

′

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As hedulingpoli y

A

isShiftTemporalInvariant(STI)iforallpossible

Φ ∈ Z

,

∀i, j, k, n

su hthat

(k, n) 6=

(i, j)

(twodistin t frames), onehas:

∀t Γ

A,ω

k,n

(t)

≻ (

resp

≺) Γ

A,ω

i,j

(t)

=⇒

Γ

A,ω

k,n

′

(t + Φ) ≻ (

resp

≺)

Γ

A,ω

′

i,j

(t + Φ)

where

Γ

A,ω

k,n

(t)

isthe priority of

f

k,n

of the on retetra streamset

ω

attime

t

.

Implementable poli ies Forbeing implementable in pra ti e, omponents of the priority ve tor must be

representablebyalimitednumberofbits(forinstan e,11bitsonstandardCAN).Inthefollowing, oordinates

of the priority ve tor take their value in the set of rational numbers

Q

; the impre ision due to the limited numberofavailablebits, alledquantizationerror asin[23℄,will betakeninto a ountin thes hedulability

analysis developedinparagraph5.

Any good MAC-levels hedulingalgorithm intended to be implemented on topof a prioritybus has to

fulll theaforementionedproperties: thepoli ymustbede idable,shifttemporalinvariantandimplementable.

Inthenextparagraph,the lassofnon-preemptivepoli iesstudiedin therest ofthepaperisdened.

3.2 Sear h spa e

Thesear hspa ebelongstothe lassofNon-PreemptiveArrivalTimeDependent poli ies. First,this lassof

poli iesisdened andthis hoi eisjustied.

Non-Preemptive Arrival TimeDependent poli ies ThetermNon-PreemptiveArrivalTime

Depen-dent (NP-ATD) poli ies omes from the fa t that, within this lass, the priority of a frame depends on its

arrivaltime. NP-ATD poli ies isasub- lassofPPEBpoli iesintrodu edearlier(seedenition1).

Denition 4 ANon-Preemptive Arrival Time Dependentpoli yisapoli ywhose priorityfun tion anbe put

under the form:

Γ

k,n

(t) =

(

₋

_→

P

k,n

= (A

k,n

+ p

k

, k, n)

if

t < B

k,n

−

→

Q

k,n

= (−∞, k, n)

if

t

≥ B

k,n

(1) where

p

k

:

k

7→ Q

+

.

p

k

isanarbitraryfun tion of thestream

f

k

, whi h returnsapositivevalueidenti al forallframesof

f

MotivationsforNon-PreemptiveArrival TimeDependentpoli ies Firstofall,NP-ATDpoli iesare

good s hedulingpoli ies:

de idabilityisensuredbythelasttwo omponentsofthepriorityve tors,

NP-ATD poli ies are Shift Temporal Invariant, see denition 3, sin ethe relativeprioritybetween two

frames

f

k,n

and

f

i,j

depends only on the oset between arrival dates and on the values returned by fun tions

p

k

and

p

i

,

thepoli iesareimplementable;se tion4isdevotedto theimplementationissues.

Se ond, NP-ATD poli ies are promising in terms of performan es. Indeed, NP-EDF, whi h is optimal with

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poli iesthatperform losetoNP-EDFintermsoffeasibilitywhilehavingamu hbetterbehaviorwithrespe t

toappli ation-dependent riteria. Theexperimentspresentedin paragraph6 onrmthisintuition. Third,asit

will beshowninparagraph4,ageneri feasibilityanalysis,throughresponsetimebound omputation, anbe

derivedforallNP-ATDpoli ies.

4 S heduling frames with NP-ATD poli ies

This se tion deals with thepra ti al issuesof implementing NP-ATD poli ies fors hedulingmessagesat the

MAC level. In the following, an implementation on top of the CAN network is onsidered. Indeed, CAN

networkiswidelyused,very heapanditisawellmasteredte hnologythatpossessesinterestingfeaturesfora

wide-rangeofreal-timesystems. Theme hanismsdis ussedinthefollowing anbeadaptedinastraightforward

mannerto allother prioritybuses, su hasVAN [6℄andtheJ1850[7℄.

Severalstudies[36,23,24,25℄haveta kledtheproblemofs hedulingmessageswithNP-EDF,whi hbelongs

toNP-ATDpoli ies,ontopofCAN.Inthefollowing,thesolutionsin[23℄andrenedlaterin[24,25℄isextended

to NP-ATDpoli ies.

4.1 Basi s of CAN MAC proto ol

OnCAN,anynodemaystartatransmissionwhenthebusisidle. Possible oni tsareresolvedbya

priority-basedarbitrationpro ess. In aseofsimultaneoustransmissions,thehighestpriorityframewillbesentdespite

the ontentionwithlowerpriorityframes. Thearbitrationisdeterminedbytheidentieroftheframes(i.e. their

priority)withthe onventionthesmallerthenumeri alvalue,thehigherthepriority. Thereis noneedthat

su essiveinstan es of are urrentframe havethesameidentier. This allowstheimplementation ofpoli ies

withdynami prioritiessu h asNP-EDF andNP-ATDpoli y.

4.2 NP-ATD poli ies s heduling on CAN

With NP-ATD poli ies, priority of frames are inversely proportionalto

A

k,n

+ p

k

. The basi idea [36℄ is to en odethepriorityoftheframeinthelargestpartoftheidentier. Sin eitisimposedonCANthatallframes

havedistin tidentiers,somebits arekepttoensuretheuniquenessof theidentier. Asolutionisto assigna

bitpatterndistin tforea htra stream. However,s hedulingmessageswithNP-ATDpoli iesonCANraises

severalproblems. These problems,expli itlystatedin [36℄,arethesameasNP-EDF:

1. priorities

A

k,n

+ p

k

be omelargerandlargerwithtimesin e

A

k,n

dependsonthe urrent lo kvalue, 2. the mapping of the priorities to a limited number of bits indu es so- alled quantization errors (see

AppendixA): twodistin tpriorities anbeassignedthesameidentierduetoitslimitednumberofbits.

Withoutlossofgenerality,thetime anbe onsidereddis retewhereitsgranularityisthebit-time(i.e.,thetime

betweentheemissionoftwosu essivebitsofthesameframe). Tosolveproblem1,theauthorsin[36,23,24,25℄

proposetoexpressthepriorityrelativelytoatimeoriginthatisin reasingovertime. Pre isely,thetimeorigin

t

start

is updatedto ea h newtransmission beginning (i.e.

t

start

=

max

k,n

(B

k,n

≤ t)

): it is equalto thevery rstbitoftheframe(StartofFramebit). Thepriorityofea hpendingmessagehastobeupdatedatea hnew

transmission start. This omputation indu es averysmalloverhead. Forinstan e, itiswasestimated[25℄ to

belessthan3%onthewidelyused SiemensC167CRmi ro ontroller.

An ee tive solution to Problem 2 was developed in [23℄. The authors use a logarithmi s ale to map

deadlines. One anexa tly reuse thiste hniquefor priorities of NP-ATD poli ies. The timeline isdivided in

so- alledblo kswhosesizeareexponentiallyin reasing(seegure5inAppendixA ). Allblo ksaresubdivided

(12)

falls. Thiss hemeallowstomapasetofframeshavingalargerangeofprioritiestoalimitednumberofpriority

bits. In [23℄, it is formally proven that this te hnique greatly improvesthe s hedulability of the system by

limiting quantizationerrors. Detailsof priorityen oding usinglogarithmi s ale applied to NP-ATD poli ies

aregiveninAppendixA. Thankstothete hniquesused,quantizationerrorsisredu edbutthey annotbefully

avoidedandthushavetobetakenintoa ount. Inthefollowing,as hedulabilityanalysisofNP-ATDpoli ies

that in ludes quantizationerrors is proposed. It is worth pointingout that, theimplementation of NP-ATD

poli ies anbeadaptedtoToken-RingandCSMA-CDnetworksasmadein[25℄forNP-EDF.

5 S hedulability analysis of NP-ATD poli ies

S hedulability an be analysed under NP-EDF using feasibility tests [34℄ and omputation of bounds on the

Worst Case Response Times(WCRT)[37, 38℄. Theaim ofthis se tion is topresentas hedulabilityanalysis

throughWCRTboundswitharbitrarydeadlinesandtakingintoa ountthequantizationerrors. Thisextends

theanalysismadein [23,24,25℄ whereonlythe asewhere deadlinesarelowerthanperiodsishandled. First,

paragraph5.1summarizesexistingresultsonWCRTandappliesthemtoNP-ATDpoli ies. Then,paragraph5.2

introdu estheoverheadbroughtbyquantizationerrors.

5.1 WCRT under NP-EDF: a re ap

The response time

r

k

(

a

)

of aframe is the time elapsed betweenits arrivala and its ompletion. The set of tra streams is feasible under a given s hedulingpoli y if the response time of ea h frame is lowerthan or

equal to the relative deadline. In general, it is not possible to omputethe response times of all frames for

all foreseeabletraje toriesof thesystem; asolutionfor assessingfeasibilityisto omputebounds onresponse

times. Su h an analysis wasderivedfor preemptiveEDF in [33℄, later for NP-EDF in [37, 38℄. In [37℄, it is

shown that the worst aseresponse time of a frame of atra stream o urs after a ertain arrivalpattern

termed the As SoonAs Possible pattern (ASAP for short). Theauthors use the on eptof deadline busy

period fora frame

f

k,n

, whi his aperiod ofnetwork utilization withoutidle-time during whi h onlyframes withdeadlinenotgreaterthan

f

k,n

areexe uted.

Lemma1 [37 ℄ Aboundontheresponsetimeofaframe,belongingtotra stream

f

k

,releasedaunitsoftime after the beginning of its deadline busy period an be found in the deadline busy period indu ed by the ASAP

pattern. TheASAPpattern isthe situationwhere:

f

k

releasesaframeattimea (otherframes mayhavebeenreleasedbeforetimea),

allframes, whi h relative deadline

D

i

issmallerthanorequal toa

+ D

k

,arereleasedfromtime

t

= 0

on attheirmaximum rate,

the largest frame of all tra streamswith relative deadline greater than a

+ D

k

, if any, is released at time

t

= −1

. This onstitutesthe so- alledblo king fa torduetonon-preemptiveness.

Thislemmaallowsto omputeaboundontheresponsetime

r

k

(

a

)

ofatra stream

f

k

releasedattimea, denoted

f

k

(

a

)

in thefollowing. Itwasproven[37℄that theexe utionof

f

k

(

a

)

starts,at thelatest,atthetime

t

solutionofthefollowingequationwhi h anbesolvedbyre urren e:

t

= W

k

(

a

, t) + B

k

(

a

) +

a

T

k

t

(13)

|

{z

}

maximumnumberofinstan esin interval

t

withadeadlinelowerthanorequalto a

+ D

k

.C

i

) = max

i=1..m

{C

i

| D

i

>

forallvalues of abutitisprovenin [33,37℄thattheonlysigni ant valuesof aaretheelementsofthefollowingset

A

k

:

A

k

= {t = n · T

i

+ D

i

− D

k

| t ≥ 0

,

t

≤ L − C

k

,

n

∈ N

,

i

= 1...m}

(7)

where

L

isthelongestbusyperiod(longestdurationoftheresour ewithoutidletime,see[33℄for omputation details). Thesigni ant valuesarethe valueswhi himply hanges in the higherpriority workload omputed

with equation3. Thenextse tionshowshowthisanalysis anbeadapted whenpriorityinversions mayo ur

due tothelimitednumberofbitsavailableforen odingthedeadlines.

5.2 WCRT under NP-EDF with quantization error

When the logarithmi s ale is used, frames whi h have distin t deadlines an belong to the same slot. The

initialpriorityorderinggivenbydeadlinesisthenlostandpriorityinversionsmaytakepla e. Intheworst- ase,

when omputingtheWCRTboundofaframe

f

k

(

a

)

,allframeshavingadeadlineinthesameslotareassumed to be of higherprioritythan

f

k

(

a

)

. Pre isely, the higher priorityworkloadis nowmade of allframes having a deadline loweror equalthan

A

k,n

+ D

k

+ S

k

(

a

, t)

, where

S

k

(

a

, t)

is abound on thesize of the slot where a

+ D

k

− t

start

falls. Thusthehigherpriorityworkload

W

k

(

a

, t)

be omes:

W

k

(

a

, t) =

X

i6=k

min

t

T

i

+ 1,

a

+ D

k

+ S

k

(

a

, t) − D

i

T

i

+ 1

+

· C

i

(8)

Ajusti ationoftheaboveformulaisgiveninAppendixB. Adi ultyisthatthelengthofslot

S

k

(

a

, t)

isnot onstantovertime: itdependsonthevalueofthedeadlinethatistobeen oded(i.e. a

+ D

k

),andonthetime originof thelogarithmi s ale(

t

start

), whi h is updatedat ea hnewtransmissionbeginning. As explainedin Appendix B,thesize of

S

k

(

a

, t)

in reases monotoni allywitha

+ D

k

− t

start

. To deriveanupperbound on

S

k

(

a

, t)

, onehastondalowerboundonthevalueof

t

start

atwhi h

f

k

(

a

)

maygainthe hannel. Sin e

f

k

(

a

)

arrivesin a, an obvious lowerbound is a but one an improvethat resulta ording to the s heme proposed

(14)

Whi hatoanalyze Asshownin[33,37℄,theonlysigni antvaluesof

a

thatarereallyneededtobeanalyzed arevalueswhi himply hangesinthehigherpriorityworkload omputedbyequation8. Pre isely,thesevalues

areintegralvaluesoftheform

n

· T

i

+ D

i

− D

k

− S

k

(

a

, t)

; aproblemisthat thelengthoftheslot

S

k

(

a

, t)

isa prioriunknownsin eitdependsalsoona. Thelogarithmi en odings hemepresentedinAppendix Aensures

that,atanytime,thelengthofaslottakesitsvaluein

[U, 2

k

max

_U

_]

,where

U

isthelengthofthersttimeslot. A solutionto apture a hange in the higherpriority workload is thus to onsider all possible values for the

slot's length;theset of atoanalysebe omes:

A

k

= {t = n · T

i

+ D

i

− D

k

− x | t ≥ 0

,

t

5.3 WCRT for NP-ATD poli ies with quantization error

Aframe

f

k,n

underNP-EDFpossessesaninitialpriorityve tor

−

→

P

k,n,

equalto

(A

k,n

+D

k

, k, n)

;NP-EDFisthus aparti ular aseofNP-ATDpoli y where

p

k

: k 7→ D

k

(see denition ofNP-ATD poli iesin paragraph3.2). Inthefollowing,itwill beshownthat Lemma1aswellastheset ofarrivaldates to onsider aftertheASAP

pattern (see equation 7) remain valid withthe onditionthat

D

k

is repla ed by

p

k

. Deadlinebusy periods be omes priority busy periods for a frame

f

k,n

whi h are intervals of network utilization without idle-time during whi h onlyinstan eswithhigherprioritythan

f

k,n

areexe uted.

Lemma2 A bound on the response time of a frame, belonging to tra stream

f

k

, released a units of time after the beginning of its priority busy period an be found in the priority busy period indu ed by the ASAP

pattern. TheASAPpattern isthe situationwhere:

f

k

releasesaframeattimea (otherframes mayhavebeenreleasedbeforetimea),

all frames, for whi h

p

i

is smaller than or equal to a

+ p

k

, are released from time

t

= 0

on at their maximumrate,

thelargestframeof alltra streamswith

p

i

greaterthana

+ p

k

,ifany,isreleasedattime

t

= −1

. This onstitutesthe so- alledblo king fa tor duetonon-preemptiveness.

Sket hOfProof:

(t) =

(

−

→

P

k,n

= (A

k,n

+ p

k

, k, n)

if

t < B

k,n

−

→

Q

k,n

= (−∞, k, n)

if

t

≥ B

k,n

,

where

p

k

,as

D

k

,possessesthepropertythatitsvalueisequalforallframesoftra stream

f

k

. Indeed,thispropertyon

p

k

isneededforlemma 4.1 in[33 ℄to hold(pre isely,whenbuildingthe ASAPpattern,shiftingleftaframemust in reasethe

higherpriorityworkload).

A ordingtolemma1 ,aresponsetimeboundfor

f

k

undervirtual-NP-EDFo ursaftertheASAPpattern,asdenedby Spuri[33 ℄,where

D

k

isrepla edby

p

k

intheequations3and7. Toassessthefeasibility,theresponsetimeboundsjusthave tobe omparedwiththea tualrelativedeadlines

D

k

.

Thewayto omputetheworst aseresponsetimes underaNP-ATD poli y isthesameasunder NP-EDF

(15)

6 Experiments

In this following, the performan es of NP-ATD s hedulingpoli ies are assessed in the ontext of Networked

ControlSystems,whi h isanimportantappli ationeldof ourproposal.

6.1 Sear h spa e

The aim is to nd s heduling poli ies that perform well in terms of feasibility, for optimizing the use of the

networkbandwidth,but that arealsoe ientwithrespe tto the riteriaimportantforNCS (denedin6.2.1

and6.3.2). Inthefollowing,simulationsaredonewithinasub- lassofNP-ATDpoli ieshavingapriorityve tor

−

→

P

k,n

expressedas:

−

→

P

k,n

= (A

k,n

+ c · C

k

+ d · D

k

, k, n)

with

c

∈ [0, 50]

and

d

∈ [0, 1]

, (10) whi h means that, in denition 4, onehas

p

k

: k → c · C

k

+ d · D

k

. Thus, apoli y belonging to our sear h spa eisdenedbyapriorityfun tionhavingtheformofequation10. Simulationshaveshownthatparameters

c

∈ [0, 50]

and

d

∈ [0, 1]

deneasear hspa ewhere, mostgenerally,thepoli iesofinterestaretobefound. This sub- lassof NP-ATD poli ies was hosen be ause itis expe tedto ontain poli ies providinga good

trade-o betweenfeasibility and the satisfa tionof the other riteria importantfor NCS (see 6.2.1). Indeed,

NP-EDF a tually belongs to this lass (

p

k

= D

k

)and poli ies whose priority fun tion is lose to NP-EDF are expe ted to perform similarly in terms of s hedulability. On the other hand, introdu ing a term that

depends onthe transmissiontimeshould help to improvetheother riteria. Inparti ular, it hasbeenshown

that preemptive Shortest Remaining Pro essing Time First (SRPTF) is optimal for average response times

(see[39,40℄quotedin[41℄)and,inoursimulations,Non-PreemptiveShortestMaximumPro essingTimeFirst

(NP-SMPTF), whi h is thenon-preemptiveversionofSRPTF dened with

−

→

P

k,n

= (C

k

, k, n)

, outperformed NP-EDF forall onsidered riteriabesidesfeasibility.

Inthefollowing,theperforman esofNP-ATD poli ies areevaluated againstNP-SMPTFand NP-EDF.In

addition, Non-PreemptiveDeadline Monotoni (NP-DM) is also onsidered be ause it is known for its good

performan esintermsofs hedulability 1

anditseaseof implementation.

6.2 Performan e wrt s heduling

6.2.1 Performan e riteria

Classi ally, a ontrol loop is omprised ofthe sampling (i.e. dataare read from sensors), the omputation of

the ontrol,and thea tuation(i.e. transmissionof the ontroloutputsto thea tuators). Spe i delayshave

beenidentied to impa tthestabilityand, moregenerally,the performan es ofthesystem (see,for instan e,

studiesin [42,1,3℄). Thesedelaysin lude:

the input-outputlaten y: thetimeelapsedbetweenthesamplingandthea tuation,

the samplinginterval: thetimeintervalbetweentwo onse utivesamplingpoints,

the samplinglaten y: thetimeelapsedbetweenthetheoreti alsamplingtimeanditsa tualo urren e.

InNCS,wherethe ontrolloopisdistributedoveranetwork,datatransferedfromthesensorstothe ontroller,

and from the ontroller to the a tuators are ex hanged over a network. The ommuni ations from sensors

to ontroller (response time denoted by

t

sc

) and from ontroller to a tuators (response time denoted by

t

ca

) indu enon- onstantdelaysintheinput-outputlaten y(equalto

t

sc

+

omputationtime

+t

ca

),anditsvariability is known to havea ru ial inuen e on the ontrol-loop performan es (see, for instan e, [1℄ and experiments

1

Non-PreemptiveDeadline Monotoni is optimal wrt to feasibilityfor the non preemptives heduling with

D

k

≤

T

k

when

(16)

of paragraph6.3). An e ient ommuni ationsystemsshouldthus minimize ommuni ationdelaysandtheir

variabilities. Intheexperimentsthatfollow,theimpa tofNP-ATDpoli iesontheresponsetimesofthetra

streams(i.e.,delaybetweenthequeuingtimeandthetransmissionend)isstudied.

6.2.2 Simulation setup

One onsiders several tra streams ex hanged on a CAN network. As to our best knowledge there are no

analyti te hniquesforevaluatingthevaluesofour riteria,simulationisemployed. Agiven riterionisevaluated

forapoli y astheaveragevalueoveralltra streams.

The byte transmission time is the smallest time unit, whi h means that the propagation delay and the

bit-stungarenegle ted(sin ebit-stungdependsonthea tualdatatransmitted,itisoutofthes opeofthis

study). Thetransmissiontimeofaframe,denotedby

C

k

,israndomly hosenbetween

,

U

m

· 1.1

where

U

isthenetworkloadand

m

thetotalnumberoftra streams. Therelativedeadlines

D

k

areuniformally hosen in the interval

T

k

−

T

k

−C

₂

k

, T

k

+

T

k

−C

In this paragraph it is assessed how the parameters

c

and

d

inuen e feasibility and average response time jitters,asmeasuredbythestandarddeviationoftheresponsetimes. Fortheformer riterion,theperforman e

for apoli y

A

is the per entageof tra streams that are feasible under NP-EDF and that remainsfeasible under

A

.

Figures 1(a) and 1(b) shows the performan es of the set of poli ies dened by equation 10 where

d

∈

{0.2, 0.5, 0.9}

and

c

takesitsvaluein

[0, 50]

withstepsequalto

0.5

. Theperforman esarepresentedin terms of feasibility in gure 1(a) and average response time jitter in gure 1(b). The average valuesare omputed

over1500simulationruns:

100

non- on retesetsof

10

tra streamswith

15

dierentosetswithaglobalload randomly hosenintheinterval

[0.6, 0.7]

. Onlytasksetsthat arefeasibleunder NP-EDF weresele ted.

50

55

60

65

70

75

80

85

90

95

100

0

10

20

30

40

50 % of feasible task sets

Parameter ’c’ in priority vector

d=0.2

d=0.5

d=0.9

(a)Feasibility

0

2

4

6

8

10

12

14

16

0

10

20

30

40

50 % of improvement over NP-EDF

Parameter ’c’ in priority vector

d=0.2

d=0.5

d=0.9

(b)Averageresponsetimejitters

Figure1: Feasibilityandaverageresponsetimejittersforpoli iesdenedbyequation10with

d

∈ {0.2, 0.5, 0.9}

and

c

rangingfrom

0

to

50

. Ea hpointistheaveragevalueover

1500

simulationsoftasksetsof ardinality

10

withanaverageloadof

0.65

.

(17)

thelarger thevalue of

c

in

−

→

P

k,n

, thebetterthe averageresponse timejitters. The ounterpartis that feasibilitysigni antlydiminisheswhen

c

in reases.

thelargerthe valueof

d

in

−

→

P

k,n

, the better thefeasibility withthe drawba kthat reponse time jitters in rease.

Indeed,when

c

be omeslargeinequation10,

c

·C

k

hasmoreinuen ethan

A

k,n

and

d·D

k

ontheframepriority. Thus, thepoli y tendstobehavein asimilar mannerasNon-PreemptiveShortestMaximumPro essingTime

First(seeparagraph6.1). Thesamereasonexplainsthatwhen

d

in reases,thepoli yperforms losetoNP-EDF forboth riteria. Itisworthnotingthatnosigni antdieren esamongpoli iesfortheaverageresponsetimes

are observed. This fa t isdue tothe relativelysmall variationsoftransmissiontime

C

k

imposed bytheCAN proto ol.

Forthesakeof omparison,table1presentstheperforman esofNP-SMPTFandNP-DMby omparisonto

NP-EDF withthesamesimulatedtasksets. Fromgure1andtable 1, oneseesthat NP-ATDpoli ies learly

Feasibility(%)

Avgresponsetimejitter

:

improvementoverNP-EDF

NP-SMPTF

57.5

17.9

NP-DM

100 −4.5

Table1: Feasibilityandaverageresponsetimejitters(in

%

)forNP-SMPTFandNP-DM.

outperformsNP-SMPTFintermsoffeasibility(when

c

lowerthan

40

),andisbetterthanNP-DMandNP-EDF fortheresponsetimejitters. Forinstan e,thepoli ydened by

−

→

P

k,n

= (A

k,n

+ 18 · C

k

+

₁₀

2 D

k

, k, n

)leadsto feasibles hedulesforabout

70

%ofthesetsoftra streamswhilea hievinganaverageimprovementof

10.2

% over NP-EDF for the response time jitter. On the same simulation setup, NP-SMPTF is feasible for

57.5

% of thetra stream sets whileimprovingjittersof

17.9

%. On theotherhand,NP-DM leadsto

100

%feasible s hedulesbutisworstthanNP-EDF regardingthejitters.

In pra ti e, most NCS will have better performan es (i.e., lower response time jitter in the simulations)

withawell- hosenNP-ATDpoli yensuringfeasibilitythanunderNP-EDForNP-DM.Furthermore,feasibility

withNP-ATD poli yismu hbetterthanunderNP-SMPTF.NP-ATD poli iesenabletheappli ationdesigner

to implement aMAClevelproto ol providing agood trade-o betweenfeasibility and appli ation-dependent

riteriasu hasjitterminimization.

6.3 Control performan e

In this part,aparti ularNCS is studied and theperforman esof thebest feasibleNP-ATD poli y foundare

ompared to the performan es of NP-EDF. The pro ess is a DC servo with a transfer fun tion dened as

G(s) =

1000

s

2 _+s

. The system, see gure 2, onsists of a ontrol-loop where the sensor, the ontroller and the a tuator nodesaredistributedoveraCAN network at

125

kbit/s. Thereisalsoaninterfering nodegenerating networktra thatmodelstheotherreal-timetra streamsex hangedonthebus. Thissystemissimulated

withthetoolboxTrueTimeunderMatlab/Simulink,andisdes ribedinmoredetailin[43℄. Forthis study,the

NP-ATDMAClayerproto oldes ribedinse tion4.1hasbeenimplementedinTrueTime.

6.3.1 Ar hite ture overview

(18)

Figure2: Thear hite tureoftheNCS onsistsofa ontrol-loopwherethesensor,the ontrollerandthea tuator

nodesaredistributed overaCANnetworkat

125

kbit/s. An interferingnodegeneratesdisturbingtra .

asensornode(onetask-periodi )wherethesensortasksamplesthepro esswithaperiodof

0.01

s. Then, thetask sendsthe result

y(k)

to the ontrolleroverthe network (

C

= 80

bits). The delay between the samplingtimeandthequeuingtimeissetto

9 · 10

−4

s.

a ontroller node (one task -eventdriven)where the ontroller task, ea h time it re eivesaframe from

thesensor, omputesthe ontrol outputsignal

u(k)

andsendsittothea tuator. Thedelaybetweenthe re eption of the frame and the queuing of the ontrol signal frame (

C

= 80

bits) is set to

5 · 10

−4

s. A

PD- ontrollerisused, andimplementeda ordingto thefollowingequations:

u(k) = P (k) + D(k),

P

(k) = 1.5 · (r(k) − y(k)),

D(k) = 3.5 · 10

−5

· D(k − 1) + 5.25 · (y(k − 1) − y(k)),

where

r(k)

,thereferen e,isaunit step(see[43℄ and[44℄formoredetails).

an a tuator node (one task - event-driven) ontrols the a tuator a ordingto the data re eived in the

ontroller'sframe. Thedelaybetweenthere eptionoftheframeandthea tuationis

5 · 10

−4

s.

theinterferingnode(onetask-time-driven)generatesthedisturbingtra . Thesetofitstra streams

israndomlygeneratedasfollows: foragivenload

U

,arandomtra streamisiterativelyaddeduntilthe loadofthenodebe omeshigherthan

U

. Theperiod ofea htra streamisset atrandoma ordingto theuniformlawin theset

{0.01, 0.02, 0.05, 0.1, 0.5}

,andthenumberofdatabytes takesarandomvalue intheinterval

[1, 8]

.

6.3.2 Performan e riteria

Theunitstepresponseofthesystem(inoursimulation,if

t <

0

,

r(t) = 0

else

r(t) = 1

)isusedwiththeovershoot

O

,thesettlingtime

T

settling

andthe

IAE

(IntegraloftheAbsolutemagnitudeError)astheperforman emetri s. Inthisstudy,

IAE

=

R

T

sim

0 |e(t)| dt

,where

T

sim

= 0.5

sisthedurationofthesimulation,andthesettlingtime

(19)

6.3.3 Simulation results

Figure 3shows aninstan e oftheunit step response ofthesystem(at aload

U

= 0.5

)with thebest feasible NP-ATDpoli yofthesear hspa eandwithNP-EDF.Asone ansee,theovershootisredu edfrom

1.155

to

1.085

,while

T

settling

dropsfrom

46 · 10

−3

to

119 · 10

−3

withthebestfeasibleNP-ATDpoli y.

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

0

0.1

0.2

0.3

0.4

0.5 time (s)

unit step reference

NP-EDF

NP-ATDP best

Figure 3: Unit step response of the NCS with NP-EDF and the best feasible NP-ATD poli y found by an

exhaustivesear h(

c

= 35

and

d

= 0.4

). Theloadisequalto

0.5

.

Figure4presentstheaverageperforman esimprovement(over

100

runsforea hpoint)ofthebestfeasible NP-ATDpoli yfoundoverNP-EDFforthethreeperforman emetri sofinterest. Itprovides lear- uteviden es

of theimprovementbroughtbyNP-ATD overNP-EDF fortheNCS under study. Indeed,theimprovementis

signi antwhatevertheloadandin reaseswithit. Forinstan e, ataloadequalto

0.6

, theimprovementover NP-EDF is

35.5

%forthesettlingtime,

30.1

%fortheovershootand

14.3

%forthe

IAE

.

7 Con lusion and future work

Thispaperintrodu esthe lassofNon-PreemptiveArrivalTimeDependent(NP-ATD)poli iesintendedforthe

s hedulingofframesattheMAClevel. Thesepoli iesareeasilyimplementableonCOTS omponents(e.g.,CAN

ontrollers)andprovideagoodtrade-obetweenfeasibilityandthesatisfa tionofotherappli ation-dependent

riteriasu hastheresponsetimejitter. AnNP-ATDpoli y anbebuilt fortheperforman erequirementsof

aparti ularappli ationorbe hosentoprovidegoodaverageperforman esonrandomsetsoftra streams.

InordertoassessthefeasibilityofsystemsusingNP-ATDpoli ies,as hedulabilityanalysisthat isgeneri

for all poli ies of the lass was proposed. In the ontext of Networked Controlled Systems where delaysand

jittersimpa ttheperforman esofthe ontrolloop,simulationshaveshownthatwell hosenpoli ies anbring

signi antimprovementsoverplainNP-EDF orNP-DM.

In a future work, a more a urate evaluation of the impa t of the s heduling poli ies on the ontrolled

(20)

5

10

15

20

25

30

35

40

45

50

0.3

0.4

0.5

0.6

0.7

0.8

0.9 % of improvement over NP-EDF

Load

Setlling time

Overshoot

IAE

Figure 4: Average improvement of overshoot, settling time and

IAE

over NP-EDF with the best feasible NP-ATDpoli yfoundbyanexhaustivesear h.

sear h te hniques for exploring the poli y sear h spa e; preliminary experiments have shown that a simple

neighbourhood te hnique su h a hill- limbing with a Tabu list is more e ient than the exhaustive sear h.

Finally, this work ould be extendedto other lassof poli iessu h astime-sharingpoli y (e.g. Round-Robin

[45℄, Pfair[46℄);themainproblemwillbehereto omeupwithas hedulabilityanalysisvalidforallpoli iesof

the lass.

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A Logarithmi s ale to map priority

This appendix presents the logarithmi s ale, introdu ed in [23℄ for the deadline ase, to map the priority

A

k,n

+ p

k

of a frame

f

k,n

to a limited number

n

of priority bits in order to redu e quantization errors by providingnerresolutionfor loserdeadline.

0

2

1 _p

min

2 h−1

p

min

2 h

p

min

slot blo k

2 h

_p

min

2 h

_S

t

start

p

min

2 h+1

p

min

t

qslots

Figure5: Logarithmi s alestartingat

t

start

,where

p

min

istheminimumrelativepriorityinthetra stream,

h

istheindexoftheblo k omputedwithequation11and

S

isthesizeofthersttimeslot(i.e.

p

min

q

).

max

(

p

max

def

=

max

τ

k

∈T

(p

k

)

)is divided in blo ksof in reasinglength. Ea h blo kis dividedinto

q

slots where

q

isanintegerknownat thedesignstage(seeequation12). Thepriorityof aframe orrespondsto the index of the slot ontainingthe urrent deadline

A

k,n

+ p

k

− t

start

. Index

i

, from [23℄, is omputedasfollow:

i

= h · q +

— A

k,n

+ p

k

− t

start

− 2

h

p

min

2 h

_S

,

where

S

isthesize ofthersttimeslot (

S

def

=

p

min

q

and

p

min

def

=

min

τ

k

∈T

(p

k

)

). Theindexoftheblo k, alled

h

,wherethe urrentdeadline

A

k,n

+ p

k

− t

start

falls,from [23℄,is:

h

=

(

j

log

2 A

k,n

+p

k

−t

start

p

min

k

if

A

k,n

+ p

k

− t

start

> p

min

0

otherwise

.

(11)

Thenumberofslots

q

inablo k(seeAppendix Aof[23℄for omputationdetails)isgivenby:

q

=

$

2 n

⌊log

2 k

max

⌋ +

₂

_{⌊log2kmax ⌋}

k

max

%

,

(12) where

k

max

def

=

p

max

p

min

. Bydenition,thesizeoftheslotwhere

A

k,n

+ p

k

− t

start

falls isequalto

2 h

_S

.

Whenthe logarithmi s aleis used (or when a limitednumber ofbits are devoted for priority en oding),

frames whi h havedistin t deadlines anbelong to the sameslot and an inversion ofprioritymay o ur. In

su ha ase,these errorsare alledquantizationerrors.

B Higher priority workload

W

k

(

a

, t

)

This appendix details omputation of higher priority workload

W

k

(

a

, t)

when priority inversions due to the limitednumberofbits availableforen odingthedeadlineso ur.

(24)

The higher priority workload

1 = max{A

i,j

| j ∈

N, A

i,j

≤ u + t}

arerespe tivelytherstandthelastframeofthetra stream

f

i

inthedeadlinebusyperiod. Thenumberofframesof

f

i

intheinterval

[u, u + t]

is

n

= j

1 − j

0 + 1

.

Thegoalistobound,forea htra stream

f

i

,theworkindu edbyframesof

f

i

in

[u, u + t]

. Theworkof thelast frameoftra stream

f

i

arrivedat

A

i,j

1

istakeintoa ounti:

1.

f

i,j

1

arrivesbefore theend oftheinterval

[u, u + t]

:

A

i,j

1 ≤ u + t

.

A

i,j1

≤ u + t

A

i,j

0 + (n − 1) · T

i

≤ u + t

Thus:

n

=

— t

T

i

+ 1 ≤

t

T

i

+ 1

be ause

n

∈ N

and

A

i,j

0 ≥ u

2. Furthermore, thelast frame

f

i,j

1

must haveaprioritygreater than

f

k

(

a

)

. Forthis purpose, we assume that allframeshavingadeadlinebelongingto thesameslotas

A

k,n

+ D

k

− t

start

are ofhigherpriority than

f

k

(

a

)

. Thus,frameshavingadeadlinelowerorequalthan

A

k,n

+ D

k

+ S

k

(

a

, t)

are onsideredtohave ahigherpriority(

S

k

(

a

, t)

isan upperbound of thesize ofthe slotto whi h

A

k,n

+ D

k

− t

start

belongs, see[24℄for omputationdetails).

A

i,j

1 + D

i

− t

start

≤ u +

a

+ D

k

+ S

k

(

a

, t) − t

n

∈ N

Hen e, an upperbound of the number of frames

n

indu ed by atra stream

f

i

in a deadline busy period of length

t

is:

min

j

t

T

i

k

+ 1,

j

a

+D

k

+S

k

(

a

,t)−D

i

T

i

k

+ 1

. A bound on thehigher priority work

W

k

(

a

, t)

, due to tra streamsdierentfrom

f

k

,isthus:

W

k

(

a

, t) =

X

i6=k

„

min

„— t

T

i

+ 1,

—

a

+ D

k

+ S

k

(

a

, t) − D

i

T

i

+ 1

««

+

· C

i

.

(25)

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