17/04/2015 What does it mean? Re spo nse -time a na lysis c o - PDF document

17/04/2015 What does it mean?  « Re spo nse -time a na lysis »  « c o nditio na l » Response-Time Analysis  « DAG ta sks » of Conditional DAG Tasks  « multipro c e sso r syste ms » in Multiprocessor Systems Alessandra Melani 1 2 What does it mean? What does it mean? DAG: Directed Acyclic Graph  « Re spo nse -time a na lysis »  « Re spo nse -time a na lysis »  « conditional »  « c o nditio na l »  « DAG tasks »  « DAG ta sks »  « multipro c e sso r syste ms »  « multipro c e sso r syste ms » If-then-else statements Switch statements 3 4 In other words Parallel task models  Ma ny pa ra lle l pro g ra mming mo de ls ha ve b e e n pro po se d  We will a na lyze a multiprocessor re a l-time syste ms… to suppo rt pa ra lle l c o mputa tio n o n multipro c e sso r pla tfo rms (e .g ., Ope nMP, Cilk, I nte l T BB)  … b y me a ns o f a schedulability test b a se d o n response- time analysis  … a ssuming Global Fixed Priority o r Global EDF sc he duling po lic ie s  … a nd a ssuming a parallel task model (i.e ., a ta sk is E a rly re a l-time sc he duling mo de ls: Re c e ntly, mo re e xpre ssive mo de lle d a s a Directed Acyclic Graph - DAG ) e a c h re c urre nt ta sk is c o mple te ly e xe c utio n mo de ls a llo w e xplo ita tio n se q ue ntia l o f pa ra lle lism within ta sks 5 6 1

17/04/2015 Fork-join Synchronous-parallel  E  Ge ne ra liza tio n o f the fo rk-jo in mo de l a c h ta sk is a n a lte rna ting se q ue nc e o f se q ue ntia l a nd pa ra lle l se g me nts  Allo ws c o nse c utive pa ra lle l se g me nts  E ve ry pa ra lle l se g me nt ha s a de g re e o f pa ra lle lism � �  Allo ws a n a rb itra ry de g re e o f pa ra lle lism o f e ve ry se g me nt (numb e r o f pro c e sso rs)  Sync hro niza tio n a t se g me nt b o unda rie s: a sub -ta sk in the ne w se g me nt ma y sta rt o nly a fte r c o mple tio n o f a ll sub - ta sks in the pre vio us se g me nt � � ���� 7 8 DAG cp-DAG  Co nditio na l - pa ra lle l DAG (c p-DAG) � � � �� � , � � �  Dire c te d a c yc lic g ra ph (DAG) � � � �� � , � � �  � � � � �,� , … , � �,� � ; � � ⊆ � � ⨯ � �  Ge ne ra liza tio n o f the pre vio us two mo de ls  E ve ry no de is a se q ue ntia l sub -ta sk  Arc s re pre se nt pre c e de nc e c o nstra ints b e twe e n sub -ta sks  T wo type s o f no de s  Regular : a ll suc c e sso rs must b e e xe c ute d in pa ra lle l  Conditional : to mo de l sta rt/ e nd o f a c o nditio na l c o nstruc t (e .g ., if-the n-e lse sta te me nt) � �,�  E a c h no de ha s a WCE T n this le c ture , we will fo c us o n this ta sk mo de l  I 9 10 Conditional pairs Why this restriction?  �� � , � � � fo rm a conditional pair  I t do e s no t ma ke se nse fo r � � to wa it fo r � � if � � is e xe c ute d  � � is a sta rting c o nditio na l no de  � � is the jo ining po int o f the c o nditio na l b ra nc he s sta rting a t � �  Ana lo g o usly, � � c a nno t b e c o nne c te d to � � sinc e o nly o ne  Restriction : the re c a nno t b e a ny c o nne c tio n b e twe e n a is e xe c ute d no de b e lo ng ing to a b ra nc h o f a c o nditio na l sta te me nt  Vio la tio n o f the c o rre c tne ss o f c o nditio na l c o nstruc ts a nd (e .g ., � � ) a nd no de s o utside tha t b ra nc h (e .g ., � � ), the se ma ntic s o f the pre c e de nc e re la tio n inc luding o the r b ra nc he s o f the sa me sta te me nt 11 12 2

17/04/2015 Formal definition (1) Formal definition (2) � ⊆ � � ⊆ � � de no te all e t � � , � � b e a pa ir o f c o nditio na l no de s in a DAG � � � � � , � � .  F o r e a c h � ∈ 1,2, … , � , le t � � a nd � � L � the no de s a nd arc s o n pa ths re a c ha b le fro m � � tha t do he pa ir �� � , � � � is a c o nditio na l pa ir if the fo llo wing ho ld: T no t inc lude � � .  Suppo se the re a re e xa c tly � o utg o ing a rc s fro m � � to the By de finitio n, � � is the so le so urc e no de o f the DAG no de s � � , � � , … , � � , fo r so me � � 1 . T he n the re a re e xa c tly � � � �� � , � � ′� . I � . � � t must ho ld tha t � � is the so le sink no de o f � � inc o ming a rc s into � � in � � , fro m so me no de s � � , � � , … , � � � � � �� � , � � ′� � � � � � � … � � � � � � � … � � � � � � � � � � … … … � � � � … … � � � � � � � � … 13 14 Formal definition (3) Motivating example (1) � ∩ � � � ∅ fo r a ll �, �, � � � .  I t must ho ld tha t �  Why is it impo rta nt to e xplic itly mo de l c o nditio nal � � sta te me nts? Additio nally, with the e xc e ptio n o f �� � , � � � , the re sho uld b e no a rc s in � � into no de s in � � ′ fro m no de s no t in � � ′ , fo r e a c h � ∈ �1,2, … , �� . � ⨯ � � � ��� � , � � �� sho uld ho ld fo r a ll � . ha t is, � � ∩ � � \� T � � � � � � � � … � � … � � � � � � … …  Whic h b ra nc h le a ds to the wo rst-c a se re spo nse -time ? � � … � � 15 16 Motivating example (2) Motivating example (3) ≥ 3 processors • 1 processor • Upper-branch U ppe r-b ra nc h 10 L o we r-b ra nc h 10 Lower-branch 18 2 processors • 6 3 processors + 1 interfering task of 6 time-units • U ppe r-b ra nc h 10 Lower-branch U ppe r-b ra nc h Lower-branch 10 12 12 17 18 3

17/04/2015 Motivating example (4) System model  T his e xa mple sho ws tha t it ma ke s se nse to e nric h the ta sk  � c o nditio nal-pa ra lle l ta sks (c p-ta sks) τ � , e xpre sse d a s c p- mo de l with c o nditio nal sta te me nts whe n de a ling with DAGs in the fo rm � � � �� � , � � � parallel task models  pla tfo rm c o mpo se d o f � ide ntic a l pro c e sso rs  De pe nding o n the numb e r o f pro c e sso rs a nd o n the  sporadic o the r ta sks, no t a lwa ys the sa me b ra nc h le a ds to the � � a rrival pa tte rn (minimum inte r-a rriva l time wo rst-c a se re spo nse -time b e twe e n jo b s o f ta sk τ � )  Why we do no t mo de l c o nditio na l sta te me nts a lso with  constrained re la tive de a dline � � � � � se q ue ntia l ta sk mo de ls?  Co nditio nal b ra nc he s a re inc o rpo ra te d in the no tio n o f WCE T Pro b le m: c o mpute a safe upper-bound o n the re spo nse -time (lo ng e st c ha in o f e xe c utio n)  T he o nly pa ra me te rs ne e de d to c o mpute the re spo nse -time o f a o f e a c h c p-ta sk, with a ny wo rk-c o nse rving a lg o rithm ta sk a re the WCE T s, pe rio ds a nd de a dline s o f e a c h ta sk in the (inc luding Glo b a l F P a nd Glo b a l E DF ) syste m 19 20  Quantities of interest 1. Chain (or path) A c hain (o r pa th) o f a c p-ta sk τ � is a se q ue nc e o f no de s 1. Cha in (o r pa th) o f a c p-ta sk λ � �� �,� , … , � �,� � suc h tha t � �,� , � �,��� ∈ � � , ∀� ∈ ��, �� . 2. L o ng e st pa th 3. Vo lume 4. Wo rst-c a se wo rklo a d 5. Critic a l c ha in 21 22 1. Chain (or path) 2. Longest path he lo ng e st pa th � � o f a c p-ta sk τ � is any so urc e -sink c ha in o f T A c hain (o r pa th) o f a c p-ta sk τ � is a se q ue nc e o f no de s the ta sk tha t a c hie ve s the lo ng e st le ng th λ � �� �,� , … , � �,� � suc h tha t � �,� , � �,��� ∈ � � , ∀� ∈ ��, �� . ������ ���� � � a lso re pre se nts the time re q uire d to e xe c ute it whe n the he le ng th o f the c hain, de no te d b y ����λ� , is the sum o f the T numb e r o f pro c e ssing units is infinite (la rg e e no ug h to a llo w WCE T s o f a ll its no de s: � ma ximum pa ra lle lism) ��� λ � � � �,� Ne c e ssa ry c o nditio n fo r fe a sib ility: � � � � � ��� 23 24 4