Aperiodic and Sporadic Jobs. Scheduling Aperiodic and Sporadic Jobs

Transcription

1 CPSC-663: Ral-Tm Sym Arodc and Soradc Job Schdulng Arodc and Soradc Job Dfnon Comaron o radonal chdulng of aynchronou vn Pollng Srvr Dfrrabl Srvr Soradc Srvr Gnralzd Procor Sharng Conan Ulzaon Srvr Toal Bandwdh Srvr Prmv Wghd Far Quung Schdulng Arodc and Soradc Job Whn varaon n nr-rla m and xcuon m ar mall: can ra a a rodc a T=(, ), and chdul accordngly. Wha abou oradc job? can arrv a any m xcuon m vary wdly dadln ar unown a ror? Gvn: n rodc a T,, T = (, ),, T n rory-drvn chdulng algorhm W wan o drmn whn o xcu arodc and oradc job,.., oradc job: accanc chdulng of accd job arodcjob: chdul job o coml ASAP.

2 CPSC-663: Ral-Tm Sym Arodc and Soradc Job Prory Quu for Prodc/Soradc/Arodc Job rjc Soradc Job Accanc T Prodc Job Procor Arodc Job Schdulng Algorhm, Ovrvw Schdulng Arodc (Soradc) Job: Non-Ral-Tm Sym Ral-Tm Sym Bacground Slac Salng Inrru-Drvn Polld Bandwdh Prrvng 2

3 CPSC-663: Ral-Tm Sym Arodc and Soradc Job Bacground/Inrru-Drvn v. Slac Salng Bacground: Arodc job quu ha alway low rory among all quu. Prodc a and accd job alway md dadln. Sml o mlmn. Excuon of arodc job may b unduly dlayd. Inrru-Drvn: Ron m a hor a obl. Prodc a may m om dadln. Slac Salng: Poon xcuon of rodc a only whn af o do o: Wll-ud for cloc-drvn nvronmn. Wha abou rory-drvn nvronmn? (qu comlcad) Examl Bacground: T T 2 = = ( 3, ) ( 0, 4 ) A : r = 0., = 2. Inrru-Drvn: Slac Salng: 3

4 CPSC-663: Ral-Tm Sym Arodc and Soradc Job Polld Excuon v. Bandwdh Prrvng Srvr Pollng rvr (, ): chduld a rodc a. : Pollr rady for xcuon vry m un. : Ur bound on xcuon m. Trmnology: (Excuon) budg: Rlnhmn: budg o a bgnnng of rod. Pollr conum budg a ra whl xcung arodc job. Pollr xhau budg whnvr ollr fnd arodc quu my. Whnvr h budg xhaud, h chdulr rmov h ollr from rodc quu unl rlnhd. Bandwdh-rrvng rvr algorhm: Imrov uon ollng aroach U rodc rvr Ar dfnd by conumon and rlnhmn rul. Examl: Pollng Srvr Ra-Monoonc: PS=(3, ) A : r = 2.8, =.7 T =(φ=2, 3.5,.5) T 2 =(φ=0, 6.5, 0.5) budg 4

5 CPSC-663: Ral-Tm Sym Arodc and Soradc Job Dfrrabl Srvr Rul: Conumon: Excuon budg conumd only whn rvr xcu. Rlnhmn: Excuon budg of rvr o a ach mull of. Prrv budg whn no arodc job rady. Any budg hld ror o rlnhmn lo (no cumulaon). Examl: Dfrrabl Srvr wh RM Ra-Monoonc: DS=(3, ) A : r = 2.8, =.7 T =(φ=2, 3.5,.5) T 2 =(φ=0, 6.5, 0.5) budg 5

6 CPSC-663: Ral-Tm Sym Arodc and Soradc Job Examl: Dfrrabl Srvr wh EDF EDF: DS=(3,) A : r = 2.8, =.7 T =(φ=2,3.5,.5) T 2 =(φ=0,6.5,0.5) budg Combnaon of Dfrabl Srvr wh Bacground Srvr DS=(3,) A : r = 2.8, =.7 rv n bacground! T =(φ=2,3.5,.5) T 2 =(φ=0,6.5,0.5) budg 6

7 CPSC-663: Ral-Tm Sym Arodc and Soradc Job Why no Incra h Budg? DS=(3,) T =(3.5,.5) T 2 =(6.5,0.5) Schdulably for Sac-Prory Sym (DS ha hgh rory) Lmma: In a ac-rory rodc ym wh D <=, wh a dfrabl rvr T DS (, ) wh hgh rory, a crcal nan for T han whn: () r,c = 0 for om job J,c n T. (2) job of hghr-rory a ar rlad a m 0. (3) budg of (bacloggd) rvr a m 0. (4) nx rlnhmn m 0. Inuvly: Low-rory a uffr from a bac-o-bac h by h dfrabl rvr. 7

8 CPSC-663: Ral-Tm Sym Arodc and Soradc Job 8 Tm-Dmand Analy Schdulabl Ulzaon: Gnrally, no nown chdulabl ulzaon. Only xcon: < < 2 < < n < 2 =D ra-monoonc chdulng n > For h ca, h chdulabl ulzaon w = = ) ( = 2 ) ( ) ( / n DS RM u u n n U Dfrabl Srvr and Arbrary Sac Prory Problm: Any budg ha no conumd a nd of rvr rod lo. Maxmum amoun of m DS can conum dnd on Rla m of all rodc job (wh rc o rlnhmn m) Excuon m of all a. Ur bound on m dmand for lowr-rory a han DS: Mull dfrabl rvr: Tm dmand for a wh rory lowr han m DS : w = ) ( q m q q q w = =,,, ) (

9 CPSC-663: Ral-Tm Sym Arodc and Soradc Job Ung h Schdulabl Ulzaon: Aum ha T ha lowr rory han rvr. T DS (, ) bhav l a rodc a (, ), xc ha may xcu for a mo addonal m un durng h nrval (r,c,r,c D ). = u u b U X ( ) Examl: chdulng algorhm T T T T DS 2 3 ( 3, 06. ) ( 4, 08. ) ( 5, 05. ) ( 7, 4. ) T : T 2 : T 3 : no affcd by T DS 3 2 = = u u = = U 2 u u = = U 3 no! RM (3) RM (4) Schdulably for Dadln-Drvn Sym Lmma: A rodc a T n a ym of n ndndn, rmv rodc a chdulabl wh a DS wh rod, xcuon m, and ulzaon u, accordng o h EDF algorhm f n = mn( D, u ) D 9

10 CPSC-663: Ral-Tm Sym Arodc and Soradc Job 0 Proof Proof: L b h dadln of om Job J,c. L - b h la on n m bfor whr hr rocor dl, or wa xcung a lowr-rory a (dadln afr ) If J c m dadln a m, oal amoun of rocor m conumd by Dfrrabl Srvr durng nrval ( -, ] boundd by : J c m dadln hr! Proof (II) Proof: Tm conumd by dfrrabl rvr: Tm conumd by Ta T : W ud floor nad of clng bcau la nvocaon ha dadln afr. W m dadln f w don hav nough m o fnh by m : Dvd by ( - ) and go from hr. ) ( ) ( DS u w = = w / ) ( ) ( = = < n u ) ( ) (

11 CPSC-663: Ral-Tm Sym Arodc and Soradc Job Soradc Srvr Problm wh Dfrabl Srvr: T DS (, ) may dlay lowr-rory job longr han rodc a T(, ). Soradc Srvr (SS): Nvr u mor m han h rodc a T(, ) wh am aramr. If o, w can ra T SS ju a a rodc a. Soradc Srvr n Sac-Prory Sym Noaon: T : Ta ym wh n a. T SS : Soradc rvr, arbrary rory. T H : Sub of T wh hghr rory han T SS. r : La rlnhmn m. f : Fr nan afr r a whch rvr bgn o xcu. : Effcv rlnhmn m. Th chdulr drmn bad on hory and nx rlnhmn m o.

12 CPSC-663: Ral-Tm Sym Arodc and Soradc Job Sml Soradc Srvr Conumon Rul: Th rvr xcuon budg conumd a h ra of on a any m afr r unl h budg xhaud whnvr h followng wo condon ar ru. Whn h condon ar no ru, h rvr hold budg: C: Th rvr xcung. C2: Th rvr ha xcud nc r and undd a h m, and T H dl. Rlnhmn Rul: R: Th xcuon budg o and h currn m r rcordd nally whn h ym bgn xcuon and ach m whn h budg rlnhd. R2: Th nx budg rlnhmn m drmnd a m f whn h rvr fr bgn o xcu nc r. A m f, o h la m nan a whch a lowr-rory a xcu n ( r, f ), and o r f T H buy hroughou h nrval. Th nx rlnhmn m a. R3: Th nx rlnhmn occur a h nx rlnhmn m, xc undr h followng condon whn h rlnhmn may b don oonr or lar. (a) If h nx rlnhmn m arlr han f, h budg rlnhd a oon a xhaud. (b) Th budg rlnhd a m whnvr h ym T ha bn dl bfor and a rodc job rlad a. Sml Soradc Srvr: Examl T = (3, 0.5) T 2 = (4,.0) T 3 = (9, 4.5) T S = (5,.5) T T 2 A (r=3, =) A 2 (r=7, =2) A 3 (r=5.5, =2) T S T Budg 2

13 CPSC-663: Ral-Tm Sym Arodc and Soradc Job A Suaon whr Rul 3a Al T H rvr T L a a f Informal Proof of Corrcn Corrcn : Th rvr nvr dmand mor m n any nrval han corrondng rodc a T = (, ). For now: T ha no bn dl, and Rul R3(b) ha nvr ald. W how ha rvr mula Ta T =(, ). For h, w vw rlnhmn m a nomnal rla m of rvr job. Rul C: Each rvr job nvr xcu for mor han budg.. Rul C2: Budg of dl oradc rvr dcra a f rvr wa xcung. Each rvr job only xcu a m whn a job of T would. On h ohr hand: C2 man ha rvr hold on o budg whn Job n T H xcung. (obvouly corrc) Srvr ha no xcud nc r. (Acual rla m can b lar han nomnal rla m.) 3

14 CPSC-663: Ral-Tm Sym Arodc and Soradc Job Informal Proof of Corrcn (con.) Rul R2 and R3(a) ma ur ha nx rlnhmn m alway m un lar han ffcv rla m. Th nx ffcv rla m nvr arlr han nx rlnhmn m. R2: Ma ffcv rla m a arly a obl. R3(a): Emula uaon whr job n T a mor m o coml han on rod. R3(b): Alcabl only whn buy nrval of rodc a ym nd, and a nw on ar. Bhavor of a n old buy rod do no affc nw buy rod. Th condon alrady accound for n chdulably analy of T and T. Emulang Gnralzd Procor Sharng Gnralzd Procor Sharng (b-by-b Round Robn): Tmng olaon. Emula GPS by (for xaml) Conan Ulzaon Srvr Toal Bandwdh Srvr Wghd Far-Quung Srucur: Run rvr algorhm on o of EDF chdulr. 4

15 CPSC-663: Ral-Tm Sym Arodc and Soradc Job Schdulng Soradc Job wh EDF Dfnon: Dny of oradc job J wh rla m r, maxmum xcuon m and dadln d : dny = /(d -r ). Thorm: A ym of ndndn, rmabl oradc job chdulabl accordng o EDF f h oal dny of all acv job n h ym no grar han a all m. Schdulng Soradc Job wh EDF (con) Thorm no ncary! Examl: dny

16 CPSC-663: Ral-Tm Sym Arodc and Soradc Job Schdulng Soradc Job wh EDF (con) Soradc a S a ram of oradc job S, S 2, S 3,... Excuon m of S j j. Prod j m bwn nvocaon of S j and S (j). Inananou ulzaon of oradc job S j : j / j. Inananou ulzaon of oradc a S : u = max j ( j / j ). Corollary: A ym of n ndndn, rmabl oradc a, whch uch ha h rlav dadln of vry job qual o rod, chdulabl on a rocor accordng o h EDF algorhm f h oal nananou ulzaon qual or l o. Conan Ulzaon Srvr Algorhm A conan ulzaon rvr mula a oradc a wh a conan nananou ulzaon. Conumon rul: A rvr conum budg only whn xcu. Rlnhmn rul (aum: rvr allocad ulzaon u ): R Inally, := 0 and d :=0. R2 Whn an arodc job wh xcuon m arrv a m o an my arodc job quu, (a) f < d, do nohng. (b) f >= d, :=, and d := /u. R3 A h dadln d of h rvr, (a) f h rvr bacloggd, := and d := d /u (b) f h rvr dl, do nohng. 6

17 CPSC-663: Ral-Tm Sym Arodc and Soradc Job Conan Ulzaon Srvr: Examl T = (3, 0.5) T 2 = (4,.0) T 3 = (9, 4.5) T T 2 T 3 A (r=3, =) A 2 (r=6.9, =2) A 3 (r=5.5, =2) T CU (u =25%) Budg Wha abou Unnown Excuon Tm? Aumon for conan ulzaon rvr: xcuon m of arodc job ar nown uon arrval. Rrcv. Pobl oluon: Agn fxd bandwdh o rvr: fxd budg fxd rod /u Uon job comlon of job wh xcuon m <, rduc currn dadln of rvr by ( -)/u bfor rlnhng agan. For xcuon m >, u mor han on rvr rod. 7

18 CPSC-663: Ral-Tm Sym Arodc and Soradc Job Problm wh Conan Ulzaon Srvr: Unud Caacy T = (3, 0.5) T 2 = (4,.0) T 3 = (9, 4.5) T T 2 T 3 A (r=3, =) A 2 (r=6.9, =2) A 3 (r=4, =2) T S Budg! d=5 Toal Bandwdh Srvr Allow rvr o u bacground m. Conumon rul: A rvr conum budg only whn xcu. Rlnhmn rul: R Inally, := 0 and d :=0. R2 Whn an arodc job wh xcuon m arrv a m o an my arodc job quu, d := max(d,) /u, and :=. R3 Uon comlon of h currn arodc job, rmov job from quu. (a) f h rvr bacloggd, d := d /u and := ; (b) f h rvr dl, do nohng. 8

19 CPSC-663: Ral-Tm Sym Arodc and Soradc Job Unud Caacy Elmnad wh Toal Bandwdh Srvr T = (3, 0.5) T 2 = (4,.0) T 3 = (9, 4.5) T T 2 T 3 A (r=3, =) A 2 (r=6.9, =2) A 3 (r=4, =2) T S Budg d=5 Corrcn of Toal Bandwdh Srvr Conan Ulzaon Srvr corrc. How do Toal Bandwdh Srvr affc rodc a dffrnly? Only nrng ca: Budg of Toal Bandwdh Srvr rlnchd a m bfor dadln. Nw dadln d = d /u. How do h affc h xcuon of rodc a? Ca : Currn rodc job J,c ha dadln bfor d xcuon of rodc job no affcd. Ca 2: Currn rodc job J,c ha dadln afr d Ca 2.: Currn rodc job J,c rady bfor m xcuon m dmandd by Toal Bandwdh Srvr from r, o d am a for Conan Ulzaon Srvr. Ca 2.2: Currn rodc job J,c rady afr m xcuon m dmandd by Toal Bandwdh Srvr from r, o d l han ha of Conan Ulzaon Srvr. 9

20 CPSC-663: Ral-Tm Sym Arodc and Soradc Job Nonrmabl Poron Nonrmabl oron hr rduc chdulabl ulzaon or nroduc ardn. Dfnon: b max (n) maxmum xcuon m of nonrmabl oron of rodc a and job xcud by rvr. ffcv xcuon m of job xcud by rvr: rao of job xcuon m and rvr z. D mn mnmum of all rlav dadln of rodc a and ffcv xcuon m of job xcud by all rvr n h ym. Corollary: Whn a ym of rodc a chduld wh on or mor oal bandwdh and conan ulzaon rvr on h EDF ba, vry rodc a and vry rvr m dadln f h um of h oal dny of h rodc a and h oal z of all rvr no grar han -b max (n)/d mn. Farn and Sarvaon Toal Bandwdh Srvr no far Examl: TB and TB 2 ach of z 0.5 If boh rvr nvr dl, rvc aroxmaly qually hard among rvr. Wh dlng rvr, h no alway h ca. 0 TB bacloggd TB 2 dl lo of hor job arrv for TB 2 dadln for TB >= 2 2 Procor m allocad farly durng (0,2), bu no durng (, 2) 20

21 CPSC-663: Ral-Tm Sym Arodc and Soradc Job Farn & Sarvaon Dfnon (Farn): w (, 2 ) = oal aand rocor m for Srvr durng m nrval (, 2 ). w (, 2 )/u = normalzd rvc. Schdulr far durng nrval (, 2 ) f normalzd rvc aand by all rvr do no dffr by mor han a farn hrhold FR. Idally, FR zro: w (, ) u 2 = w (, ) = u ( ) 2 2 w (, ) u j 2 j Elmnang Sarvaon Problm wh Toal Bandwdh rvr: Whn rocng m avalabl, allow o ndfnly u dadln no h fuur. Conan Ulzaon rvr dadln clo : d - <=,max / u, whr,max max. xcuon m of job rvd by Srvr. Rlnhmn rul for arvaon-fr Conan Ulzaon / Bacground rvr: R - R3 : Sam a Conan Ulzaon rvr. R4 : Whnvr buy nrval nd, rlnh budg of all bacloggd rvr. No: Bacground m no drbud o bacloggd rvr accordng o hr z => arvaon lmnad, bu no unfarn. 2

22 CPSC-663: Ral-Tm Sym Arodc and Soradc Job Prmv Wghd Far-Quung Algorhm Rlnhmn rul mlar o Toal Bandwdh rvr; xc for comuaon of dadln a ach rlnhmn m. WFQ algorhm bound farn. Rlnhmn rul of WFQ rvr ma mula GPS rvr wh am z. u =/ u 2 =/8 u 3 =/ u 4 =3/8 8 3 Vrual Tm: Enquu job n ordr of fnh numbr: numbr of round for GPS rvr o xhau budg. Rul for WFQ Schdulng Rul: Agn ror n ordr of ncrang fnh numbr. Conumon Rul: WFQ rvr conum budg only whn xcu. Inalzaon Rul: I: Whn ym dl, FN = 0, U b = 0, - = 0. Budg of all rvr ar zro. I2: Whn fr job arrv a m wh xcuon m a om rvr FQ whn ym dl: (a) - :=, and U b := U b u, and (b) budg of FQ o and fnh numbr fn := /u. Rul for udang Fnh Tm durng Sym Buy Inrval: R: Whn job arrv a quu FQ whl FQ dl (a) ncrmn ym fnh numbr FN := FN (- - )/U b (b) - :=, and U b := U b u, and (c) budg of FQ o and fnh numbr fn := FN /u, nquu rvr R2: Whnvr FQ coml job (a) f rvr rman bacloggd, rvr budg o and ncrmn fnh numbr: fn := fn /u. (b) f rvr bcom dl, uda Ub and FN a follow: FN := FN (- - )/U b, - :=, and U b := U b -u. 22

23 CPSC-663: Ral-Tm Sym Arodc and Soradc Job Schdulng Soradc Ta n EDF Sym Toal dny of rodc a. A long a oal dny of oradc job do no xcd -, all dadln can b manand by EDF. Accanc T: Manan l of m nrval and hr dn: Accanc hn add nw dny: dny of nw job 23