Quantifying the Sub-optimality of Uniprocessor Fixed Priority Pre-emptive Scheduling for Sporadic Tasksets with Arbitrary Deadlines

Transcription

1 Quantfyng the Sub-optmalty of Unprocessor Fxed Prorty Pre-emptve Schedulng for Sporadc Tasksets wth Arbtrary Deadlnes Robert Davs, Sanjoy Baruah, Thomas Rothvoss, Alan Burns To cte ths verson: Robert Davs, Sanjoy Baruah, Thomas Rothvoss, Alan Burns Quantfyng the Sub-optmalty of Unprocessor Fxed Prorty Pre-emptve Schedulng for Sporadc Tasksets wth Arbtrary Deadlnes Laurent George and Marylne Chetto andmkael Sjodn 17th Internatonal Conference on Real-Tme and Network Systems, Oct 2009, Pars, France pp23-34, 2009 <nra > HAL Id: nra Submtted on 17 Dec 2009 HAL s a mult-dscplnary open access archve for the depost and dssemnaton of scentfc research documents, whether they are publshed or not The documents may come from teachng and research nsttutons n France or abroad, or from publc or prvate research centers L archve ouverte plurdscplnare HAL, est destnée au dépôt et à la dffuson de documents scentfques de nveau recherche, publés ou non, émanant des établssements d ensegnement et de recherche franças ou étrangers, des laboratores publcs ou prvés

2 In Proc of the 17th Internatonal Conference on Real-Tme and Network Systems RTNS'2009, Pars, ECE, October, 2009 Quantfyng the Sub-optmalty of Unprocessor Fxed Prorty Pre-emptve Schedulng for Sporadc Tasksets wth Arbtrary Deadlnes Robert I Davs ( ) Real-Tme Systems Research Group, Department of Computer Scence, Unversty of York, York, UK robdavs@csyorkacuk Sanjoy K Baruah Department of Computer Scence, Unversty of North Carolna, Chapel Hll, NC , Carolna, USA baruah@csuncedu Thomas Rothvoß Ecole Polytechnque Federale de Lausanne, Insttute of Mathematcs, Staton 8 - Bâtment MA, CH-1015 Lausanne, Swtzerland thomasrothvoss@epflch Alan Burns Real-Tme Systems Research Group, Department of Computer Scence, Unversty of York, York, UK alanburns@csyorkacuk Abstract Ths paper examnes the relatve effectveness of fxed prorty pre-emptve schedulng n a unprocessor system, compared to an optmal algorthm such as Earlest Deadlne Frst (EDF) The quanttatve metrc used n ths comparson s the processor speedup factor, defned as the factor by whch processor speed needs to ncrease to ensure that any taskset that s schedulable accordng to an optmal schedulng algorthm can be scheduled usng fxed prorty pre-emptve schedulng For mplct-deadlne tasksets, the speedup factor s 1/ln(2) For constraned-deadlne tasksets, the speedup factor s 1/ Ω In ths paper, we show that for arbtrarydeadlne tasksets, the speedup factor s lower bounded by 1/ Ω and upper bounded by 2 Further, when deadlne monotonc prorty assgnment s used, we show that the speedup factor s exactly 2 1 Introducton In ths paper, we are nterested n determnng the largest factor by whch the processng speed of a unprocessor would need to be ncreased, such that any feasble taskset (that was prevously schedulable accordng to an optmal schedulng algorthm) could be guaranteed to be schedulable accordng to fxed prorty pre-emptve schedulng We refer to ths resource augmentaton factor as the processor speedup factor [14] In 1973, Lu and Layland [18] consdered fxed prorty pre-emptve schedulng of synchronous 1 tasksets comprsng ndependent perodc tasks, wth bounded executon tmes, and deadlnes equal to ther perods We refer to such tasksets as mplct-deadlne tasksets Lu and Layland showed that rate monotonc prorty orderng (RMPO) s the optmal fxed prorty assgnment polcy for mplct-deadlne tasksets, and that usng rate monotonc prorty orderng, fxed prorty pre-emptve schedulng can schedule any mplct-deadlne taskset wth a total utlsaton U ln(2) 0693 Lu and Layland also showed that Earlest Deadlne Frst (EDF) s an optmal dynamc prorty schedulng algorthm for mplct-deadlne tasksets, and that EDF can schedule any such taskset wth a total utlsaton U 1 In 1974, Dertouzos [11] showed that EDF s n fact an optmal pre-emptve unprocessor schedulng algorthm, n the sense that f a vald schedule exsts for a taskset, then the schedule produced by EDF wll also meet all deadlnes Combnng the result of Dertouzos [11] wth the results of Lu and Layland [18] for both EDF and fxed prorty pre-emptve schedulng, we can see that the processor speedup factor requred to guarantee that fxed prorty preemptve schedulng can schedule any feasble mplctdeadlne taskset s 1/ ln(2) Research nto real-tme schedulng durng the 1980 s and early 1990 s focussed on lftng many of the restrctons of the Lu and Layland task model Task arrvals were permtted to be sporadc, wth known 1 A taskset s synchronous f all of ts tasks share a common release tme

3 In Proc of the 17th Internatonal Conference on Real-Tme and Network Systems RTNS'2009, Pars, ECE, October, 2009 mnmal nter-arrval tmes, (stll referred to as perods), and task deadlnes were permtted to be less than or equal to ther perods (so called constraned deadlnes) or less than, equal to, or greater than ther perods (so called arbtrary deadlnes) In 1982, Leung and Whtehead [15] showed that deadlne monotonc 2 prorty orderng (DMPO) s the optmal fxed prorty orderng for constraned-deadlne tasksets Exact fxed prorty schedulablty tests for constraned-deadlne tasksets were ntroduced by Joseph and Pandya n 1986 [13], Lehoczky et al n 1989 [17], and Audsley et al n 1993 [1] In 1990, Lehoczky [16] showed that deadlne monotonc prorty orderng s not optmal for tasksets wth arbtrary deadlnes; however, an optmal prorty orderng for such tasksets can be determned, n at most n ( n +1) / 2 task schedulablty tests, usng Audsley s optmal prorty assgnment algorthm 3 [1] Exact schedulablty tests for tasksets wth arbtrary deadlnes were developed by Lehoczky [16] n 1990 and Tndell et al n 1994 [20] Exact EDF schedulablty tests for both constraned and arbtrary-deadlne tasksets were ntroduced by Baruah et al n 1990 [6], [7] In 2008, Baruah and Burns [5] showed that the processor speedup factor for constraned-deadlne tasksets s lower bounded by 15 and upper bounded by 2 In 2009, Davs et al [10] derved the exact speedup factor for constraned-deadlne tasksets; 1/ Ω (where Ω s the mathematcal constant defned by the transcendental equaton ln( 1/ Ω) = Ω, hence, Ω ) In ths paper, we derve the speedup factor for fxed prorty pre-emptve schedulng of arbtrary-deadlne tasksets We are able to gve an exact speedup factor when deadlne monotonc prorty assgnment s used, and upper and lower bounds assumng an optmal prorty assgnment It s known that an exact condton for the schedulablty of a constraned or arbtrary-deadlne taskset under an optmal pre-emptve unprocessor schedulng algorthm, such as EDF [11], s that a quantty referred to as the processor LOAD (see Secton 23) does not exceed the capacty of the processor (e LOAD 1 ) [6], [7] The processor speedup factor derved n ths paper shows that every arbtrary-deadlne taskset wth LOAD 05 s guaranteed to be schedulable accordng to fxed prorty pre-emptve schedulng usng ether 2 Deadlne monotonc prorty orderng assgns prortes n order of task deadlnes, such that the task wth the shortest deadlne s gven the hghest prorty 3 Ths algorthm s optmal n the sense that t fnds a schedulable prorty orderng whenever such an orderng exsts deadlne-monotonc prorty assgnment or an optmal prorty assgnment algorthm Ths result complements the earler results of Davs et al [10] that every constraned-deadlne taskset wth LOAD Ω s guaranteed to be schedulable accordng to fxed prorty pre-emptve schedulng usng deadlne-monotonc prorty assgnment; and the semnal result of Lu and Layland [18] ( U ln( 2) ), that apples to mplct-deadlne tasksets Whle the results presented n ths paper are manly theoretcal, they also have practcal utlty n enablng system desgners to quantfy the maxmum penalty for usng fxed prorty pre-emptve schedulng n terms of the addtonal processng capacty requred Ths performance penalty can then be weghed aganst other factors such as mplementaton overheads when consderng whch schedulng algorthm to use 11 Related work on average case sub-optmalty Ths paper examnes the sub-optmalty of fxed prorty pre-emptve schedulng n the worst-case, other research has examned ts behavour n the average-case In 1989, Lehoczky et al [17] ntroduced the breakdown utlsaton metrc: A taskset s randomly generated, and then all task executon tmes are scaled untl a deadlne s just mssed The utlsaton of the scaled taskset gves the breakdown utlsaton Lehoczky et al showed that the average breakdown utlsaton, for mplct-deadlne tasksets of large cardnalty under fxed prorty preemptve schedulng s approxmately 88%, correspondng to a penalty of approxmately 12% of processng capacty wth respect to an optmal algorthm such as EDF In 2005, Bn and Buttazzo [8] showed that breakdown utlsaton suffers from a bas whch tends to penalse fxed prorty schedulng by favourng tasksets where the utlsaton of ndvdual tasks s smlar Bn and Buttazzo ntroduced the optmalty degree metrc, defned as the number of tasksets n a gven doman that are schedulable accordng to some algorthm A dvded by the number that are schedulable accordng to an optmal algorthm Usng ths metrc, they showed that the penalty for usng fxed prorty-pre-emptve schedulng for mplct-deadlne tasksets s typcally sgnfcantly lower than that assumed by determnng the average breakdown utlsaton 12 Organsaton The remander of ths paper s organsed as follows Secton 2 descrbes the system model and notaton used, and recaptulates exact schedulablty analyss for both fxed prorty and EDF schedulng Secton 3 llustrates the processor speedup factor va a smple example Secton 4

4 In Proc of the 17th Internatonal Conference on Real-Tme and Network Systems RTNS'2009, Pars, ECE, October, 2009 derves the processor speedup factor requred for arbtrarydeadlne tasksets under fxed prorty pre-emptve schedulng Secton 5 concludes wth a summary of the results 2 Schedulng model and schedulablty analyss In ths secton, we outlne the schedulng model, notaton and termnology used n the rest of the paper We then recaptulate the exact schedulablty analyss for both fxed prorty pre-emptve schedulng and EDF schedulng 21 Schedulng model, termnology and notaton In ths paper, we consder the pre-emptve schedulng of a set of tasks (or taskset) on a unprocessor Each taskset comprses a statc set of n tasks ( τ 1 τ n ), where n s a postve nteger We assume that the ndex of task τ also represents the task prorty used n fxed prorty pre-emptve schedulng, hence τ 1 has the hghest fxed-prorty, and τ n the lowest Each task τ s charactersed by ts bounded worst-case executon tme C, mnmum nter-arrval tme or perod T, and relatve deadlne D Each task τ therefore gves rse to a potentally nfnte sequence of nvocatons, each of whch has an executon tme upper bounded by C, an arrval tme at least T after the arrval of ts prevous nvocaton, and an absolute deadlne D tme unts after ts arrval In an mplct-deadlne taskset, all tasks have D = T In a constraned-deadlne taskset, all tasks have D T, whle n an arbtrary-deadlne taskset, task deadlnes are ndependent of ther perods, thus each task may have a deadlne that s less than, equal to, or greater than, ts perod The set of arbtrary-deadlne tasksets s therefore a superset of the set of constraned-deadlne tasksets, whch s tself a superset of the set of mplct deadlne tasksets The utlsaton U, of a task s gven by ts executon tme dvded by ts perod ( U = C / T ) The total utlsaton U, of a taskset s the sum of the utlsatons of all of ts tasks: n C U = (1) = 1 T The followng assumptons are made about the behavour of the tasks: o The arrval tmes of the tasks are ndependent and hence the tasks may share a common release tme o Each task s released (e becomes ready to execute) as soon as t arrves o The tasks are ndependent and so cannot block each other from executng by accessng mutually exclusve shared resources, wth the excepton of the processor o The tasks do not voluntarly suspend themselves A task s sad to be ready f t has outstandng computaton and so s awatng executon by the processor A taskset s sad to be schedulable wth respect to some schedulng algorthm and some system, f all possble sequences of task nvocatons (or jobs) that may be generated by the taskset can be scheduled on the system by the schedulng algorthm wthout any deadlnes beng mssed Under Earlest Deadlne Frst (EDF) schedulng, at any gven tme, the ready task nvocaton wth the earlest absolute deadlne s executed by the processor In contrast, under fxed prorty pre-emptve schedulng, at any gven tme, the hghest prorty ready task s executed by the processor When a taskset s scheduled accordng to fxed prortes, task prortes need to be assgned accordng to some algorthm Optmal prorty assgnment algorthms are known for mplct-deadlne [18], constraned-deadlne [15], and arbtrary-deadlne [1] tasksets A prorty assgnment polcy P s sad to be optmal wth respect to some class of tasksets f there are no tasksets n the class that are schedulable accordng to fxed prorty pre-emptve schedulng usng any other prorty orderng polcy that are not also schedulable usng the prorty assgnment determned by polcy P A taskset s sad to be feasble wth respect to a gven system model f there exsts some schedulng algorthm that can schedule all possble sequences of task actvatons that may be generated by the taskset on that system wthout mssng any deadlnes Note, n ths paper, we are prmarly nterested n a reference system model that conssts of a pre-emptve unprocessor wth unt processng speed A schedulng algorthm s sad to be optmal wth respect to a system model and a taskng model f t can schedule all of the tasksets that comply wth the taskng model and are feasble on the system We note that EDF s known to be an optmal preemptve unprocessor schedulng algorthm for tasksets complant wth the taskng model descrbed n ths secton [11] Least Laxty Frst s another such optmal algorthm [19] A schedulablty test s termed suffcent, wth respect to a schedulng algorthm and system model, f all of the tasksets that are deemed schedulable accordng to the test are n fact schedulable on the system under the schedulng algorthm Smlarly, a schedulablty test s termed necessary, f all of the tasksets that are deemed unschedulable accordng to the test are n fact unschedulable on the system under the schedulng

5 In Proc of the 17th Internatonal Conference on Real-Tme and Network Systems RTNS'2009, Pars, ECE, October, 2009 algorthm A schedulablty test that s both suffcent and necessary s referred to as exact 22 Schedulablty analyss for fxed prorty preemptve schedulng In ths secton, we gve a bref summary of Response Tme Analyss [2] used to provde an exact schedulablty test for fxed prorty pre-emptve schedulng of constraned-deadlne tasksets We then recaptulate on response tme analyss for arbtrary-deadlne tasksets Frst, we ntroduce the concepts of worst-case response tme, synchronous arrval sequence, and busy perods, whch are fundamental to response tme analyss For a gven taskset scheduled under fxed prorty preemptve schedulng, the worst-case response tme R of task τ s gven by the longest possble tme from release of the task untl t completes executon Thus task τ s schedulable f and only f R D, and the taskset s schedulable f and only f R D A synchronous arrval sequence refers to a pattern of arrval such that all tasks arrve smultaneously, and then subsequently as early as possble gven the constrants on mnmum nter-arrval tmes The term prorty level- busy perod refers to a perod of tme [ t 1, t2 ) durng whch the processor s busy executng computaton at prorty or hgher, that was released at the start of the busy perod at t 1, or durng the busy perod but strctly before ts end at t 2 The synchronous arrval sequence generates the longest possble prorty level- busy perod For constraneddeadlne tasksets, the length w of ths busy perod corresponds drectly to the worst-case response tme of task τ In the remander of ths paper, when we refer to a prorty level- busy perod, we mean the longest such busy perod Further, when t s clear whch prorty level s referred to we use the more concse term, busy perod The busy perod comprses two components, the executon tme of the task tself, and so called nterference, equal to the tme for whch task τ s prevented from executng by hgher prorty tasks For constraned-deadlne tasksets, the length of the busy perod w, can be computed usng the followng fxed pont teraton [2], wth the summaton term gvng the nterference due to the set of hgher prorty tasks hp() m m w w = C C j (2) j hp( ) T j 0 Iteraton starts wth an ntal value w, typcally 0 m+1 m w = C, and ends when ether w = w n whch case m+1 the worst-case response tme R, s gven by w, or m+1 when w > D n whch case the task s unschedulable The fxed pont teraton s guaranteed to converge provded that the overall taskset utlsaton s less than or equal to 1 Equaton (2) gves an exact schedulablty test for the fxed prorty pre-emptve schedulng of constraneddeadlne tasksets wth any fxed prorty orderng For arbtrary-deadlne tasksets, executon of one nvocaton of a task may not necessarly be complete before the next nvocaton s released Hence a number of nvocatons of task τ may be present wthn the longest prorty level- busy perod, wth earler nvocatons delayng the executon of later ones In general t s therefore necessary to compute the response tmes of all nvocatons wthn the busy perod n order to determne the worst-case response tme [20] The length of the busy perod w (q), startng at the smultaneous arrval of all tasks and extendng untl the completon of the qth nvocaton of τ (where q = 0 s the frst nvocaton) s gven by the fxed pont teraton: n n+ 1 w ( q) w ( q) = ( q + 1) C + C j (3) j hp( ) T j Iteraton starts wth an ntal value w 0 ( q), typcally 0 n+ 1 n w ( q) = ( q + 1) C, and ends when ether w ( q) = w ( q) n whch case the worst-case response tme R (q), of n+ 1 nvocaton q, s gven by w ( q) qt or when n+ 1 w ( q) qt > D n whch case nvocaton q s unschedulable Invocaton q can only mpnge upon the executon of subsequent nvocatons f ts completon occurs after ther release Hence, response tmes need to be calculated for nvocatons q=0,1,2,3 untl an nvocaton q s found that completes at or before the earlest possble release of the next nvocaton q+1, e where: w ( q) ( q + 1) T The worst-case response tme of task τ s then gven by: R = max q ( w ( q) qt ) (4) Agan, the task s schedulable provded that R D Equatons (3) and (4) gve an exact schedulablty test for the fxed prorty pre-emptve schedulng of arbtrarydeadlne tasksets wth any fxed prorty orderng The exact schedulablty test gven by Equatons (3) and (4) potentally requres the examnaton of a large number of nvocatons of the task of nterest A smpler suffcent schedulablty test for a task τ n an arbtrary-deadlne taskset can be derved by consderng the maxmum amount of task executon at prorty and hgher released wthn an nterval of length D startng wth smultaneous arrval of all tasks If all of ths executon can be completed by D, then ths ndcates that the length of the longest prorty level- busy perod s at most D, and hence that all nvocatons of τ released n that busy perod meet ther deadlnes, and so τ s schedulable Ths suffcent schedulablty test s gven by

6 In Proc of the 17th Internatonal Conference on Real-Tme and Network Systems RTNS'2009, Pars, ECE, October, 2009 Equaton (5): D C j D (5) j= hep( ) T j Where hep() s the set of tasks wth prortes hgher than or equal to 23 Exact schedulablty analyss for EDF The schedulablty of an arbtrary-deadlne taskset under EDF can be determned va the processor demand bound functon h(t) gven below: n t D h( t) = max 0, + C 1 (6) = T 1 Baruah et al [6], [7] showed that a taskset s schedulable under EDF f and only f a quantty referred to as the processor LOAD s 1 where the processor LOAD s gven by: h ( t) LOAD = max (7) t t Further, they showed that the maxmum value of h ( t) / t occurs for some value of t n the nterval ( 0, L], where L s defned as follows, thus lmtng the number of values of t that need to be checked to determne schedulablty U L = max D1, D2, Dn, max ( T D ) (8) 1 U The only values of t that need to be checked n the nterval ( 0, L] are those where the processor LOAD can change, e t = kt + D for nteger values of k Sgnfcant developments have been made, extendng the scope of the schedulablty tests for both fxed prorty pre-emptve schedulng and EDF; however, these basc forms are suffcent for the purposes of ths paper 24 Defntons Defnton 1: Let Ψ be a taskset that s feasble (e schedulable accordng to an optmal schedulng algorthm) on a processor of speed 1 Now assume that f (Ψ) s the lowest speed of any smlar processor that wll schedule taskset Ψ usng schedulng algorthm A The processor A speedup factor f for schedulng algorthm A s gven by the maxmum processor speed requred to schedule any such taskset Ψ f A = max( f ( Ψ)) Ψ For any schedulng algorthm A, we have f 1, wth A smaller values of f ndcatve of a more effectve A schedulng algorthm, and f = 1 mplyng that A s an optmal algorthm A In the remander of the paper, unless otherwse stated, when we refer to the processor speedup factor, we mean the processor speedup factor for fxed prorty pre-emptve schedulng usng an optmal prorty assgnment polcy Defnton 2: A taskset s sad to be speedup-optmal f t requres the processor to be speeded up by the processor speedup factor n order to be schedulable under fxed prorty pre-emptve schedulng Hence for a speedupoptmal taskset Ψ, f (Ψ) = f A 3 Example The concept of processor speedup factor defned n the prevous secton can be llustrated by means of an example Consder the arbtrary-deadlne taskset S comprsng the two tasks defned n Table 1 The parameters of these tasks appear to have some unusual values; however, ths s because they have been chosen so that the taskset s just schedulable accordng to EDF, yet requres a speedup factor of 18 n order to be schedulable accordng to fxed prorty pre-emptve schedulng, wth prortes ordered va deadlne monotonc prorty assgnment Task τ 1 τ 2 C Table 1 T D We now show that taskset S s schedulable accordng to EDF Under EDF schedulng, the processor demand bound functon h(t) for taskset S s the sum of the processor demand bound functons h ( t, τ1) and h ( t, τ 2 ) for tasks τ 1 and τ 2 respectvely, where h(t, τ ) s the processor demand bound at tme t for a sngle task τ, gven below: t D h t (, τ ) = max 0, + C 1 (9) T Thus: 0 t < 16 τ ) = t t 16 (10) 2 as x / y x / y, we have: 0 t < 16 τ ) 18( t 16) t 16 (11) 2 Smlarly, the processor demand bound functon for task τ s: 2

7 In Proc of the 17th Internatonal Conference on Real-Tme and Network Systems RTNS'2009, Pars, ECE, October, t < 17 τ 2 ) = (12) 144 t 17 Recall that any arbtrary-deadlne taskset s schedulable accordng to EDF, provded that: h( t) LOAD = max 1 (13) t t Now, gven the followng: () The value of h ( t) / t at tmes t = 16, t = 17, and t = 18 are 18, 162 and 18 respectvely () From Equatons (11) and (12), an upper bound on the value of h ( t) / t at tme t = 18 s 18 () From Equaton (11), the rate of ncrease of the upper bound on h ( t) / t for t 18 s 09 Hence, the maxmum value of h ( t) / t occurs at tme t = 18 The processor LOAD of taskset S s therefore 1, ndcatng that the taskset s just schedulable accordng to EDF We now consder the schedulablty of taskset S when scheduled accordng to fxed prorty pre-emptve schedulng, usng deadlne monotonc prorty assgnment, on a processor that has been speeded up by a factor of 18 The parameters of the taskset on ths faster processor are gven n Table 2 We refer to ths taskset as V Table 2 Task C T D τ τ Fgure 1 llustrates the executon of taskset V under fxed prorty pre-emptve schedulng, assumng a synchronous arrval sequence Fgure 1 We note that the worst-case response tme of task τ 1 s 1 and that of task τ 2 s 16 Taskset V s only just schedulable under fxed prorty pre-emptve schedulng, usng deadlne monotonc prorty assgnment Any reducton n processor speed would result n the taskset beng unschedulable The processor speedup factor requred s therefore 18 4 Processor speedup factor for arbtrarydeadlne tasksets In ths secton, we derve the exact processor speedup factor requred for the (non-optmal) case where deadlne monotonc prorty orderng s used n conjuncton wth arbtrary-deadlne tasksets Further, we provde upper and lower bounds on the processor speedup factor requred for the general case where an optmal prorty assgnment algorthm [1] s used to determne task prortes 41 Arbtrary-deadlne tasksets wth deadlne Monotonc prorty orderng Intally, we consder the case of arbtrary-deadlne tasksets where task prortes are assgned n deadlne monotonc prorty order (DMPO) Recall that DMPO s not optmal n ths case [16]; nevertheless, fxed prorty pre-emptve schedulng usng DMPO s a smple combnaton of schedulng algorthm and prorty assgnment polcy that s used n many real-tme systems We now derve an exact processor speedup factor for ths combnaton Lemma 1: An upper bound on the processor speedup factor for fxed prorty pre-emptve schedulng of arbtrary-deadlne tasksets usng deadlne monotonc prorty assgnment s 2 Proof: Let S be any taskset that s schedulable on a processor of unt speed accordng to an optmal schedulng polcy such as EDF For each task τ k, n S, consder the processor demand bound durng an nterval of length 2Dk As taskset S s schedulable accordng to EDF, t follows that: n 2Dk D s max 0, 1 C 2D + k (14) 1 T = Where s = 1 s the speed of the processor Next, consder taskset S scheduled accordng to fxed prorty pre-emptve schedulng on a processor of speed s = 2 usng deadlne monotonc prorty assgnment DMPO mples that k D Dk From Equaton (14) above, assumng speed s = 2, and dscardng the contrbuton from all tasks of lower prorty than k we have: k Dk D 2 max 0, + C D 1 k (15) T =1 As x + 1 x and k D Dk then: k Dk C Dk (16) =1 T

8 In Proc of the 17th Internatonal Conference on Real-Tme and Network Systems RTNS'2009, Pars, ECE, October, 2009 Equaton (16) s recognsable as the suffcent schedulablty test for task τ k n an arbtrary-deadlne taskset S, scheduled under fxed prorty pre-emptve schedulng (see Equaton (4) n Secton 22) Repeatng the above argument for each task τ k n S proves that the taskset s schedulable on a processor of speed 2 under fxed prorty pre-emptve schedulng usng deadlne monotonc prorty assgnment Theorem 1: An exact bound on the processor speedup factor for fxed prorty pre-emptve schedulng of arbtrary-deadlne tasksets usng deadlne monotonc prorty orderng s 2 Proof: Consder taskset V wth the followng parameters on a processor of speed f : τ 1 : C1 = 1/ 2k, T 1 = 1/ k, D 1 = 1 τ 2 : C 2 = 1/ 2, T 2 =, D 2 = 1+1/ 2k where k s an nteger, and task τ 1 has a hgher prorty than task τ 2 e deadlne monotonc prorty orderng The executon of taskset V under fxed prorty pre-emptve schedulng s llustrated n Fgure 2 (Note the smlarty to the taskset used as an example n Secton 3) 0 1/2k 1/k T 1 2T 1 3T 1 Task 1 Task 2 Fgure 2 D 1 1 D 2 1+1/2k We observe that wth fxed prorty pre-emptve schedulng, any ncrease n the executon tme of ether task wll cause task τ 2 to mss ts frst deadlne followng smultaneous release of the two tasks We now consder the executon of taskset V under EDF on a processor of unt speed Let taskset S be formed from taskset V by ncreasng the executon tmes of tasks τ 1 and τ 2 by a scalng factor f to form tasks τ 1 and τ 2, thus accountng for the reducton n processor speed We observe that f = 2 s an upper bound on the maxmum scalng factor that could possbly result n a schedulable taskset under EDF as ths scalng factor results n task τ 1 havng a utlsaton of 100% Under EDF schedulng, the processor demand bound functon h(t) for taskset S s the sum of the processor demand bound functons h ( t, τ 1 ) and h ( t, τ 2 ) for tasks τ 1 and τ 2 respectvely 0 t < 1 τ ) = t 1+ (1/ k) f 1 t 1 (17) (1/ k) 2k as x / y x / y, we have the followng upper bound: 0 t < 1 τ ) f ( t 1) f 1 + t 1 (18) 2 2k Smlarly, the processor demand bound functon for task τ 2 s: 0 t < 1+ (1/ 2k) τ 2 ) = (19) f / 2 t 1+ (1/ 2k) Recall that any arbtrary-deadlne taskset s schedulable accordng to EDF, provded that: h( t) LOAD = max 1 (20) t t Now, gven the followng: () The value of h ( t) / t at tme t = 1 s f / 2k () An upper bound, from Equatons (18) and (19), on the value of h ( t) / t at tme t = 1+ (1/ 2k) s: h(1 + (1/ 2k)) ( f / 2) + ((1 + (1/ 2k)) 1)( f / 2) + ( f / 2k) (1 + (1/ 2k)) (1 + (1/ 2k)) f ( k + (3/ 2)) = (21) 2( k + (1/ 2)) () The rate of ncrease of the upper bound on h ( t) / t for t > 1+ (1/ 2k) s f / 2 (from Equaton (18)) Then for values of f 2, the maxmum value of the upper bound on h ( t) / t occurs at tme t = 1+1/ 2k, therefore: h( t) f ( k + (3/ 2)) k f max = = (22) t t 2( k + (1/ 2)) 2 From Equaton (22), the mnmum value for the processor LOAD s acheved n the lmt as k, and ths value s f / 2 From Equaton (22), for k =, taskset V s schedulable accordng to EDF when ts task executon tmes are scaled up by a factor of f = 2 to form taskset S Hence taskset S requres a processor speedup factor of 2 n order to be schedulable under fxed prorty pre-emptve schedulng wth deadlne monotonc prorty orderng As the processor speedup factor for fxed prorty pre-emptve schedulng of arbtrary-deadlne tasksets usng deadlne monotonc prorty orderng s also upper bounded by 2 (Lemma 1), the exact processor speedup factor s 2 Corollary 1: Taskset S defned n the proof of Theorem 1 (wth k = ), s a speedup-optmal taskset for fxed prorty pre-emptve schedulng of arbtrary-deadlne tasksets usng deadlne monotonc prorty orderng It s nterestng to note that the speedup-optmal taskset (requrng the largest speedup factor), ncludes a task τ 1, wth a deadlne much larger than ts nfntesmal perod, lm

9 In Proc of the 17th Internatonal Conference on Real-Tme and Network Systems RTNS'2009, Pars, ECE, October, 2009 and a task τ 2, wth a deadlne much smaller than ts nfnte perod Theorem 2: An upper bound on the processor speedup factor for fxed prorty pre-emptve schedulng of arbtrary-deadlne tasksets usng an optmal prorty assgnment algorthm s 2 Proof: Follows drectly from the fact that usng an optmal prorty assgnment algorthm, fxed prorty pre-emptve schedulng can schedule any taskset that s schedulable usng deadlne monotonc prorty orderng Hence the processor speedup factor requred can be no greater wth optmal prorty assgnment than the exact processor speedup factor gven by Theorem 1 for deadlne monotonc prorty orderng Theorem 3: A lower bound on the processor speedup factor for fxed prorty pre-emptve schedulng of arbtrary-deadlne tasksets usng an optmal prorty assgnment algorthm s 1/ Ω = Proof: Follows drectly from the fact that the set of arbtrary-deadlne tasksets s a superset of the set of constraned-deadlne tasksets, and the proof gven by Davs et al [10] that the exact speedup factor requred for constraned-deadlne tasksets s 1/ Ω 5 Summary and conclusons In ths paper, we have examned the relatve effectveness of fxed prorty pre-emptve schedulng for tasksets wth arbtrary deadlnes Our metrc for measurng the effectveness of ths schedulng algorthm s a resource augmentaton factor known as the processor speedup factor The processor speedup factor s defned as the mnmum amount by whch the processor needs to be speeded up so that any taskset that s feasble (e schedulable by an optmal algorthm such as EDF) can be guaranteed to be schedulable under fxed prorty pre-emptve schedulng Table 3 shows the processor speedup factor needed for fxed prorty pre-emptve schedulng gven the dfferent taskset classfcatons (mplct-, constraned-, and arbtrary-deadlne) and dfferent prorty assgnment polces In Table 3, when a sngle value s shown for both the upper and lower bounds, ths mples that the bounds are the same and the value s exact (Note the results shown are for tasksets of arbtrary cardnalty) Table 3: Fxed prorty pre-emptve schedulng processor speedup factors Taskset constrants [Prorty orderng] Implct-deadlne [Optmal (RMPO)] Constraned-deadlne [Optmal (DMPO)] Arbtrary-deadlne [Not optmal (DMPO)] Arbtrary-deadlne [Optmal algorthm] Lower Upper Bound Bound 1 / ln(2) = / Ω = / Ω = In concluson, the major contrbutons of ths paper are as follows: o Provng that the exact processor speedup factor for fxed prorty pre-emptve schedulng of arbtrarydeadlne tasksets wth prortes assgned accordng to deadlne monotonc prorty assgnment s 2 o Provng that the processor speedup factor for fxed prorty pre-emptve schedulng of arbtrarydeadlne tasksets wth prortes assgned accordng to Audsley s optmal prorty assgnment algorthm, s upper bounded by 2 and lower bounded by 1 / Ω = The semnal work of Lu and Layland [18] characterses the maxmum performance penalty ncurred when an mplct-deadlne taskset s scheduled usng ratemonotonc, fxed prorty pre-emptve schedulng nstead of an optmal algorthm such as EDF The research n ths paper provdes an analogous charactersaton of the maxmum performance penalty ncurred when arbtrary-deadlne tasksets are scheduled usng fxed prorty pre-emptve schedulng nstead of an optmal algorthm such as EDF Table 4 summarses the maxmum extent of these performance penaltes, when deadlne monotonc prorty assgnment s used Table 4: Sub-optmalty of fxed prorty preemptve schedulng usng deadlne monotonc prorty assgnment Implctdeadlne Constraneddeadlne Arbtrarydeadlne Optmal (eg EDF) Fxed Prorty (DMPO) U ln( Ω Speedup factor U 1 ) 1/ ln(2) LOAD 1 LOAD 1/ Ω LOAD 1 LOAD 0 5 2

10 In Proc of the 17th Internatonal Conference on Real-Tme and Network Systems RTNS'2009, Pars, ECE, October, 2009 Note that although n ths paper, we have made numerous references to EDF as an example of an optmal pre-emptve unprocessor schedulng algorthm, and made use of results about EDF n our proofs, our results are vald wth respect to any optmal pre-emptve unprocessor schedulng algorthm, for example Least Laxty Frst [19] Ths s because all such optmal algorthms can by defnton schedule exactly the same set of tasksets: all those that are feasble In concluson, ths paper provdes for the frst tme, bounds on the sub-optmalty of fxed prorty pre-emptve schedulng for unprocessor systems wth arbtrarydeadlnes Future work Although ths paper provdes upper and lower bounds, the exact sub-optmalty of fxed prorty pre-emptve schedulng wth respect to arbtrary-deadlne tasksets assumng optmal prorty assgnment remans an open queston To the best of our knowledge, no research has yet been done to characterse the average-case sub-optmalty of fxed prorty pre-emptve schedulng for arbtrarydeadlne tasksets Ths s also an nterestng area for future research Acknowledgements Ths work was funded n part by the EU FP7 projects Jeopard (project number ) and emuco (project number ) References [1] Audsley NC, "Optmal prorty assgnment and feasblty of statc prorty tasks wth arbtrary start tmes", Techncal Report YCS 164, Dept Computer Scence, Unversty of York, UK, 1991 [2] Audsley NC, Burns A, Rchardson M, Wellngs AJ, Applyng new Schedulng Theory to Statc Prorty Pre-emptve Schedulng Software Engneerng Journal, 8(5), pages , 1993 [3] Baker TP, Stack-based Schedulng of Real-Tme Processes Real-Tme Systems Journal (3)1, pages [4] Baruah S, Burns A Sustanable Schedulng Analyss In Proceedngs of the IEEE Real-Tme Systems Symposum, pages , 2006 [5] Baruah S, Burns A, Quantfyng the sub-optmalty of unprocessor fxed prorty schedulng In Proceedngs of the IEEE Internatonal conference on Real-Tme and Network Systems, pages 89-95, 2008 [6] Baruah SK, Mok AK, Roser LE, Preemptvely Schedulng Hard-Real-Tme Sporadc Tasks on One Processor In Proceedngs of the IEEE Real-Tme System Symposum, pages , 1990 [7] Baruah SK, Roser LE, Howell RR, Algorthms and Complexty Concernng the Preemptve Schedulng of Perodc Real-Tme Tasks on one Processor Real-Tme Systems, 2(4), pages , 1990 [8] Bn E, Buttazzo GC, Measurng the Performance of Schedulablty Tests, Real-Tme Systems 30 (1-2), pages , 2005 [9] Bn E, Buttazzo GC, Buttazzo GM, Rate Monotonc Schedulng: The Hyperbolc Bound IEEE Transactons on Computers, 52(7), pages , 2003 [10] Davs RI, Rothvoß T, Baruah SK, Burns A, Exact Quantfcaton of the Sub-optmalty of Unprocessor Fxed Prorty Pre-emptve Schedulng Real-Tme Systems to appear 2009 [11] Dertouzos ML, Control Robotcs: The Procedural Control of Physcal Processes In Proceedngs of the IFIP congress, pages , 1974 [12] Fneberg MS, Serln O, Multprogrammng for hybrd computaton In Proceedngs of AFIPS Fall Jont Computng Conference, pages 1-13, 1967 [13] Joseph M, Pandya PK, Fndng Response Tmes n a Real-tme System The Computer Journal, 29(5), pages , 1986 [14] Kalyanasundaram B, Pruhs K, Speed s as powerful as clarvoyance In Proceedngs of the 36th Symposum on Foundatons of Computer Scence, pages , 1995 [15] Leung JY-T, Whtehead J, "On the complexty of fxedprorty schedulng of perodc real-tme tasks" Performance Evaluaton, 2(4), pages , 1982 [16] Lehoczky J, Fxed prorty schedulng of perodc task sets wth arbtrary deadlnes In Proceedngs 11th IEEE Real-Tme Systems Symposum, pages , 1990 [17] Lehoczky JP, Sha L, Dng Y, The rate monotonc schedulng algorthm: Exact characterzaton and average case behavour In Proceedngs of the IEEE Real-Tme Systems Symposum, pages , 1989 [18] Lu CL, Layland JW, "Schedulng algorthms for multprogrammng n a hard-real-tme envronment", Journal of the ACM, 20(1) pages 46-61, 1973 [19] Mok AK, Fundamental Desgn Problems of Dstrbuted Systems for the Hard-Real-Tme Envronment, PhD Thess, Department of Electrcal Engneerng and Computer Scence, Massachusetts Insttute of Technology, Cambrdge, Massachusetts, 1983 [20] Tndell KW, Burns A, Wellngs AJ, An extendble approach for analyzng fxed prorty hard real-tme tasks Real- Tme Systems Volume 6, Number 2, pages , 1994 [21] Zuhly A, Burns A, Optmalty of (D-J)-monotonc Prorty Assgnment Informaton Processng Letters Number 103, pages , 2007