Quantifying the Sub-optimality of Uniprocessor Fixed Priority Non-Pre-emptive Scheduling

Transcription

1 Quantfyng the Sub-optmalty of Unprocessor Fxed Prorty Non-Pre-emptve Schedulng Robert I Davs Real-Tme Systems Research Group, Department of Computer Scence, Unversty of York, York, UK robdavs@csyorkacuk Laurent George OSTE Team, INRI Rocquencourt, BP 105, Domane de Voluceau, 78153, Le Chesnay Cedex, France lgeorge@eeeorg Perre Courbn Ecole Centrale d'electronque de Pars (ECE), LCSC, 37 Qua de Grenelle, Pars, France courbn@ecefr bstract Ths paper examnes the relatve effectveness of fxed prorty non-pre-emptve schedulng (FP-NP) n a unprocessor system, compared to an optmal workconservng non-pre-emptve algorthm; Earlest Deadlne Frst (EDF-NP) 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 EDF-NP can be scheduled usng FP-NP schedulng For sporadc tasksets wth mplct, constraned, or arbtrary deadlnes, the speedup factor s shown to be lower bounded by 1/ Ω and upper bounded by 2 We also report the results of emprcal nvestgatons nto the speedup factor requred to ensure schedulablty n the non-pre-emptve case 1 Introducton In ths paper, we are nterested n determnng the largest factor by whch the processng speed of a unprocessor needs to be ncreased, to ensure that any taskset that was prevously schedulable accordng to an optmal work-conservng (e non-dlng), non-preemptve schedulng algorthm s schedulable accordng to fxed prorty non-pre-emptve (FP-NP) schedulng We refer to ths resource augmentaton factor as the processor speedup factor [17] 11 Pre-emptve schedulng In 1973, Lu and Layland [22] consdered fxed prorty pre-emptve (FP-P) 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, FP-P can schedule any mplct-deadlne taskset that has a total utlsaton U ln( 2) Lu and Layland [22] also showed that Earlest Deadlne Frst (EDF-P) s an optmal dynamc prorty 1 taskset s synchronous f all of ts tasks share a common release tme pre-emptve schedulng algorthm for mplct-deadlne tasksets, and that EDF-P can schedule any such taskset that has a total utlsaton U 1 In 1974, Dertouzos [9] showed that EDF-P s an optmal unprocessor schedulng algorthm, n the sense that f a vald schedule exsts for a taskset, then the schedule produced by EDF-P wll also meet all deadlnes 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 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 [19] showed that deadlne monotonc 2 prorty orderng (DMPO) s the optmal fxed prorty orderng for constraned-deadlne tasksets Exact schedulablty tests for FP-P schedulng of constraned-deadlne tasksets were ntroduced by Joseph and Pandya n 1986 [16], Lehoczky et al n 1989 [21], and udsley et al n 1993 [1] In 1990, Lehoczky [20] showed that DMPO 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 udsley s optmal prorty assgnment (OP) algorthm 3 [1], [2] Exact schedulablty tests for tasksets wth arbtrary deadlnes were developed by Lehoczky [20] n 1990 and Tndell et al [23] n 1994 Exact EDF-P schedulablty tests for both constraned and arbtrary-deadlne tasksets were ntroduced by Baruah et al [3], [4] n Non-pre-emptve schedulng In 1980, Km and Naghbdadeh [18], and n 1991, Jeffay et al [15], gave exact schedulablty tests for mplct-deadlne tasksets under Earlest Deadlne Frst non-pre-emptve (EDF-NP) schedulng These tests were 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

2 extended by George et al [12] n 1996, to the general case of sporadc tasksets wth arbtrary deadlnes Whle EDF-P s an optmal unprocessor schedulng algorthm, n the non-pre-emptve case no workconservng algorthm s optmal Ths s because n general t s necessary to nsert dle tme to acheve a feasble schedule The nterested reader s referred to [12] for examples of ths behavour In 1995, Howell and Venkatrao [14] showed that for non-concrete 4 perodc tasksets, the problem of determnng a feasble non-pre-emptve schedule s NP hard Further they showed that for sporadc tasksets, no optmal on-lne nserted dle tme algorthm can exst In other words, clarvoyance s needed to determne a feasble non-pre-emptve schedule Whle no work-conservng algorthm s optmal n the strong sense that t can schedule any taskset for whch a feasble non-pre-emptve schedule exsts; n 1995, George et al [13] showed that EDF-NP s optmal n the weak sense that t can schedule any taskset for whch a feasble work-conservng, non-pre-emptve schedule exsts Hence we can regard EDF-NP as an optmal work-conservng, non-pre-emptve schedulng algorthm, for non-concrete tasksets George et al [12] derved an exact schedulablty test for fxed prorty non-pre-emptve (FP-NP) schedulng of arbtrary-deadlne tasksets, based on the approach of Tndell et al [23] for the pre-emptve case George et al showed that unlke n the pre-emptve case, deadlne monotonc prorty orderng s not optmal for constraned-deadlne tasksets scheduled by FP-NP Further, they showed that udsley s optmal prorty assgnment algorthm [2] s applcable, and can be used to determne an optmal prorty orderng for tasksets wth arbtrary-deadlnes scheduled usng FP-NP Subsequent research by Brl et al [5] has refned exact analyss of FP-NP, correctng ssues of both pessmsm and optmsm, and extendng the schedulablty tests to co-operatve schedulng where each task s made up of a number of non-pre-emptve sectons 13 Sub-optmalty and speedup factors Combnng the result of Dertouzos [9] wth the results of Lu and Layland [22], shows that the processor speedup factor requred to guarantee that FP-P schedulng can schedule any mplct-deadlne taskset schedulable by EDF-P s 1/ ln(2) In 2009, Davs et al [7] derved the exact speedup factor for FP-P schedulng of constraned-deadlne tasksets; 1/ Ω (where Ω s the mathematcal constant defned by the transcendental equaton ln( 1/ Ω) = Ω, hence, Ω ) lso n 2009, Davs et al [8] showed that the speedup factor for FP-P schedulng of arbtrary-deadlne tasksets s lower bounded by 1/ Ω and upper bounded by 2 Further, f deadlne monotonc prorty assgnment s assumed (whch s not optmal for arbtrary-deadlne tasksets), then the exact speedup factor requred s 2 4 perodc taskset s referred to as non-concrete f the tmes at whch each task s frst released are unknown In ths paper, we derve upper and lower bounds on the speedup factor requred such that any taskset that was prevously schedulable accordng to an optmal work-conservng (e non-dlng) non-pre-emptve schedulng algorthm (e EDF-NP) can be guaranteed to be schedulable accordng to fxed prorty non-preemptve (FP-NP) schedulng These bounds are vald for all three classes of non-concrete taskset; Implctdeadlne, constraned-deadlne, and arbtrary-deadlne Whle these results are manly theoretcal, they also have practcal utlty n enablng system desgners to quantfy the maxmum penalty for usng fxed prorty non-preemptve 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 14 Organsaton The remander of ths paper s organsed as follows Secton 2 descrbes the system model, notaton and analyss used Secton 3 llustrates the concept of processor speedup factors va a smple example Secton 4 derves a lower bound on the processor speedup factor requred for FP-NP schedulng, whle Secton 5 derves the correspondng upper bound Secton 6 reports on an emprcal nvestgaton nto the speedup factor requred for FP-NP schedulng, verfyng the theoretcal lower bound Secton 7 concludes wth a summary of the results and suggestons for future work 2 System model, notaton, and analyss In ths secton, we outlne the schedulng model, notaton and termnology used n the rest of the paper We then recaptulate schedulablty analyss for both FP- NP and EDF-NP schedulng 21 Schedulng model, termnology and notaton In ths paper, we consder the non-pre-emptve schedulng of a set of sporadc tasks (or taskse 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 schedulng, hence τ 1 has the hghest fxedprorty, and τ n the lowest Each task τ s charactersed by ts bounded worstcase 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 (or jobs), 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 = In a constraned-deadlne taskset, all tasks T 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

3 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: U = n U = 1 The followng assumptons are made about the behavour of the tasks: o The arrval tmes of the tasks are ndependent and unknown a pror (non-concrete), 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 task s sad to be ready f t has outstandng computaton awatng executon by the processor taskset s sad to be schedulable wth respect to some schedulng algorthm and some system, f all vald 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 EDF-NP schedulng, whenever a job completes executon, or when the processor s dle and a job becomes ready to execute, the ready job wth the earlest absolute deadlne s selected to execute By contrast, under FP-NP the hghest prorty ready job s selected When a taskset s scheduled accordng to FP-NP, task prortes need to be assgned accordng to some algorthm udsley s Optmal Prorty ssgnment (OP) algorthm [1], [2], (see Fgure 1 below) provdes the optmal polcy for sporadc tasksets wth mplctdeadlnes, constraned-deadlnes or arbtrary-deadlnes Optmal Prorty ssgnment lgorthm for each prorty level k, lowest frst { for each unassgned task τ { f(τ s schedulable at prorty k wth all other unassgned tasks assumed to have hgher prortes) { assgn τ to prorty k break (contnue outer loop) } } return unschedulable } return schedulable Fgure 1: OP algorthm prorty assgnment polcy P s sad to be optmal wth respect to some class of tasksets, and a fxed prorty schedulng algorthm FP- f there are no tasksets n the class that are schedulable accordng to FP- usng any other prorty orderng polcy that are not also schedulable usng the prorty assgnment determned by polcy P taskset s sad to be feasble wth respect to a gven (1) 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 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-NP s optmal n the weak sense that t can schedule any sporadc taskset for whch feasble work-conservng, non-pre-emptve schedules exst [13] 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 algorthm schedulablty test that s both suffcent and necessary s referred to as exact 22 Schedulablty analyss for FP-NP Exact schedulablty analyss for an arbtrarydeadlne sporadc taskset under FP-NP was gven by George et al [12] and Brl et al [5] Below, we provde a smple suffcent schedulablty test for FP-NP schedulng, derved from these exact tests Ths suffcent test s used n the dervaton of an upper bound on the speedup factor for FP-NP schedulng gven n Secton 5 Frst, we ntroduce the concepts of worst-case response tme, prorty level- actve perod, and - crtcal nstant, whch are fundamental to analyss of FP- NP schedulng For a taskset scheduled under FP-NP schedulng, the worst-case response tme R of a 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 The term prorty level- actve perod refers to a contnuous perod of tme [ t 1, t2 ) durng whch tasks, of prorty or hgher, that were released at the start of the actve perod at t 1, or durng the actve perod but strctly before ts end at t 2, are ether executng or ready to execute -crtcal nstant for a task τ refers to a pattern of task arrvals such that task τ s released smultaneously wth all tasks of hgher prorty than, and then subsequent releases of task τ and the hgher prorty tasks occur as early as possble gven the constrants on mnmum nter-arrval tmes Further, some nfntesmal amount of tme pror to ths smultaneous release, a lower prorty task τ k s released, and ths task has the longest executon tme of any such lower prorty task Thus the longest tme that task τ and hgher prorty tasks can be blocked from executng by lower prorty tasks s gven by : max ( Ck Δ) < n B = k lp( ) (2) 0 = n

4 where lp() s the set of tasks wth prortes lower than Brl et al [5] showed that for FP-NP schedulng, the longest response tme of a task τ occurs for some nvocaton of the task wthn the prorty level- actve perod startng at the -crtcal nstant for task τ Lemma 3 n [5] states that the worst-case length of a prorty level- actve perod s gven by the mnmum soluton to the followng fxed pont teraton: m m + 1 = B + C j (3) j hep( ) T j where hep() s the set of tasks wth prortes hgher than 0 or equal to Iteraton starts wth an ntal value guaranteed to be no larger than the mnmum soluton, 0 m+1 m for example = C, and ends when = From Equaton (3), we can form the followng smple suffcent test for FP-NP schedulng of arbtrary-deadlne tasksets Each task τ s schedulable provded that: D D B Ck k hep Tk + (4) ( ) If Equaton (4) holds, then ths ndcates that the soluton to the fxed pont teraton of Equaton (3) must be D s the worst-case length of a prorty level- actve perod s then D, t follows that the worst-case response tme R of task τ must also be D, and hence the task must be schedulable If all of the tasks are schedulable accordng to Equaton (4), then the taskset s schedulable (For an exact schedulablty test for FP-NP, see [5]) 23 Schedulablty analyss for EDF-NP Baruah et al [3], [4] gave an exact schedulablty test for EDF-P based on the concept of the processor demand bound functon h( George et al [12] extended ths test to EDF-NP va the addton of a blockng factor B( n t D h t ( ) = max 0, + C 1 (5) = T 1 max ( C Δ < ) t max( D ) : D > t B( = (6) 0 t max( D ) George et al [12] showed that an arbtrary-deadlne taskset s schedulable under EDF-NP f and only f U 1 and a quantty referred to as the processor LOD s 1, where the processor LOD s gven by: h( LOD = max (7) t t Further, f U 1 and the value of ( h ( ) / t s 1 for all values of t n the nterval ( 0, L ], (where L s the length of the synchronous prorty level-n actve perod, gven by the mnmum soluton to Equaton (3)), then LOD 1 Thus the only values of t that need to be checked are those n the nterval ( 0, L ] that correspond to tmes when h ( can change, e t = kt + D for nteger values of k We note that recently sgnfcant developments have been made, reducng the number of values of t that need to be checked [11]; however, ths smple descrpton of the analyss s suffcent for the purposes of ths paper (We note that for schedulable tasksets, the maxmum value of ( h ( ) / t does not necessarly occur n the nterval ( 0, L ], but does occur n the nterval [ D mn, H ] where D mn s the shortest task deadlne and H s the taskset hyperperod or Least Common Multple of task perods [10]) 24 Defntons OPT Defnton 1: Let f (Ψ) be the lowest processor speed such that taskset Ψ s schedulable accordng to an optmal schedulng algorthm ssume that f (Ψ) s smlarly the lowest processor speed that wll schedule taskset Ψ usng schedulng algorthm The processor speedup factor f for algorthm s gven by the maxmum ncrease n processor speed requred over an optmal algorthm for any taskset Ψ OPT f = max f ( Ψ) / f ( Ψ) (8) Ψ ( ) For any schedulng algorthm, we have f 1, wth smaller values of f ndcatve of a more effectve schedulng algorthm, and f = 1 mplyng that s an optmal algorthm In the remander of the paper, unless otherwse stated, when we refer to the processor speedup factor, we mean the processor speedup factor for FP-NP schedulng usng an optmal prorty assgnment polcy, as compared to EDF-NP, an optmal (n the weak sense [13]), workconservng non-pre-emptve schedulng algorthm Defnton 2: 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 usng FP-NP schedulng Hence for a speedup-optmal taskset Ψ, OPT f ( Ψ ) / f ( Ψ) = f Defnton 3: Let S be some arbtrary taskset, now FPNP assume that α (S) s the maxmum factor by whch the executon tmes of all of the tasks n S can be scaled, such that the taskset s schedulable under FP-NP EDF NP Smlarly, let α (S) be the maxmum scalng factor under EDF-NP The speedup factor FPNP f (S) for the taskset s gven by: FPNP EDF NP FPNP f ( S) = / (9) 3 Example The concept of a speedup factor for a gven taskset S can be llustrated by means of the followng example Consder the taskset S comprsng the tasks defned n Table 1, wth prortes assgned n the order that the tasks appear n the table (e τ has the hghest prorty, and τ D the lowes Table 1 Task C D = T τ 1 6 τ B 1 7 τ C 1 8 τ D 3 + Δ The worst-case arrval pattern for tasks τ, τ B, and τ C under FP-NP schedulng s shown n Fgure 2 Note the 1 st job of each task s shaded n grey, whle the 2 nd job of each task s un-shaded

5 Fgure 2: FP-NP schedule Now consder the maxmum factor by whch the executon tmes of the tasks can be scaled and the taskset reman schedulable accordng to FP-NP Ths factor s FPNP α (S) = (6 / 5) (e a value nfntesmally less than 6/5) Fgure 3: FP-NP schedule, maxmal scalng Fgure 3 shows the FP-NP schedule for the scaled taskset Scalng by any larger factor, for example, a factor equal to 6/5 would result n the frst job of task τ C beng unable to start executng before the 2 nd job of task τ s released at tme t = 6 It would then be further delayed by the 2 nd job of task τ B, and hence fal to met ts deadlne at tme t = 8 In fact, there s no prorty orderng whch results n taskset S, scaled by a factor of 6/5, beng schedulable Ths can be seen by consderng the behavour of the OP algorthm Whle task τ D s schedulable at the lowest prorty, and can therefore be assgned prorty 4, none of the other tasks are schedulable at prorty 3 Fgure 4: EDF-NP schedule, maxmal scalng Wth EDF-NP schedulng, the maxmum scalng factor commensurate wth taskset S remanng EDFNP schedulable s = 8/ 6 Under EDF-NP, the frst job of task τ C has a later absolute deadlne than the frst jobs of tasks τ and τ B, and therefore executes after those jobs and after the frst job of τ D whch s released at tme t = Δ The frst job of task τ C s not however delayed by the 2 nd jobs of tasks τ and τ B, as these jobs have later absolute deadlnes Wth a scalng factor of 8/6, the frst job of task τ C just completes by ts deadlne (see Fgure 4) Further analyss s requred to prove that the scaled taskset s schedulable under EDF- NP; however, as the prorty level 3 actve perod ends at t = 12, we need only check all deadlnes n the nterval [0, 12] to show schedulablty Note, Fgure 4 shows the -crtcal nstant for tasks τ, τ B, τ C, and all deadlnes are met n the nterval [0, 12] Further, task τ D s trvally schedulable as t has an nfnte deadlne, and the taskset utlsaton s less than 1 Usng Equaton (9), the speedup factor for the taskset FPNP gven n Table 1 s f (S) = (8/6)/(6/5) = + + ( 40 / 36) = ( 10 / 9) (e a value nfntesmally larger than 10/9) In the next secton, we generalse ths example and show how tasksets wth a smlar structure but wth a large number of tasks requre a much larger speedup factor 4 Lower bound speedup factor for FP-NP In ths secton, we derve a lower bound on the processor speedup factor requred for FP-NP schedulng usng optmal prorty assgnment [1], [2] Ths lower bound s vald for sporadc and non-concrete perodc tasksets wth mplct-, constraned-, and arbtrarydeadlnes In Secton 5, we derve an upper bound wth the same scope To derve a lower bound, we need only select a sngle taskset and determne the requred speedup factor for that taskset The taskset S that we use s a generalsaton of the taskset used as an example n Secton 3 In ths case, there are n tasks, wth the parameters gven n Table 2 Tasks τ 1 to τ n1 are represented by τ n the frst row of the table ll of these tasks have the same small executon tme ε <<, and related perods/deadlnes Further, all of the tasks have perods equal to ther deadlnes, so ths s an mplct-deadlne taskset Task τ n has an executon tme of + Δ, where s a free varable that we can alter to maxmse the requred speedup factor Table 2 Task C D = T τ = 1 ( 1) ε + ( n 1) ( n 1) τ n + Δ The executon of taskset S under FP-NP s depcted n Fgure 5 below Note, jobs of task τ n1 are marked wth an ε Fgure 5: FP-NP schedule

6 Fgure 6: FP-NP schedule, maxmal scalng FPNP Lemma 1: The maxmum factor α (S) by whch the executon tmes of the tasks n taskset S (Table 2) can be scaled and the taskset reman schedulable accordng to FP-NP s gven by: FPNP + = = 1 ε ε 0 (10) Fgure 6 depcts the FP-NP schedule for the scaled taskset Proof: Scalng by a factor equal to ( ) /(1 + + ε ) would result n the frst job of task τ n1 beng unable to start executng before the 2 nd job of task τ 1 s released at tme t = 1 + It would then be further delayed by the 2 nd jobs of tasks τ 1 to τ n2, and hence fal to met ts deadlne at tme t = 2 + ε In fact, there s no prorty orderng whch results n taskset S, scaled by a factor of ( ) /(1 + + ε ), beng schedulable Ths can be seen by consderng the behavour of the OP algorthm, gven the scaled taskset: Task τ n s schedulable at the lowest prorty, and can therefore be assgned that prorty However, consderng the remanng tasks n turn, none of them are schedulable at prorty n-1 Task τ 1 s not schedulable at prorty n-1, as ts 1 st job would mss ts deadlne at t = 1 + Task τ 2 s not schedulable at prorty n-1, as ts 1 st job s then unable to start before the 2 nd job of task τ 1 arrves, and so msses ts deadlne at t = ε In general, wth a scalng factor of ( ) /(1 + + ε ), for each task wth ndex from 2 to n-1, assumng that task τ s assgned prorty n-1, ensures that the 1 st job of task τ s unable to start before the 2 nd job of task τ 1 arrves, and so the 1 st job of task τ msses ts deadlne By contrast, wth a scalng factor of ( ) /(1 + + ε ), task τ n1 s schedulable at prorty n-1, as t s able to start executng just pror to the arrval of the 2 nd job of τ 1 at t = 1 + Further, wth ths scalng factor, all of the other tasks are schedulable wth prortes assgned accordng to ther ndces (e n Deadlne Monotonc prorty order) Ths can be seen by checkng the deadlnes of all jobs up to the end of the prorty level n-1 actve perod, whch occurs at: + t (1 ) = ( 2 + ) (11) + ε s t < 2(1 + ) = 2D1, the prorty level n-1 actve perod comprses the 1 st job of task τ n and the 1 st and 2 nd jobs of tasks τ 1 to τ n1 ll of these are schedulable (see Fgure 6) The prorty level-n actve perod s of smlar length, and hence task τ n s trvally schedulable gven ts nfnte deadlne EDF NP Lemma 2: The maxmum factor α (S) by whch the executon tmes of the tasks n taskset S (Table 2) can be scaled and the taskset reman schedulable accordng to EDF-NP s gven by: EDF NP α (S) = (1/ Ω) (12) Proof: There are two key condtons whch lmt the maxmum scalng factor under EDF-FP (otherwse the taskset would become unschedulable): 1 The 1 st jobs of all tasks must be complete by the deadlne of task τ n1, D n = 2 + ε 2 Utlsaton of the scaled taskset must not exceed 100% Consderng the frst condton, we have: 1 EDF NP 2 + ε (13) The utlsaton of the un-scaled taskset s gven by the sum of the utlsaton of each task: n = U (14) ( 1) 1 1 = n (1 + + ) n 1 The RHS of Equaton (14) s recognsable as the left Remann sum of the functon 1/z, over the nterval [,2 + ), hence: n U = dz = ln (15) z Thus, consderng the second condton, we have: EDFNP 2 + 1/ ln (16) s Equaton (13) s monotoncally non-ncreasng n and tends to 2 for small, and Equaton (16) s monotoncally non-decreasng n and tends to 1/ln(2) for small, then the maxmum value s obtaned when the RHSs of Equatons (13) and (16) are equal, e when: EDF NP 2 + ε 2 + = = 1 / ln (17) Scalng factor a1() a2() Fgure 7: Constrants on the scalng factor as a functon of Fgure 7 plots Equatons (13) and (16) (labelled a1() and a2() respectvely) aganst s n, ε 0,

7 the soluton to Equaton (17) s gven by the ntersecton of the lnes plotted n Fgure 7, thus EDF NP = (1/ Ω) , (where Ω s the mathematcal constant defned by the transcendental equaton ln( 1/ Ω) = Ω, hence, Ω ) Further, 2Ω 1 ε = (18) 1 Ω We now show that taskset S (Table 2) s schedulable under EDF-NP, when scaled by a factor of EDF NP α (S) = (1/ Ω) Proof s made sgnfcantly easer by the commonalty between taskset S and the speedup-optmal taskset V for the constraned-deadlne case of FP-P schedulng, descrbed n Theorem 2 of [7] In fact, tasks τ 1 to τ n1 are dentcal n these two tasksets, only task τ n dffers In taskset V, the parameters of task τ n are: C n =, D n = 2 +, and T n =, whereas n taskset S, the parameters of task τ n are: C n = + Δ, and D n = T n = Theorem 4 n [7] proves that taskset V s schedulable under EDF-P when scaled by a factor of 1 / Ω Hence for taskset V, P h( LOD = max 1 (19) t t We make use of ths result to show that taskset S, scaled by a factor of ( 1/ Ω) s schedulable under EDF- NP s tasks τ 1 to τ n1 are dentcal, ther contrbuton to the processor demand bound h ( s the same for any tme t We now compare the contrbuton from task τ n n each case In the pre-emptve case, (taskset V), τ n contrbutes to h ( as follows: 0 0 t < (2 + ) h P n ( = (20) / Ω t (2 + ) whereas, n the non-pre-emptve case, (taskset S), τ n contrbutes only to the blockng factor: B ( = ( / Ω) t 0 (21) Recall that n the non-pre-emptve case, a taskset s schedulable provded that U 1 and: NP h( LOD = max 1 (22) t t Comparng Equatons (19) and (22), and the contrbutons of task τ n n each case (Equatons (20) and (21)), t follows that Equaton (22) holds for all values of t ( 2 + ) for taskset S scaled by a factor of ( 1/ Ω) Ths s because, for all values of t ( 2 + ) the value of ( h ( ) / t s the same as that for taskset V, assumng both tasksets are scaled by the same factor To prove the schedulablty of taskset S scaled by a factor of ( 1/ Ω), t remans only to show that ( h ( ) / t 1 for all values of t n the nterval [ 0,(2 + )) Here, we need only check values of t that correspond to task deadlnes s 2 + > 2D1, ths amounts to checkng the 1 st deadlne of each of the n-1 hghest prorty tasks t each of these deadlnes D, we have: n ( ) max 0, = 1 D Dk h D + Ω k= 1 Tk n 1 (23) as, k D = T and D < 2Dk t follows that: 1 h( D ) = Ω n 1 (24) Hence the scaled taskset s schedulable provded that, from 1 to n-1, D h D ) D ) ( n 1 Ω n 1 (25) Substtutng for ( 1/ Ω) = (2 + ε ) /(1 + ) and ε = 1/( n 1), and rearrangng, we have: (1 + ) n 1 n 1 n 1 (26) whch smplfes to: n 1 n 1 2 ( n 1) (27) and then to: n n 1 2 ( n 1) (28) For n 2, the frst term n nequalty (28) s nonnegatve from 1 to n-2, whle the second term s always postve Further, for = n-1, the frst and second terms cancel out, thus the nequalty holds from 1 to n-1 Taskset S s therefore schedulable accordng to EDF-NP when scaled by a factor of ( 1/ Ω) Theorem 1: lower bound on the speedup factor requred for FP-NP schedulng of an mplct-deadlne taskset s: EDFNP FPNP (1/ Ω) f = = = (1/ Ω) (29) FPNP + 1 Proof: Follows from Lemmas 1 and 2 and the defnton of the speedup factor Corollary 1: We observe that as taskset S s an mplctdeadlne taskset, and all mplct-deadlne tasksets are also constraned-deadlne and arbtrary-deadlne tasksets, the lower bound of Theorem 1 apples to all three classes of taskset It remans an open queston whether or not the lower bound gven n Theorem 1 s tght Whle the taskset used to derve the bound s vald, whether or not t s a speedup optmal taskset (see Defnton 2) remans to be proved / dsproved If the bound s not tght, then there exsts a dfferent taskset constructon that requres a larger speedup factor 5 Upper bound speedup factor for FP-NP In ths secton, we derve an upper bound on the speedup factor requred for FP-NP schedulng of arbtrary-deadlne sporadc and non-concrete perodc tasksets Theorem 2: n upper bound on the processor speedup factor requred such that FP-NP schedulng, usng optmal prorty assgnment can schedule any arbtrarydeadlne sporadc or non-concrete perodc taskset schedulable under EDF-NP accordng to Equaton (7), s 2

8 Proof: Let S be any taskset that s schedulable accordng to Equaton (7) on a processor of unt speed under EDF- NP For each task τ k, n S, consder the processor demand bound and blockng factor for an nterval of length 2 Dk s taskset S s schedulable accordng to EDF-NP, t follows that: n 2Dk D B(2Dk ) max + 0, 1 C 2Dk + 1 T = (30) Next, consder taskset S scheduled accordng to FP-NP schedulng on a processor of speed 2 usng Deadlne Monotonc prorty assgnment (rather than OP) DMPO mples that k D Dk From Equaton (30) above, assumng speed 2, and separatng out the contrbuton from all tasks of lower or equal prorty to k we have: 2Dk D B( 2D + + k ) max 0, + 1 C lep( k) T Dk D 2 max 0, C D + 1 k (31) hp k T ( ) where lep (k) s the set of tasks wth prortes lower than or equal to k s the tasks are n DMPO, we note that all of the tasks n lep (k) have deadlnes Dk We now consder just the frst and second terms n Equaton (31) Observe that the contrbuton to the second term from every task τ n lep (k) wth D > 2Dk s zero Further, there s a contrbuton from each task τ wth Dk D 2Dk of at least C From the defnton of B ( (Equaton (6)), the defnton of B k (Equaton (2)), and the fact that the tasks are n DMPO, t follows that the sum of the frst two terms n Equaton (31) are Bk, the blockng factor for FP-NP schedulng: max ( C Δ) 2Dk < max( D ) : D > 2Dk + C Bk 0 2Dk max( D ) : Dk D 2D k max ( C Δ) k < n where B = lp( k) k 0 k = n (32) Substtutng B k for the frst two terms n Equaton (31) and transformng the thrd term by notng that x +1 x and hp( k) D Dk we have: Dk Bk + C Dk (33) hp( k) T Equaton (33) s dentcal to Equaton (4); the suffcent schedulablty test for task τ k n an arbtrary-deadlne taskset S, scheduled under FP-NP Repeatng the above argument for each task τ k n S therefore proves that the taskset s schedulable on a processor of speed 2 under FP-NP, wth Deadlne Monotonc prorty assgnment s optmal prorty assgnment for FP-NP can schedule any taskset that s schedulable usng FP-NP wth Deadlne Monotonc prorty orderng Corollary 2: We observe that as the upper bound n Theorem 2 holds for arbtrary-deadlne tasksets, t must also hold for mplct-deadlne, and constraned-deadlne tasksets 6 Emprcal results In ths secton, we confrm by experment the results presented n Secton 4 concernng the lower bound speedup factor for FP-NP We consder the task set proposed n Table 2 where n 1 tasks have worst-case executon tmes equal to 1/( n 1), and task τ n has a worst-case executon tme equal to + Δ and an nfnte perod and deadlne Our experments ndcate that a value of 0 31 results n the maxmum speedup factor for a gven number of tasks Based on ths value, we verfy the lower bound gven n ths paper for a large number of tasks Taskset S nvestgated n ths secton s based on the task model presented n Table 2 Ths model s theoretcal, whle our emprcal assumptons are as follows: o Δ s equal to 0001 o The nfnte values of perod and deadlne for task τ n are replaced n the experments by values such that τ n does not nterfere wth other tasks Fnally, to avod the problem of roundng, we have used nteger values, thus gven that Δ s the tme granularty, each task parameter { C, D, T } s normalzed accordng to Δ For example, wth three tasks and = 1, Table 3 gves the task set studed Table 3: Example of taskset studed wth Δ = 0001 and = 1 Task C D = T τ τ τ FPNP In order to determne the speedup factor f (S), EDF NP we compute the maxmum factors α (S) and FPNP α (S) va an approach based on bnary search (dchotomy) lgorthm 1 descrbes ths approach, as used n our experments To check the schedulablty of task set S, the functon checkfeasblty() uses followng exact tests: For EDF-NP [12]: h( max 1 (34) t [0, L] t For FP-NP [6]: R = max ( W + C qt ) < D (35), max, q q [0, Q ] max = / T where 1 Q, s computed accordng to Equaton (3) and W, q s gven by the mnmum soluton to the followng fxed pont teraton: m W q + Δ m+ 1, W, q = C + B (36) T 0 wth B determned by Equaton (2) and W, q = qt Fnally, the speedup factor s computed va Equaton FPNP EDF NP FPNP (9) f ( S) = /

9 Fgure 8: Constrants on the scalng factor as a functon of for n = Speedup factor for FP-NP Lower Bound Speedup factor (Emprcal value) Number of tasks lgorthm 1: Determnes the maxmum scalng factor of a taskset S wth a precson Δ va bnary search (dchotomy) Fgure 8 shows the maxmum speedup factor obtaned for n = 5 tasks It shows the value of for whch the speedup factor was emprcally found to be a maxmum e = 0 31 Fgure 9 represents, for ths optmum value = 031, the speedup factor obtaned as a functon of the number of tasks (from 10 to 400 tasks) We observe that the maxmum speedup factor for FP-NP tends FPNP towards , the lower bound of f (S) characterzed n Secton 4, as the number of tasks ncreases Note that the saw tooth appearance of the curve s an artefact of the quantsaton of the executon tme values used n the experment When the number of tasks s large, ths causes a notceable quantsaton of the scalng factors that can be explored Fgure 9: Speedup factor for = 0 31 as a functon of the number of tasks 7 Conclusons and future work In ths paper, we examned the relatve effectveness of fxed prorty non-pre-emptve schedulng Our metrc for measurng the effectveness of ths schedulng algorthm s a resource augmentaton factor known as the processor speedup factor In ths case, 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 schedulable by an optmal workconservng non-pre-emptve schedulng algorthm (eg EDF-NP) can be guaranteed to be schedulable under FP- NP schedulng Recall that EDF-NP s optmal n the weak sense [13], n that t can schedule all sporadc or non-concrete perodc tasksets for whch a feasble nonpre-emptve, work-conservng schedule exsts It s not optmal n the strong sense as t cannot schedule all tasksets for whch a feasble non-work-conservng, nonpre-emptve schedulable exsts The speedup factors derved n ths paper are wth respect to ths weak form of optmalty

10 Table 4: FP schedulng speedup factors Taskset constrants Implctdeadlne Constraneddeadlne rbtrarydeadlne Pre-emptve Lower Upper Bound Bound 1 / ln(2) / Ω / Ω Non-pre-emptve Lower Upper Bound Bound ( 1/ Ω) ( 1/ Ω) ( 1/ Ω) Table 4 shows the processor speedup factor needed for fxed prorty schedulng wth optmal prorty assgnment, for both the pre-emptve case (FP-P v EDF- P), see Davs et al [7], [8], and for the non-pre-emptve case (FP-NP v EDF-NP), derved n ths paper The major contrbuton of ths paper s n provng that the processor speedup factor for fxed prorty nonpre-emptve schedulng of sporadc or non-concrete perodc tasksets wth optmal prorty assgnment, s upper bounded by 2, and lower bounded by ( 1/ Ω) = We note that these bounds hold for tasksets wth mplct-, constraned-, and arbtrary-deadlnes The semnal work of Lu and Layland [22] characterses the maxmum performance penalty ncurred when an mplct-deadlne taskset s scheduled usng Rate-Monotonc, fxed prorty pre-emptve schedulng nstead of an optmal algorthm such as EDF- P The research n ths paper provdes an analogous charactersaton of the maxmum performance penalty ncurred when tasksets are scheduled usng fxed prorty non-pre-emptve schedulng nstead of an optmal workconservng non-pre-emptve schedulng algorthm eg EDF-NP We note that the two cases n Table 4 where a tght bound s known correspond to the only cases where optmal prorty assgnment can be acheved ndependently of schedulablty testng In the arbtrarydeadlne case for FP-P schedulng and all cases of FP- NP schedulng, udsley s OP algorthm s requred to fnd the optmal prorty orderng Ths dependence of prorty orderng on schedulablty testng makes t more dffcult to reason about the propertes of a theoretcal speedup-optmal taskset that requres the exact speedup factor to be schedulable In these cases, the exact suboptmalty of fxed prorty schedulng remans an open queston 71 cknowledgements Ths work was funded n part by the EPSRC project TEMPO (EP/G055548/1) and the EU funded rtstdesgn Network of Excellence The authors would lke to thank lan Burns for hs nsghtful comments on an earler draft References [1] udsley 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] udsley NC On prorty assgnment n fxed prorty schedulng, Informaton Processng Letters, 79(1): 39-44, May 2001 [3] Baruah SK, Mok K, Roser LE, Preemptvely Schedulng Hard-Real-Tme Sporadc Tasks on One Processor In Proc RTSS, pages , 1990 [4] Baruah SK, Roser LE, Howell RR, lgorthms and Complexty Concernng the Preemptve Schedulng of Perodc Real-Tme Tasks on one Processor Real-Tme Systems, 2(4), pages , 1990 [5] Brl, RJ, Lukken, JJ, and Verhaegh, WF, Worst-case response tme analyss of real-tme tasks under fxed-prorty schedulng wth deferred pre-empton Real-Tme Systems 42, 1-3 (ug 2009), [6] Davs, R I, Burns,, Controller area network (CN) schedulablty analyss: Refuted, revsted and revsed, 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