Quantifying the Exact Sub-Optimality of Non-Preemptive Scheduling

Transcription

1 Quantfyng the Exact Sub-Optmalty of Non-Preemptve Schedulng Robert I. Davs 1, Abhlash Thekklakattl 2, Olver Gettngs 1, Radu Dobrn 2, and Saskumar Punnekkat 2 1 Real-Tme Systems Research Group, Unversty of York, UK and AOSTE team, Inra, Pars-Rocquencourt, France 2 Mälardalen Real-Tme Research Center, Mälardalen Unversty, Sweden Abstract Fxed prorty schedulng s used n many real-tme systems; however, both preemptve and non-preemptve varants (FP-P and FP-NP) are known to be sub-optmal when compared to an optmal unprocessor schedulng algorthm such as preemptve Earlest Deadlne Frst (EDF-P). In ths paper, we nvestgate the sub-optmalty of fxed prorty non-preemptve schedulng. Specfcally, we derve the exact processor speed-up factor requred to guarantee the feasblty under FP-NP (.e. schedulablablty assumng an optmal prorty assgnment) of any task set that s feasble under EDF-P. As a consequence of ths work, we also derve a lower bound on the sub-optmalty of non-preemptve EDF (EDF-NP), whch snce t matches a recently publshed upper bound gves the exact sub-optmalty for EDF-NP. It s known that nether preemptve, nor non-preemptve fxed prorty schedulng domnates the other,.e., there are task sets that are feasble on a processor of unt speed under FP-P that are not feasble under FP-NP and vce-versa. Hence comparng these two algorthms, there are non-trval speedup factors n both drectons. We derve the exact speed-up factor requred to guarantee the FP-NP feasblty of any FP-P feasble task set. Further, we derve upper and lower bounds on the speed-up factor requred to guarantee FP-P feasblty of any FP-NP feasble task set. Emprcal evdence suggests that the lower bound may be tght, and hence equate to the exact speed-up factor n ths case. Keywords-real-tme; unprocessor; resource augmentaton; speedup factor; sub-optmalty; non-preemptve schedulng; preemptve schedulng; EDF; fxed prorty. I. INTRODUCTION Real-tme systems are prevalent n a wde varety of applcaton areas ncludng telecommuncatons, consumer electroncs, aerospace systems, automotve electroncs, robotcs, and medcal systems. The functonalty of these systems s typcally mapped to a set of perodc or sporadc real-tme tasks, wth each task gvng rse to a potentally unbounded sequence of jobs. Tmely executon of the tasks and ther jobs s supported by the use of real-tme schedulng algorthms. Real-tme schedulng algorthms for sngle processor systems may be classfed nto two man types: fxed prorty and dynamc prorty. Fxed prorty schedulng s the defacto standard approach used n many applcatons. Here, a unque statc prorty s assgned to each task and nherted by all of ts jobs. At runtme, the scheduler uses these prortes to determne whch job to execute. Earlest Deadlne Frst (EDF) s the most common example of a dynamc prorty schedulng algorthm. EDF uses prortes based on the absolute deadlne of each job to make schedulng decsons. Real-tme schedulng algorthms may also be classfed n terms of when and f preempton s permtted. Thus we have preemptve and non-preemptve varants of both fxed prorty (FP-P and FP-NP) and EDF (EDF-P and EDF-NP) schedulng. There are a number of dfferent ways n whch the performance of real-tme schedulng algorthms can be compared. Emprcal technques typcally rely on generatng a large number of task sets wth parameters chosen from some approprate dstrbutons. The performance of the schedulng algorthms are then compared by determnng task set schedulablty accordng to exact or suffcent schedulablty tests and plottng a graph of the success rato,.e. the proporton of task sets that are deemed schedulable, at dfferent utlsaton levels. More advanced approaches use a weghted schedulablty metrc [8] to llustrate how schedulablty vares wth a further parameter, for example task set cardnalty, or the range of task perods. Smlar comparsons may be obtaned by usng a smulaton of each algorthm as a necessary schedulablty test, hence showng the proporton of task sets found to be defntely unschedulable due to a deadlne mss n the smulaton. These emprcal approaches tend to focus on the average-case behavour over large numbers of task sets rather than hghlghtng those task sets that are partcularly dffcult to schedule usng one algorthm, but may be easy to schedule usng another. Metrcs such as breakdown utlsaton [26] and optmalty degree [9] can also be used to examne average-case performance. In ths paper, we focus on a theoretcal method of comparng the worst-case performance of real-tme schedulng algorthms based on a resource augmentaton metrc referred to as the processor speedup factor [25]. Specfcally, we derve bounds on the factor by whch the speed of the processor needs to be ncreased to ensure that any task set that s feasble under some schedulng algorthm A s guaranteed to be feasble under another algorthm B. When A s an optmal algorthm then ths speedup factor provdes a measure of the sub-optmalty of algorthm B. Note, when we refer to a task set as beng feasble under a partcular schedulng algorthm, f that algorthm uses fxed

2 prortes, then we mean that the task set s schedulable under that algorthm wth an optmal prorty assgnment. In ths paper, we use speedup factors to compare fxed prorty non-preemptve schedulng (FP-NP) wth both fxed prorty preemptve (FP-P) and Earlest Deadlne Frst (EDF-P) schedulng. Our nterest n FP-NP schedulng stems from the fact that n modern unprocessor systems, pre-empton can sgnfcantly ncrease overheads due to a number of factors. These nclude context swtch costs and cache related pre-empton delays (CRPD) whch have to be accounted for n both FP-P [2] and EDF-P [30] schedulng. CRPD can have a substantal mpact, ncreasng task executon tmes by as much as 33% [11]. One way of reducng or elmnatng CRPD s to partton the cache; however, allocatng each task a cache partton, whch s some fracton of the overall sze of the cache, has an mpact on the task s worst-case executon tme (WCET) whch may be sgnfcantly nflated. Such parttonng rarely mproves upon schedulablty compared to accountng for CRPD and allowng tasks to use the entre cache [3]. An alternatve method whch elmnates CRPD wthout ncreasng WCETs s to employ a fully non-preemptve scheduler. Non-preemptve schedulng has the addtonal advantage of reducng memory requrements, as well as mprovng the dependablty of real-tme systems [32]. It s however well known that non-preemptve schedulng can be nfeasble at low processor utlzaton levels due to the long task problem [32], where some task has a WCET greater than the deadlne of another task. When consderng the theoretcal optmalty of unprocessor schedulng algorthms (.e., wthout accountng for overheads), then EDF-P s optmal n the sense that any task set that s feasble on a unprocessor under some other schedulng algorthm s also feasble usng EDF-P [21]. As a result, EDF-P domnates other unprocessor schedulng algorthms such as FP-P, FP-NP, and EDF-NP. When usng fxed prorty schedulng, prorty assgnment has a sgnfcant mpact on schedulablty. For FP-P schedulng, Deadlne Monotonc Prorty Orderng (DMPO) s optmal for constraned-deadlne task sets [28]. In other words, any constraned-deadlne task set that s schedulable under FP-P wth some other prorty orderng s also guaranteed to be schedulable wth DMPO. DMPO s not however optmal f task deadlnes are arbtrary [27] (.e. may be larger than ther perods). In that case, Audsley s algorthm [5] can be used to provde an optmal prorty assgnment. Wthn the class of non-preemptve schedulng algorthms, no work-conservng algorthm s optmal. Ths s because n general t s necessary to nsert dle tme to acheve a feasble schedule [22]. EDF-NP s however weakly optmal n the sense that f a work conservng non-preemptve schedule exsts for a task set, then EDF-NP can schedule t [23], hence EDF- NP domnates FP-NP. Wth FP-NP schedulng, DMPO s not optmal for constraned-deadlne task sets; however, Audsley s algorthm [5] can agan be appled [22]. Comparng the preemptve and non-preemptve paradgms, EDF-P domnates EDF-NP; however, the same s not true wth fxed prortes, FP-P does not domnate FP-NP. Instead, they are ncomparable. In other words, task sets exst that are feasble under FP-NP that are not feasble under FP-P and vce-versa [39]. Ths lack of any domnance relatonshp means that when fxed prortes are used, some systems are easer to schedule preemptvely, whle others are easer to schedule non-preemptvely. (Optmalty for fxed prorty schedulng requres lmted preempton wth fnal non-preemptve regons [14]; consderaton of that more complex model s however beyond the scope of ths paper). A. Speedup Factors In 2009, Davs et al. [20] derved the exact sub-optmalty S = 1/Ω 1.76 of FP-P schedulng for constraned-deadlne task sets. Ths exact bound complements the one for mplctdeadlne task sets S =1/ln(2) 1.44 that may be derved from Lu and Layland s famous results [29]. In 2009, Davs et al. [19] also derved upper and lower bounds of S =1/Ω and S =2on the sub-optmalty of FP-P schedulng for arbtrarydeadlne task sets. In 2015, Davs et al. [15] completed the exact characterzaton of the sub-optmalty of FP-P schedulng by provng that the exact speedup factor requred for arbtrarydeadlne task sets s n fact S =2. In the same paper, the authors also extended these results to the case where tasks share resources under mutual excluson accordng to the Stack Resource Polcy (SRP) [6] or the Dead Floor Protocol (DFP) [12], thus provdng exact speedup factors comparng FP-P + SRP to EDF + SRP or EDF + DFP. In 2010, Davs et al. [17] derved upper and lower bounds on the speedup factor requred to guarantee FP-NP feasblty of all EDF-NP feasble task sets. These bounds are S =1/Ω and S = 2 respectvely for all three classes of task set (mplct, constraned and arbtrary deadlne). In 2015, von der Brüggen et al. [38] proved upper bounds of S =1/Ω for the mplct and constraned deadlne cases, thus along wth the pror results, showng that these values are exact. Later n 2015, Davs et al. [15] also completed the exact characterzaton of the speedup factors requred to guarantee FP-NP feasblty of EDF-NP feasble task sets by showng that the exact speedup factor for the arbtrary deadlne case s S =2(the same as n the preemptve case for FP-P v. EDF-P). In 2013, Thekklakattl et al. [35][36] quantfed the sub-optmalty of EDF-NP (wth respect to EDF-P), brdgng between the preemptve and non-preemptve paradgms. (Ths result was subsequently extended to the case of global deadlne based schedulng [34]). In 2015, Abugchem et al. [1] subsequently provded a tghter upper bound on the sub-optmalty of EDF-NP. In ths paper, we focus on quantfyng the sub-optmalty of unprocessor FP-NP schedulng wth respect to an optmal algorthm such as EDF-P. As a consequence of ths work, we also quantfy the exact sub-optmalty of unprocessor EDF-NP schedulng. Further, we use the speedup factor

3 metrc to compare the performance of FP-P and FP-NP schedulng n both drectons, gven the lack of any domnance relaton between them. The man contrbutons of ths paper are n determnng for unprocessor systems: S1: The exact speedup factor requred to guarantee FP-NP feasblty of any EDF-P feasble task set (.e. the exact sub-optmalty of FP-NP). S2: The exact speedup factor requred to guarantee FP-NP feasblty for any task set that s FP-P feasble. S3: The exact speedup factor requred to guarantee EDF-NP feasblty of any EDF-P feasble task set (.e. the exact sub-optmalty of EDF-NP). S4: Upper and lower bounds on the speedup factor requred to guarantee FP-P feasblty for any task set that s FP-NP feasble. Note, where we refer to the exact sub-optmalty, orexact speedup factor for a non-preemptve schedulng algorthm compared to a preemptve one, then t s mportant to clarfy precsely what we mean. Snce non-preemptve schedulng suffers from the long task problem [32], whereby a task set may be trvally unschedulable because the longest executon tme C max of one task exceeds the shortest deadlne of another, then assumng freely determned task parameters no fnte speedup factor exsts. Ths s the case, because C max / can be made arbtrarly large. Instead, n ths paper we provde exact speedup factors that are parametrc n the rato C max /, and thus hold wth ths mnmal constrant on task parameters such that a fnte speedup factor exsts. We note that wth further nformaton about task set characterstcs t may be possble to determne more precse speedup factors wth narrower scope,.e. more constrants on ther valdty. In the extreme, each ndvdual task set effectvely has a precse speedup factor whch may be computed by referrng to all of the parameters of ts component tasks. In ths paper, as n prevous work on speedup factors [20], [19], [17], [35], [15] we assume that changes n processor speed have a lnear effect on the tme requred to execute each task. Consderng a unprocessor system n more detal, our assumpton s that the clock frequency may be changed and that ths has a lnear effect on the speed of all hardware components (processor, memory etc.) thus producng a lnear scalng of executon tmes. Such behavour s a reasonable approxmaton for smple systems. Whle the results presented n ths paper are manly theoretcal, they may also have practcal utlty n enablng system desgners to quantfy the maxmum penalty for usng FP-NP schedulng n terms of the addtonal processng capacty requred as compared to FP-P or EDF-P. Ths performance penalty can then be weghed aganst other factors such as the addtonal overheads (such as context swtch costs and CRPD) ncurred by preemptve schedulng, when consderng whch algorthm to use. B. Organzaton The rest of the paper s organzed as follows: the system model s presented n Secton II. Secton III recaps on the schedulablty analyses for preemptve and non-preemptve EDF and fxed prorty schedulng. Our man results on sub-optmalty and speedup factors are presented n Sectons IV and V, wth the results of an emprcal nvestgaton reported n Secton VI. Secton VII concludes wth a summary and a dscusson of open problems. II. SYSTEM MODEL In ths secton we descrbe the system model, termnology, and notaton used n the rest of the paper. A. Task Model We consder the schedulablty of a set of sporadc tasks on a unprocessor system. A task set Γ comprses a statc set of n tasks {τ 1,τ 2,...τ n }. Each task τ s characterzed by ts mnmum nter-arrval tme T, bounded worst-case executon tme C, and relatve deadlne D. Deadlnes may be mplct (D = T ), constraned (D T ), or arbtrary (ndependent of the task s perod). The longest executon tme of any of the tasks s denoted by C max = max C. Smlarly, the τ Γ shortest deadlne s denoted by = mn D. In the case τ Γ of fxed prorty schedulng, we use hp() and hep() to denote respectvely the set of tasks wth prortes hgher than, and hgher than or equal to that of task τ. Smlarly, we use lp() to denote the set of tasks wth prortes lower than that of task τ. (Note, we assume that prortes are unque). Further, we use B to denote the longest tme for whch task τ may be blocked by a lower prorty task that s executng non-preemptvely. The utlzaton U of a task τ s gven by U = C T and the utlzaton of the task set s the sum of the utlzatons of the ndvdual tasks U = n =1 U. B. Executon Tme Model To ease readablty, and wthout loss of generalty, we assume that the task set of nterest s ntally executng on a processor of unt speed. Accordngly, we assume that C represents the WCET of task τ on a processor of speed S =1. We assume a lnear relatonshp between executon tme and processor speed. The WCET of task τ on a processor of speed S s therefore gven by C S = C /S. Conversely, the speed S requred to obtan an executon tme of C S s gven by S = C /C S. Ths model allows us to use processor speedup factors and processor speeds nterchangeably. In other words, changng the processor speed from S =1to S = x, s equvalent to speedng up the processor by a factor of x. C. Schedulng Model In ths paper, we consder four schedulng algorthms EDF-P, EDF-NP, FP-P, and FP-NP. Wth EDF-P, at any gven tme the ready task wth the job that has the earlest absolute deadlne s executed by the processor. Smlarly, wth FP-P

4 schedulng, at any gven tme the processor executes the job of the ready task wth the hghest prorty. By contrast, wth EDF-NP, whenever a job s released that has an earler absolute deadlne than the currently executng job, nstead of preemptng the executng job the scheduler blocks the new job untl the currently executng job completes. Only at that pont s the ready job wth the earlest absolute deadlne dspatched for executon. Smlarly, wth FP-NP schedulng, whenever a hgher prorty task s released durng the executon of a lower prorty task τ, nstead of preemptng τ the scheduler blocks the hgher prorty task untl τ completes ts executon. Only at that pont s the hghest prorty ready task dspatched for executon. We note that all four schedulng algorthms are work-conservng and so never dle the processor when there s a task ready to execute. D. Defntons We now provde formal defntons for the terms speedup factor, speedup optmal task set and sub-optmalty. Recall that when we use the term feasble, then n the case of fxed prorty schedulng, we mean schedulable wth an optmal prorty assgnment. Defnton II.1. The speed-up factor of a schedulng algorthm A wth respect to a schedulng algorthm B s defned as the mnmum factor S, S 1, such that any task set that s feasble under B on a processor of unt speed, s guaranteed to be feasble under A on a processor that s S tmes faster. Defnton II.2. A task set s sad to be speed-up optmal for the comparson between schedulng algorthms A and B f t s feasble on a processor of unt speed under B and requres the processor speed to be ncreased by the speedup factor S n order to be feasble under A. Defnton II.3. The sub-optmalty of a schedulng algorthm A s defned by ts speedup factor wth respect to an optmal schedulng algorthm. Defnton II.4. A schedulng algorthm s sad to be optmal f t can schedule every task set that s feasble under some other schedulng algorthm, on a processor of equvalent speed. The lower the sub-optmalty of a partcular schedulng algorthm, the closer t s to beng optmal, wth a value of S =1mplyng optmalty. We note that FP-P, FP-NP, and EDF-NP are all sub-optmal wth respect to an optmal unprocessor schedulng algorthm such as EDF-P. III. SCHEDULABILITY ANALYSIS In ths secton, we recaptulate schedulablty analyss for fxed prorty and EDF schedulng under both preemptve and non-preemptve paradgms. A. Fxed Prorty Preemptve Schedulng The schedulablty of a set of arbtrary-deadlne sporadc tasks under FP-P can be determned usng response tme analyss [37] [27]. Response tme analyss nvolves calculatng the worst-case response tme R P of each task τ and comparng t to ts deadlne D. To determne schedulablty, the analyss must check each job of task τ n the longest prorty level- busy perod. Ths busy perod starts wth a crtcal nstant correspondng to the synchronous arrval of a job of task τ and jobs of all hgher prorty tasks. Jobs of these tasks are then re-released as soon as possble. The length of the prorty level- busy perod s gven by the soluton to the followng recurrence relaton: A P = A P C j (1) τ j hep() The number of jobs of task τ n the busy perod s gven by Q P = AP T. The completon tme W P (q) of job q of task τ relatve to the start of the busy perod s gven by the followng recurrence relaton: W P (q) =(q +1)C + W P (q) C j (2) τ j hp() Iteraton starts wth W P (q) =(q +1)C and ends ether on convergence or when W P (q) qt >D n whch case the job and therefore the task s unschedulable. Assumng that all Q P jobs n the busy perod are schedulable, then the worst-case response tme of the task s gven by: R P = max (W (q) qt ) (3) q=0,1,2,...q P 1 For task sets wth constraned deadlnes, only the response tme of the frst job n the busy perod need be checked, leadng to a smpler exact test [4], [24], based on the followng recurrence relaton: R P = C + τ j hp() τ j hep() R P C j (4) Iteraton starts wth R P = C and ends ether on convergence or when R P >D n whch case the task s unschedulable. B. Fxed Prorty Non-Preemptve Schedulng Determnng exact schedulablty of a task τ under FP-NP also requres checkng all of the jobs of task τ wthn a prorty level- busy perod [10]. In ths case, the busy perod starts wth an nterval of blockng and so ts length s gven by the soluton to the followng recurrence relaton: A NP = B + A NP C j (5) where B s the blockng factor: { max B = C k Δ <n τ k lp() 0 = n and Δ s the tme granularty 1. 1 Wthout loss of generalty, we assume that Δ s the granularty of the processor clock and that Δ C k for every task τ k even when we ncrease the processor speed. (6)

5 The number of jobs of task τ n the busy perod s gven by Q NP = ANP T. The start tme W NP (q) of job q of task τ relatve to the start of the busy perod s gven by the followng recurrence relaton: W NP (q) =B + qc + W NP (q)+δ C j (7) τ j hp() Iteraton starts wth W NP (q) = B + qc and ends ether on convergence or when W NP (q)+c qt >D n whch case the job and therefore the task s unschedulable. Assumng that all Q NP jobs n the busy perod are schedulable, then the worst-case response tme of the task s gven by: R NP = max q=0,1,2,...q NP (W NP (q)+c qt ) (8) 1 Note, n the above formulaton we use a celng functon wth +Δ, rather than the alternatve of a floor functon +1, snce ths asssts n the proofs gven later n the paper. The two formulatons are however equvalent. We make use of the followng suffcent schedulablty tests for each task τ under FP-NP. The frst s based on a lnear equaton [17]: B + D C j D (9) τ j hep() The second, whch s only applcable to constraned-deadlne task sets s based on a recurrence relaton [16]: W NP = C max + R NP τ j hp() W NP +Δ C j = W NP + C (10) where W NP s an upper bound on the longest tme from release to the start of any job of task τ. C. Preemptve Earlest Deadlne Frst Schedulng A task set s schedulable under preemptve EDF f and only f n every tme nterval, the total processor demand requested by the task set s no greater than the length of the nterval [7]. A task set s EDF-P feasble f and only f : DBF (t) t (11) where τ Γ t = k + D j, k N,j [1,n] t A P n ( DBF (t) =max 0, 1+ t D T ) C (12) and A P n s the length of the longest busy perod, gven by (1) [31] [33]. IV. EXACT SUB-OPTIMALITY AND SPEEDUP FACTORS In ths secton, we compare the effectveness of fxed prorty non-preemptve schedulng (FP-NP) wth that of preemptve schedulng; both FP-P and EDF-P. We determne the exact sub-optmalty of FP-NP. Specfcally, we derve the exact speedup factor S1 requred to guarantee feasblty under FP-NP of all EDF-P feasble task sets. Further, we derve the exact speedup factor S2 requred to guarantee feasblty under FP-NP of all FP-P feasble task sets. Surprsngly these two speedup factors are the same (S1 = S2). We also derve an exact speedup factor for the case of FP-NP v. FP-P, when tasks have constraned deadlnes. Ths speedup factor s smaller than n the arbtrary-deadlne case. We obtan the exact speedup factors by dervng upper bounds va analyss and lower bounds from example task sets and then showng that they are the same. The example task set we use to provde a lower bound for FP-NP v. EDF-P also apples to EDF-NP v. EDF-P, hence we also obtan S3, the exact sub-optmalty of EDF-NP, snce our lower bound s the same as the upper bound recently publshed by Abugchem et al. [1]. Lemma IV.1. An upper bound on the speedup factor requred such that FP-NP, usng optmal prorty assgnment can schedule any arbtrary-deadlne sporadc task set that s feasble under EDF-P s gven by: Proof: We show that the speedup factor n the lemma s enough to ensure schedulablty under FP-NP accordng to the suffcent test gven by (9) usng DMPO, snce that suffces to also prove schedulablty wth an exact test and optmal prorty assgnment. Comparng (9) and (12) and assumng DMPO we observe that: D DBF j (2D ) C j (13) τ j Γ τ j hep() τ j:d j D D C j From (9), (13), and the fact that B C max then schedulablty under FP-NP s assured on a processor of speed S provded that for every task τ : C max + τ k Γ DBF k(2d ) D (14) S Snce the task set s schedulable under EDF-P on a processor of unt speed, then t follows from (11) that τ k Γ DBF k(2d ) 2D. Substtutng nto (14) and re-arrangng, we have: S 2+ C max D Substtutng for D gves an upper bound on the speedup factor requred.

6 Lemma IV.2. An upper bound on the speedup factor requred such that FP-NP, usng optmal prorty assgnment can schedule any arbtrary-deadlne sporadc task set that s feasble under FP-P schedulng s gven by: Proof: Follows drectly from Lemma IV.1 and the fact that EDF-P can schedule all task sets that are feasble under FP-P schedulng. [21]. Lemma IV.3. A lower bound on the speedup factor requred such that FP-NP, usng optmal prorty assgnment can schedule any mplct, constraned, or arbtrary-deadlne sporadc task set that s feasble under EDF-P (or FP-P) s gven by: Proof: Consder the followng task set: τ 1 : C 1 = k 1, D 1 = k, T 1 = k τ 2 : C 2 = k 2 +1, D 2 =, T 2 = We note that the task set s trvally schedulable on a processor of unt speed usng ether EDF-P or FP-P. For the task set to be schedulable wth FP-NP effectvely requres that the executon tme of both tasks 2 (.e. k 2 + k) can be accommodated wthn the smallest deadlne D 1 = k. Hence we have S (k 2 +k)/k = k+1. Snce Cmax = k+ 1 k we obtan: S 1+ C max 1 k and so as k we have a lower bound of: S 1+ C max Theorem IV.1. The exact sub-optmalty (S3) of EDF-NP,.e, the exact speedup factor requred such that EDF-NP can schedule any mplct, constraned, or arbtrary-deadlne sporadc task set that s feasble under EDF-P s gven by: Proof: Follows from a consderaton of the task set n Lemma IV.3. For the task set to be schedulable under EDF- NP also requres that the total executon tme of both tasks can be accommodated wthn the smallest deadlne resultng n the same requrement on the speedup factor. Snce the lower bound from Lemma IV.3 matches the upper bound gven by Abugchem et al. [1] the value s exact. Lemma IV.4. An upper bound on the speedup factor requred such that FP-NP schedulng, usng optmal prorty assgnment can schedule any constraned-deadlne sporadc task set that s feasble under FP-P schedulng s gven by: 2 For ease of presentaton, and snce t does not affect the result, we omt the small reducton n blockng due to the tme granularty Δ 1. (Note ths Lemma does not apply to arbtrary-deadlne tasks sets). Proof: Let Γ be a task set that s schedulable under FP-P schedulng on a processor of unt speed, usng DMPO, whch s optmal n the constraned deadlne case. We wll prove that Γ s schedulable on a processor of speed S under FP-NP schedulng usng the same prorty orderng. We note that ths orderng s not necessarly optmal for FP-NP schedulng, but suffces to prove feasblty. Let W P be the completon tme of the frst job of task τ n the prorty level- busy perod under FP-P schedulng. Snce all tasks are schedulable and have constraned deadlnes, then W P = R P D. We consder two cases. Case 1: W P Let E P (t) equate to C plus the maxmum amount of executon from tasks of hgher prorty than τ released n an nterval of length t: E P (t) =C + t C j (15) τ j hp() From (4), t follows that E P (W P )=W P = R P where R P s the exact response tme of task τ under FP-P schedulng. Let E NP (t) be the maxmum amount of executon from tasks of hgher prorty than τ released n an nterval of length t ncludng any releases at the end of the nterval: E NP (t) = t +Δ C j (16) τ j hp() From the suffcent test for constraned-deadlne task sets under FP-NP schedulng (10) we have E NP (W NP )+C max + C = W NP + C where W NP s an upper bound on the tme from the release of a job of task τ untl t starts to execute, under FP-NP schedulng, and W NP + C s an upper bound on the task s response tme. From (15) and (16), observe that the followng holds x Δ and t x: E NP (t x)+c E P (t) (17) To ensure schedulablty under FP-NP schedulng, we speed up the processor by some factor S 1 such that the latest completon tme of task τ under FP-NP schedulng s no greater than W P the completon tme under FP-P schedulng on a processor of unt speed. It follows that the start tme of τ must be at the latest W P C S. An upper bound on the nterference from hgher prorty tasks n an nterval of ths length s gven by E NP (W P C S ). Schedulablty under FP-NP s then ensured provded that: C max + E NP (W P C S )+C W P (18) S Ths follows, snce f (18) holds then the upper bound response tme for task τ computed va (10) wll be W P. Snce even on the faster processor of speed S, the executon tme of τ cannot be less than the tme granularty ( C S

7 Δ), then from (17), t follows that E NP (W P C S )+C E P (W P ).AsEP (W P )=W P, substtutng nto (18) and re-arrangng we have: S 1+ C max W p (19) From the assumpton of ths case (Case 1) W P and hence the task set s guaranteed to be schedulable on a processor of speed S, where: < S 1+ C max (20) Case 2: W P Snce deadlnes are constraned, there are no tasks wth perods that are less than, and so under FP-P schedulng on a processor of unt speed, we have: W P = C + C j (21) j hp() In ths case, to ensure schedulablty under FP-NP on a processor of speed S, we smply requre that task τ completes before hence, we requre that: C max + E NP ( C S )+C (22) S where S s the processor speed. Followng the same logc as n Case 1, we observe that E NP ( C S )+C E P (W P )=W P. Snce n ths case (Case 2) W P < substtutng nto (22) and re-arrangng we obtan the speed S at whch the task set s guaranteed to be schedulable: S 1+ C max (23) Theorem IV.2. The exact speedup factor requred such that FP-NP, usng optmal prorty assgnment can schedule any mplct, or constraned-deadlne sporadc task set that s feasble under FP-P schedulng s gven by: Proof: Proof follows from the lower bound gven by Lemma IV.3 and the upper bound gven by Lemma IV.4 whch have the same value. Lemma IV.5. A lower bound on the speedup factor requred such that FP-NP schedulng, usng optmal prorty assgnment can schedule any arbtrary-deadlne sporadc task set that s feasble under FP-P schedulng s gven by: Proof: Consder the followng task set: τ wth =1,...,k 1: C =1, D = k +1, T = k τ k : C k =1, D k = k +1, T k = k +1 τ k+1 : C k+1 = k 2, D k+1 =, T k+1 = Ths task set s trvally schedulable on a processor of unt speed under FP-P. In the prorty order shown, then for j =1 to k, task τ j has a response tme of j. Further, task τ k+1 executes n the one spare unt of executon tme n each Least Common Multple k(k +1) of the perods of tasks τ 1 to τ k and therefore has a worst-case response tme of k 3 (k +1). Under FP-NP on a processor of speed S 1 consder the operaton of Audsleys OPA algorthm, whch s optmal n ths case [22]. Frst, task τ k+1 s assgned as t s trvally schedulable at the lowest prorty on a processor of unt speed or hgher. There are then two cases to consder 3. Case 1: τ k s assgned the next hgher prorty level above τ k+1. In ths case, task τ k s subject to blockng due to task τ k+1 and nterference (before t starts to execute) from tasks τ 1 to τ k 1. Consderng the crtcal nstant for task τ k, there are two possble scenaros whch could result n the task beng schedulable. In the frst scenaro, the frst jobs of all tasks except τ k must complete ther executon strctly before the second jobs of tasks τ 1 to τ k 1 are released at tme k. Ths allows task τ k to start executng before tme k, thus avodng nterference from the second job of each hgher prorty task. For ths to happen mples the followng constrant: S>(k 2 + k 1)/k = k +(k 1)/k. Further, task τ k must also complete by tme k +1, whch gves the weaker constrant S (k 2 + k)/(k +1)=k. The alternatve scenaro s that task τ k does not get to start before the second jobs of tasks τ 1 to τ k 1 are released at tme k. In ths scenaro, for task τ k to be schedulable, the frst job of task τ k+1, the frst and second jobs of tasks τ 1 to τ k 1, and the frst job of task τ k must complete ther executon by tme k + 1, whch leads to the constrant that S (k 2 +2k 1)/(k +1)=k +(k 1)/(k +1). Case 2: τ k 1 s assgned the next hgher prorty level above τ k+1 (snce τ 1 to τ k 1 are dentcal ths s effectvely the only other opton asde from Case 1 for ths prorty level). Consderng the crtcal nstant for task τ k 1, there are two possble scenaros whch could result n the task beng schedulable. In the frst scenaro, the frst jobs of all tasks except τ k 1 must complete ther executon strctly before the second jobs of tasks τ 1 to τ k 2 are released at tme k. Ths allows task τ k 1 to start executng before tme k, thus avodng nterference from the second job of each hgher prorty task. As n Case 1, ths mples the followng constrant: S>(k 2 + k 1)/k = k +(k 1)/k. In addton task τ k 1 must also complete by tme k +1, whch agan gves S (k 2 + k)/(k +1)=k. The alternatve scenaro s that the frst job of task τ k 1 does not get to start before the second jobs of tasks τ 1 to τ k 1 are released at tme k. In ths scenaro, for task τ k 1 to be schedulable, then the frst job of task τ k+1, the frst and second jobs of tasks τ 1 to τ k 2, and the frst job of task τ k 1 must complete ther executon by tme k +1, whch leads to the constrant S (k 2 +2k 2)/(k +1)=k +(k 2)/(k +1). 3 Agan, for ease of presentaton, and snce t does not affect the result, we omt the small reducton n blockng due to the tme granularty Δ 1.

8 Consderng both Case 1 and Case 2, then the mnmum speed necessary for FP-NP schedulablty s S (k 2 +2k 2)/(k +1) = k +(k 2)/(k +1). Snce C max / = k 2 /(k +1) we obtan: S C max + k2 +2k 2 k2 k +1 k +1 = C max + 2k 2 k +1 As k ths gves a lower bound of S =2+ Cmax for the speedup factor Theorem IV.3. The exact speedup factor (S2) requred such that FP-NP schedulng, usng optmal prorty assgnment can schedule any arbtrary-deadlne sporadc task set that s feasble under FP-P schedulng s gven by: Fg. 1. Fxed Prorty Non-Preemptve Schedule. Proof: Proof follows from the lower bound gven by Lemma IV.5 and the upper bound gven by Lemma IV.2 whch have the same value. From Theorems IV.2 and IV.3, t s nterestng to note that when comparng FP-NP aganst FP-P schedulng, then the relaxaton from constraned-deadlne task sets to the general case of arbtrary-deadlne tasks results n an ncrease n the exact speedup factor requred from to 2+ C max Theorem IV.4. The exact sub-optmalty (S1) of FP-NP.e. the exact speedup factor requred such that FP-NP schedulng, usng optmal prorty assgnment can schedule any arbtrarydeadlne sporadc task set that s feasble under EDF-P s gven by: Proof: Lemma IV.1 shows that the speedup factor n the theorem s a vald upper bound. Lemma IV.5 and the fact that EDF-P domnates FP-P shows that t s also a vald lower bound and hence exact for arbtrary-deadlne task sets. V. PREEMPTIVE FPS VS. NON-PREEMPTIVE FPS Preemptve and non-preemptve fxed prorty schedulng are ncomparable,.e., there are task sets that FP-P can schedule that FP-NP cannot and vce versa, hence there are non-trval speed-up factors n both drectons between these two schedulng algorthms. In ths secton, we derve upper and lower bounds on the processor speed-up factor S4 that guarantees FP-P feasblty of FP-NP feasble task sets. Theorem V.1. An upper bound on the speed-up factor that guarantees FP-P feasblty of all FP-NP feasble task sets s gven by: S =2 Fg. 2. Fxed Prorty Preemptve Schedule. Proof: Snce EDF-P domnates FP-NP and Theorem 2 from [20] states that an upper bound on the speed-up factor requred to guarantee FP-P feasblty of any EDF-P feasble task set s S =2then such an ncrease n processor speed must also be suffcent to guarantee FP-P feasblty of all FP-NP feasble task sets. Theorem V.2. A lower bound on the speed-up factor that guarantees FP-P feasblty of all FP-NP feasble task sets s gven by: S = 2 Proof: Consder the followng task set scheduled on a processor of unt speed under FP-NP schedulng. τ 1 : C 1 =2 2, D 1 =1, T 1 =1 τ 2 : C 2 = 2 1, D 2 = 2, T 2 = τ 3 : C 3 = 2 1, D 3 = 2, T 3 = Ths task set s schedulable wth DMPO under FP-NP as evdenced by the exact schedulablty test emboded n (8). The response tmes of the three tasks are as follows: R 1 =1 Δ, R 2 = 2 Δ, R 3 = 2. Note, that n each case we need only examne the response tme of the frst job. For task τ 1, the prorty level-1 busy perod s of length 1 and so ncludes only one job of the task, whle tasks τ 2 and τ 3 have nfnte perods and so only gve rse to a sngle job. The schedule startng wth task τ 1 s llustrated n Fgure 1. Next, consder the same task set scheduled on a processor of speed S = 2 under FP-P schedulng, agan usng DMPO whch s optmal n ths case. The scaled task executon tmes are now C S 1 = 2 1, C S 2 =(2 2)/2, and C S 3 =(2 2)/2. The schedule s as llustrated n Fgure 2, agan startng wth task τ 1. In ths case, the worst-case response tme of task τ 3 s 1. Further, any ncrease n the executon tmes of the tasks (.e. by usng a smaller speedup factor) would result n task τ 3 mssng ts deadlne, due to preempton by the second job of task τ 1 whch s released at tme t =1. Hence the speedup factor requred by ths task set s S = 2. In the next secton, we descrbe the results of an emprcal

9 study whch hnts that the lower bound derved above may be tght (.e., the exact speed-up factor may be 2 rather than some larger value such as the upper bound of 2). VI. EMPIRICAL INVESTIGATION In ths secton, we descrbe the results of an emprcal nvestgaton nto the speed-up factor needed to ensure that FP-NP feasble task sets are schedulable under FP-P. Ths was done by usng a genetc algorthm to explore the search space of task parameters. The operaton of the genetc algorthm s outlned below. Frst, an ntal populaton of N task sets each wth n tasks were created. The task utlsatons were assgned accordng to the UUnfast [9] algorthm. Task perods were chosen n the range [10 4, 10 7 ] accordng to a log unform dstrbuton. For task sets wth constraned deadlnes, deadlnes were chosen accordng to a log unform dstrbuton n the range [C, T] and for arbtrary deadlne task sets from the range [C, 10T ]. Computaton tmes were determned accordng to C = U T. All parameters were dscrete and represented usng 64-bt ntegers. The speed-up factor for each task set was found by performng two bnary searches to determne respectvely, the scalng factors f FP NP and f FP P requred such that that the task set was just schedulable accordng to FP-NP and just unschedulable accordng to FP-P. Audsleys algorthm was used n both cases to determne the optmal prorty assgnment, along wth exact schedulablty tests for arbtrary deadlne task sets under FP-NP (8), and FP-P (3). The speed-up factor for the task set was then gven by S = f FP P. Ths approach ensured f FP NP that any lack of precson n the computed speed-up factor caused by mprecson n the bnary searches could only result n a small underestmate (and no overestmate) of the precse speed-up factor for the specfc task set. The precson of the bnary searches was 0.01% (.e., the termnaton condton was such that the low and hgh values for the scalng factor were wthn 0.01% of each other). Crossover type 1-pont Crossover probablty 0.5 Mutaton 20% of C, D, or T Mutaton probablty 0.6 Parent selecton Tournament Survval selecton Tournament on combned old and new populaton Populaton sze 20,000 Tournament sze 50 Generatons 400 TABLE I EXPERIMENTAL PARAMETERS The computed speed-up factor for each task set was used as the ftness functon n the genetc algorthm. The genetc algorthm operated as follows. Frst a populaton of N random task sets each wth n tasks were created, as descrbed above. The ftness (speed-up factor) of each of these task sets was Fg. 3. Emprcal results evaluated, wth the maxmum speed-up factor of any task set n the populaton recorded at ths and each subsequent stage. A new generaton of N chld task sets was then produced. Parent task sets were selected from the exstng populaton usng a tournament selecton process. These parents produced chldren va crossover and mutaton operatons, whch occurred wth specfed probabltes. Frst copes were made of the parent task sets, whch became the chldren. If crossover occurred, then a random poston n the lst of tasks was chosen and the chld task sets splt and recombned at that pont. (The head of one beng joned to the tal of the other). Each chld task set was then potentally subject to mutaton. If mutaton occurred, then a sngle parameter C, D, ort was selected at random, and ncreased or decreased by a random value n the range [0, 20%]. The parameters were then repared as necessary to ensure that any constrants on task deadlnes contnued to be met. For example, wth constraned deadlnes, f the perod of a task was decreased below ts deadlne, then the deadlne was adjusted to be equal to the perod. (Repars were also made to avod parameters gong out of range). Once N chld nodes had been produced, then ther ftness was evaluated, and a further tournament selecton used to reduce the overall populaton (parents and chldren) to N task sets. Ths overall process was then repeated for further generatons, wth task sets wth hgher speed-up factors more lkely to survve and produce offsprng. The tournament selecton process nvolved random selecton of 50 task sets wth the one wth the hghest speed-up factor selected for the next part of the process. Here a large tournament sze ncreases selecton pressure, but reduces dversty n the populaton. The parameters enumerated n Table I were found emprcally to be effectve for ths problem. Fgure 3 shows the results obtaned for task sets wth mplct, constraned and arbtrary deadlnes. These results were produced usng the genetc algorthm descrbed above. 400 generatons of a populaton of 20, 000 task sets,.e., 8 mllon task sets were generated for each task set cardnalty and deadlne type. We note that wth both mplct and constraned deadlnes, task sets of cardnalty two have a

10 speed-up factor of 1. Ths s because wth FP-NP both tasks have the same worst-case response tme whch must be less than the smaller of ther deadlnes, and hence perods. Ths response tme s the same as that of the lowest prorty task under FP-P, hence all task sets of cardnalty two that are schedulable under FP-NP are also schedulable under FP-P, mplyng a speed-up factor of 1. For three or more tasks, then wth constraned or arbtrary deadlnes, the maxmum speed-up factor found by the genetc algorthm s very close to 2 = In fact the values range from to (constraned deadlnes) and from to (arbtrary deadlnes) for task sets of cardnalty 3 to 10. Wth mplct-deadlne task sets, the largest speed-up factor found was somewhat lower at The fact that the maxmum value found emprcally (1.4139) s very close to but does not exceed 2= gves credence to the hypothess that the theoretcal lower bound (of 2 ) on the speed-up factor s the exact value. It remans an nterestng open queston whether or not ths s the case [18]. VII. SUMMARY AND CONCLUSIONS The man contrbuton of ths paper s the dervaton of resource augmentaton bounds for preemptve and non-preemptve schedulng algorthms on a unprocessor. Specfcally, we derved the followng sub-optmalty and speedup factor results: S1: Exact sub-optmalty of FP-NP for tasks wth arbtrary deadlnes: For task sets wth mplct or constraned-deadlnes: Lower Bound Upper Bound S2: Exact speedup factor requred for FP-NP feasblty of any arbtrary deadlne task set that s FP-P feasble: For task sets wth mplct or constraned deadlnes: S3: Exact sub-optmalty of EDF-NP for mplct, constraned or arbtrary deadlne task sets: S4: Speedup factor requred for FP-P feasblty of any constraned or arbtrary deadlne task set that s FP-NP feasble. Lower Bound S = 2 Upper Bound S =2 A summary of the results derved n ths paper (underlned), together wth the state-of-the-art for arbtrary deadlne task sets s presented n Fgure 4. (The dashed arrows on the fgure represent domnance relatonshps, where the exact speed-up factor n the reverse drecton s 1). The major remanng open problems nvolve tghtenng the upper and lower bounds where exact values are not yet known. These nclude determnng the exact sub-optmalty of FP-NP for the case of mplct and constraned deadlne task sets, and determnng the exact speedup factor requred for FP-P feasblty of any task set that s FP-NP feasble for the mplct, constraned and arbtrary deadlne cases. Whle the speedup factor results derved n ths paper are manly of nterest n provdng a theoretcal comparson focusng on the worst-case behavour of the dfferent schedulng algorthms, these results also help provde practcal gudance. For example, the majorty of real-tme operatng systems support fxed prorty schedulng, wth those mandated for automotve systems by the OSEK and AUTOSAR standards supportng both FP-P and FP-NP schedulng. Here, t s nterestng to consder the comparson between FP-P and FP-NP; even though the two schedulng polces are ncomparable. The exact speedup factor requred for FP-NP feasblty of any constraned deadlne task set that s FP-P feasble s S =1+ Cmax (see Theorem IV.2). Thus f we have a system where the longest executon tme of any task s substantally less than the shortest deadlne (C max ), we can quantfy the small processng speed penalty for usng non-preemptve schedulng. Ths can then be weghed aganst the addtonal overheads (e.g. preempton costs, cache related preempton delays, support for mutually exclusve resource accesses etc.) ncurred n usng preemptve schedulng; as well as other consderatons such as the addtonal complexty nvolved n accurately modellng and testng a preemptve system. When C max then t s clear that the penalty for usng fully non-preemptve schedulng s very hgh (the long task problem [32]), n such cases methods that support lmted preempton, effectvely breakng long tasks nto a set of non-preemptve regons may be preferable. A further avenue for the extenson of ths work s to systems that support lmted preempton [13], n partcular ncludng fnal non-preemptve regons, snce that paradgm domnates both FP-P and FP-NP schedulng [14]. ACKNOWLEDGMENTS Ths work was funded n part by the EPSRC project MCC (EP/K011626/1) and the Inra Internatonal Char program. EPSRC Research Data Management: No new prmary data was created durng ths study. REFERENCES [1] Fath Abugchem, Mchael Short, and Dongla Xu. A note on the suboptmalty of non-preemptve real-tme schedulng. Embedded Systems Letters, IEEE, PP(99):1 1, [2] Sebastan Altmeyer, Robert I. Davs, and Clare Maza. Improved cache related pre-empton delay aware response tme analyss for fxed prorty pre-emptve systems. Real-Tme Systems, 48(5): , [3] Sebastan Altmeyer, Roeland Douma, Wll Lunnss, and Robert I. Davs. Evaluaton of cache parttonng for hard real-tme systems. In proceedngs Euromcro Conference on Real-Tme Systems (ECRTS), pages 15 26, July 2014.

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