Worst-case response time analysis of real-time tasks under fixed-priority scheduling with deferred preemption

Transcription

1 Real-Tme Syst (2009) 42: DOI /s z Worst-case response tme analyss of real-tme tasks under fxed-prorty schedulng wth deferred preempton Render J. Brl Johan J. Lukken Wm F.J. Verhaegh Publshed onlne: 28 Aprl 2009 The Author(s) Ths artcle s publshed wth open access at Sprngerlnk.com Abstract Fxed-prorty schedulng wth deferred preempton (FPDS) has been proposed n the lterature as a vable alternatve to fxed-prorty pre-emptve schedulng (FPPS), that obvates the need for non-trval resource access protocols and reduces the cost of arbtrary preemptons. Ths paper shows that exstng worst-case response tme analyss of hard real-tme tasks under FPDS, arbtrary phasng and relatve deadlnes at most equal to perods s pessmstc and/or optmstc. The same problem also arses for fxed-prorty nonpre-emptve schedulng (FPNS), beng a specal case of FPDS. Ths paper provdes a revsed analyss, resolvng the problems wth the exstng approaches. The analyss s based on known concepts of crtcal nstant and busy perod for FPPS. To accommodate for our schedulng model for FPDS, we need to slghtly modfy exstng defntons of these concepts. The analyss assumes a contnuous schedulng model, whch s based on a parttonng of the tmelne n a set of non-empty, rght sem-open ntervals. It s shown that the crtcal nstant, longest busy perod, and worst-case response tme for a task are suprema rather than maxma for all tasks, except for the lowest prorty task. Hence, that nstant, perod, and response tme cannot be assumed for any task, except for the lowest prorty task. Moreover, t s shown that the analyss s not unform for all tasks,.e. the analyss for the lowest prorty task dffers from the Excerpts of ths document have been publshed as Brl et al. (2007). R.J. Brl ( ) J.J. Lukken Department of Mathematcs and Computer Scence, Technsche Unverstet Endhoven (TU/e), Den Dolech 2, 5600 AZ, Endhoven, The Netherlands e-mal: r.j.brl@tue.nl J.J. Lukken e-mal: j.j.lukken@tue.nl W.F.J. Verhaegh Phlps Research Laboratores, Hgh Tech Campus 11, 5656 AE, Endhoven, The Netherlands e-mal: wm.verhaegh@phlps.com

2 64 Real-Tme Syst (2009) 42: analyss of the other tasks. These anomales for the lowest prorty task are an mmedate consequence of the fact that only the lowest prorty task cannot be blocked. To buld on earler work, the worst-case response tme analyss for FPDS s expressed n terms of known worst-case analyss results for FPPS. The paper ncludes pessmstc varants of the analyss, whch are unform for all tasks, llustrates the revsed analyss for an advanced model for FPDS, where tasks are structured as flow graphs of subjobs rather than sequences, and shows that our analyss s sustanable. Keywords Level- actve perod Level- busy perod Worst-case response tme Worst-case occuped tme Perodc tasks Fxed-prorty schedulng Deferred preempton Real-tme systems 1 Introducton 1.1 Motvaton Based on the semnal paper of Lu and Layland (1973), many results have been acheved n the area of analyss for fxed-prorty preemptve schedulng (FPPS). Arbtrary preempton of real-tme tasks has a number of drawbacks, though. In systems requrng mutual access to shared resources, arbtrary preemptons nduce the need for non-trval resource access protocols, such as the prorty celng protocol (Sha et al. 1990). In systems usng cache memory, e.g. to brdge the speed gap between processors and man memory, arbtrary preemptons nduce addtonal cache flushes and reloads. As a consequence, system performance and predctablty are degraded, complcatng system desgn, analyss and testng (Burns and Wellngs 1997; Gopalakrshnan and Parulkar 1996; Lee et al. 1998; Mok and Poon 2005; Smonson and Patel 1995). Although fxed-prorty non-preemptve schedulng (FPNS) may resolve these problems, t generally leads to reduced schedulablty compared to FPPS. Therefore, alternatve schedulng schemes have been proposed between the extremes of arbtrary preempton and no preempton. These schemes are also known as deferred preempton or co-operatve schedulng (Burns 1994), and are denoted by fxedprorty schedulng wth deferred preempton (FPDS) n the remander of ths paper. Worst-case response tme analyss of perodc real-tme tasks under FPDS, arbtrary phasng, and relatve deadlnes wthn perods has been addressed n a number of papers (Brl et al. 2004; Burns 1994; Burns and Wellngs 1997; Lee et al. 1998). The exstng analyss s not exact, however. In Brl et al. (2004), t has already been shown that the analyss presented n Burns (1994), Burns and Wellngs (1997), Lee et al. (1998) s pessmstc. More recently, t has been shown n Brl (2006) that the analyss presented n Brl et al. (2004), Burns (1994), Burns and Wellngs (1997) s optmstc. Unlke the mplct assumptons n those latter papers, the worst-case response tme of a task under FPDS and arbtrary phasng s not necessarly assumed for the frst job of that task upon ts crtcal nstant. Hence, the exstng analyss may provde guarantees for tasks that n fact mss ther deadlnes n the worst-case. In Brl et al. (2006), t has been shown that the latter problem also arses for FPNS, beng a specal case of FPDS, and ts applcaton for the schedulablty analyss of controller area networks (CAN) (Tndell and Burns 1994; Tndell et al. 1994, 1995). Revsed

3 Real-Tme Syst (2009) 42: analyss for CAN resolvng the problem wth the orgnal approach n an evolutonary fashon can be found n Davs et al. (2007). 1.2 Contrbutons Ths paper resolves the problems wth the exstng approaches by presentng a novel worst-case response tme analyss for hard real-tme tasks under FPDS, arbtrary phasng and arbtrary relatve deadlnes. The analyss assumes a contnuous schedulng model rather than a dscrete schedulng model (Baruah et al. 1990b), e.g. all task parameters are taken from the real numbers. The motvaton for ths assumpton stems from the observaton that a dscrete vew on tme s n many stuatons nsuffcent; see for example Baeten and Mddelburg (2002), Hooman (1991), Koymans (1990). The schedulng model s based on a parttonng of the tmelne n a set of non-empty, rght sem-open ntervals (Buttazzo 2005; Hooman 1991). The analyss s based on the concepts of crtcal nstant (Lu and Layland 1973) and busy perod (Lehoczky 1990). To accommodate for our schedulng model for FPDS, we need to slghtly modfy the exstng defntons of these concepts. To prevent confuson wth the exstng defnton of busy perod, we use the term actve perod n ths document, for whch we gve a formal defnton. In ths document, we dscuss condtons for termnaton of an actve perod, and present a suffcent condton wth a formal proof. Moreover, we show that the crtcal nstant, longest actve perod, and worst-case response tme for a task are suprema rather than maxma for all tasks, except for the lowest prorty task. Hence, that nstant, perod, and response tme cannot be assumed for any task, except for the lowest prorty task. Our worst-case response tme analyss s not unform for all tasks. In partcular, the analyss for the lowest prorty task dffers from the analyss for the other tasks. These anomales for the lowest prorty task are an mmedate consequence of the fact that, unlke the other tasks, the lowest prorty task cannot be blocked. To buld on earler results, worst-case response tmes under FPDS are expressed n terms of worst-case response tmes and worst-case occuped tmes (Brl 2004) under FPPS. We also present pessmstc varants of the analyss, whch are ndeed unform for all tasks, and show that the revsed analyss for CAN presented n Davs et al. (2007) conforms to a pessmstc varant. We llustrate our analyss for an advanced model for FPDS, where tasks are structured as flow graphs of subjobs rather than sequences, and we show that our analyss for FPDS s sustanable (Baruah and Burns 2006) and therefore also applcable for sporadc task systems (Mok 1983; Baruah et al. 1990a). 1.3 Related work Next to contnuous schedulng models, one can fnd dscrete schedulng models n the lterature, e.g. n George et al. (1996), Hermant et al. (1996), and models n whch domans are not explctly specfed (Buttazzo 2005; Klen et al. 1993; Lu2000). Because the equatons for response tme analyss depend on the model, we prefer to be explct about the domans n our model. As mentoned above, our schedulng model s based on a parttonng of the tmelne n a set of non-empty, rght sem-open ntervals. Alternatvely, the schedulng model n Lu (2000) s based on left sem-open ntervals.

4 66 Real-Tme Syst (2009) 42: In ths paper, we assume that each job (or actvaton) of a task conssts of a sequence of non-preemptable subjobs, where each subjob has a known worst-case computaton tme, and we present a novel worst-case response tme analyss to determne schedulablty of tasks under FPDS. Smlarly, George et al. (1996) assume that the worst-case computaton tme of each non-preemptve job s known, and present worst-case response tme analyss of tasks under FPNS. Conversely, Baruah (2005) determnes the largest non-preemptve chunks nto whch jobs of a task can be broken up to stll ensure feasblty under earlest deadlne frst (EDF). For worst-case response tme analyss of tasks under FPPS, arbtrary phasng, and relatve deadlnes at most equal to perods, t suffces to determne the response tme of the frst job of a task upon ts crtcal nstant. For tasks wth relatve deadlnes larger than ther respectve perods, Lehoczky (1990) ntroduced the concept of a busy perod, and showed that all jobs of a task n a busy perod need to be consdered to determne ts worst-case response tme. Hence, when the relatve deadlne of a task s larger than ts perod, the worst-case response tme of that task s not necessarly assumed for the frst job of a task when released at a crtcal nstant. Smlarly, González Harbour et al. (1991) showed that f relatve deadlnes are at most equal to perods, but prortes vary durng executon, then agan multple jobs must be consdered to determne the worst-case response tme. Intal work on pre-empton thresholds (Wang and Saksena 1999) faled to dentfy ths ssue. The resultng flaw was later corrected by Regehr (2002). Worst-case response tme analyss of tasks under EDF and relatve deadlnes at most equal to perods descrbed by Spur (1996) s also based on the concept of busy perod. 1.4 Structure Ths paper has the followng structure. Frst, n Sect. 2, we present basc real-tme schedulng models for FPPS and FPDS, n whch we assume fxed values for computaton tmes and perods. These assumptons ease the presentaton n subsequent sectons, and wll be lfted n Sect. 8. Next, worst-case analyss for FPPS s brefly recaptulated n Sect. 3. Secton 4 presents varous examples refutng the exstng worst-case response tme analyss for FPDS. The noton of actve perod s the topc of Sect. 5. We present a formal defnton of actve perod and theorems wth a recursve equaton for the length of an actve perod and an teratve procedure to determne ts value. Worst-case analyss for FPDS s addressed n Sect. 6. We present a theorem for crtcal nstant and theorems to determne the worst-case response tme of a task under FPDS, arbtrary phasng, and arbtrary relatve deadlnes. Secton 7 llustrates the worst-case response tme analyss by applyng t to some examples presented n Sect. 4. Secton 8 compares the noton of level- actve perod wth smlar defntons n the lterature, and presents pessmstc varants of the worst-case response tme analyss for FPDS. Moreover, ths secton llustrates the revsed analyss for an advanced model for FPDS and brefly dscusses the sustanablty of the analyss,.e. we show that our schedulablty analyss remans vald f we replace fxed computaton tmes by worst-case computaton tmes and fxed perods by mnmal nter-arrval tmes (or worst-case perods) of tasks n our real-tme schedulng models. The paper s concluded n Sect. 9.

5 Real-Tme Syst (2009) 42: Real-tme schedulng models Ths secton starts wth a presentaton of a basc real-tme schedulng model for FPPS. Next, that basc model s refned for FPDS. The secton s concluded wth remarks. 2.1 Basc model for FPPS We assume a sngle processor and a set T of n perodcally released, ndependent tasks τ 1,τ 2,...,τ n wth unque, fxed prortes. At any moment n tme, the processor s used to execute the hghest prorty task that has work pendng. So, when a task τ s beng executed, and a release occurs for a hgher prorty task τ j, then the executon of τ s preempted, and wll resume when the executon of τ j has ended, as well as all other releases of tasks wth a hgher prorty than τ that have taken place n the meantme. A schedule s an assgnment of the tasks to the processor. A schedule can be defned as an nteger step functon σ : R {0, 1,...,n}, where σ(t) = wth >0 means that task τ s beng executed at tme t, whle σ(t)= 0 means that the processor s dle. More specfcally, we defne σ(t) as a rght-contnuous and pece-wse constant functon,.e. σ parttons the tmelne n a set of non-empty, rght sem-open ntervals {[t j,t j+1 )} j Z. At tmes t j, the processor performs a context swtch. Fgure 1 shows an example of the executon of a set T of three perodc tasks and the correspondng value of the schedule σ(t). Each task τ s characterzed by a (release) perod T R +,acomputaton tme C R +,a(relatve) deadlne D R +, where C mn(d,t ), and a phasng ϕ R + {0}. In ths paper, we assume arbtrary deadlnes,.e. the deadlne of a task may exceed ts perod. An actvaton (or release) tme s a tme at whch a task τ becomes ready for executon. A release of a task s also termed a job. The frst job of task τ s released at tme ϕ and s referred to as job zero. The release of job k of τ therefore takes place at tme a k = ϕ + kt, k N. The(absolute) deadlne of job k of τ takes place at d k = a k + D.Thebegn tme b k and fnalzaton (or Fg. 1 An example of the executon of a set T of three ndependent perodc tasks τ 1, τ 2,andτ 3,where task τ 1 has hghest prorty, and task τ 3 has lowest prorty, and the correspondng value of σ(t)

6 68 Real-Tme Syst (2009) 42: Fg. 2 Basc model for task τ completon) tme f k of job k of τ s the tme at whch τ actually starts and ends the executon of that job, respectvely. The set of phasngs ϕ s termed the phasng ϕ of thetasksett. The actve (or response) nterval of job k of τ s defned as the tme span between the actvaton tme of that job and ts fnalzaton tme,.e. [a k,f k ).Theresponse tme R k of job k of τ s defned as the length of ts actve nterval,.e. R k = f k a k. Smlarly, the begn (or start) nterval s defned as [a k,b k ), and the start tme S k as S k = b k a k. Fgure 2 llustrates the above basc notons for an example job of task τ. The worst-case response tme WR and the worst-case start tme of a task τ are the largest response tme and the largest start tme of any of ts jobs, respectvely,.e. WR = sup R k (ϕ), (1) ϕ,k WS = sup S k (ϕ). (2) ϕ,k Note that the response tme R k and the start tme S k have been parameterzed n these equatons to denote ther dependency on the phasng ϕ. In many cases, we are not nterested n the worst-case response tme of a task for a partcular computaton tme, but n the value as a functon of the computaton tme C R +. We wll therefore use a functonal notaton when needed, e.g. WR (C). A crtcal nstant of a task s defned to be an (hypothetcal) nstant that leads to the worst-case response tme for that task. Typcally, such an nstant s descrbed as a pont n tme wth partcular propertes. As an example, a crtcal nstant for tasks under FPPS s gven by a pont n tme for whch all tasks have a smultaneous release. We assume that we do not have control over the phasng ϕ, for nstance snce the tasks are released by external events, so we assume that any arbtrary phasng may occur. Ths assumpton s common n real-tme schedulng lterature (Joseph and Pandya 1986; Klen et al. 1993; Lu and Layland 1973). We also adopt other standard basc assumptons (Lu and Layland 1973),.e. tasks are ready to run at the start of each perod and do no suspend themselves, tasks wll be preempted nstantaneously when a hgher prorty task becomes ready to run, a job of task τ does not start before ts prevous job s completed, and the overhead of context swtchng and task schedulng s gnored. Fnally, we assume that the deadlnes are hard,.e. each job of a task must be completed at or before ts deadlne. Hence, a set T of n perodc tasks

7 Real-Tme Syst (2009) 42: can be scheduled f and only f WR D (3) for all = 1,...,n. For notatonal convenence, we assume that the tasks are gven n order of decreasng prorty,.e. task τ 1 has hghest prorty and task τ n has lowest prorty. The (processor) utlzaton factor U s the fracton of the processor tme spent on the executon of the task set (Lu and Layland 1973). The fracton of processor tme spent on executng task τ s C /T, and s termed the utlzaton factor U τ of task τ,.e. U τ = C. (4) T The cumulatve utlzaton factor U for tasks τ 1 tll τ s the fracton of processor tme spent on executng these tasks, and s gven by U = j U τ j. (5) Therefore, U s equal to the cumulatve utlzaton factor U n for n tasks: U = U n = j n U τ j = j n C j. (6) In Lu and Layland (1973), the followng necessary condton s determned for the schedulablty of a set T of n perodc tasks under any schedulng algorthm: U 1. (7) Unless explctly stated otherwse, we assume n ths document that task sets satsfy ths condton. 2.2 Refned model for FPDS For FPDS, we need to refne our basc model of Sect Each job of task τ s now assumed to consst of a sequence of m subjobs. The kth subjob of τ s characterzed by a computaton tme C k R +, where C = m k=1 C k. For convenence, we wll use the term F to denote the computaton tme C,m of the fnal subjob of τ,.e. F = C,m. (8) We assume that subjobs are non-preemptable. Hence, a task can only be preempted at subjob boundares,.e. at so-called preempton ponts. A task can therefore defer the preempton and executon of a hgher prorty task. We wll use the term blockng of a task τ to denote the tme that the executon of τ s deferred by lower prorty tasks. Note that when m = 1 for all, we have FPNS as specal case.

8 70 Real-Tme Syst (2009) 42: Concludng remarks In ths document, we wll use the superscrpt P to denote FPPS, e.g. WR P denotes the worst-case response tme of task τ under FPPS and arbtrary phasng. Smlarly, we wll use the superscrpts D and N to denote FPDS and FPNS, respectvely. In our basc model for FPPS, we ntroduced notons for ponts n tme wth a subscrpt dentfyng a task and optonally a job of that task, e.g. a k s the absolute release tme of job k of task τ. We wll need smlar notons that are expressed relatve to a partcular moment n tme, e.g. the relatve release tme of the frst job of a task at or after tme t s. We wll therefore also use relatve versons of the notons, where relatve can refer to the dentfcaton of the job and/or to a partcular moment n tme, dependng on the noton. As an example, let φ (t) denote the earlest absolute actvaton of a job of task τ at or after tme t,.e. ( ) t + ϕ φ (t) = ϕ + T. Here, the notaton x + stands for max(x, 0), whch s used to ndcate that there are no releases of τ before tme ϕ. Because ϕ 0, the term ( t ϕ T ) + s equal to the number of releases of τ n [0,t).Gvenφ (t), therelatve phasng ϕ (t) s gven by ϕ (t) = φ (t) t. The release of job k of task τ relatve to t takes place at the relatve actvaton tme a k (t) = ϕ (t) + kt, k N. Fora k (t), both the dentfcaton of the job and the tme are therefore relatve to t. Smlarly, the notons relatve begn tme b k (t) and relatve fnalzaton tme f k (t) denote a tme relatve to t and concern the job k of task τ relatve to t. Fortherelatve response tme R k (t), only the dentfcaton of the job s relatve to t. We wll use abbrevated representatons for the relatve notons usng a prme ( ) when the partcular moment n tme s clear from the context. As an example, n a context concernng a partcular moment t s,the relatve actvaton tme a k denotes a k(t s ). T 3 Recaptulaton of worst-case analyss for FPPS For the analyss under FPPS, we only consder cases where the deadlnes of tasks are less than or equal to the respectve perods. For llustraton purposes, we wll use a set T 1 of two ndependent perodc tasks τ 1 and τ 2 wth characterstcs as gven n Table 1. Fgure 3 shows an example of the executon of the tasks τ 1 and τ 2 under FPPS. Note that even an nfntesmal ncrease of the computaton tme of ether task τ 1 or τ 2 wll mmedately cause the job of task τ 2 released at tme 0 to mss ts deadlne at tme 7. Table 1 Task characterstcs of T 1 T = D C τ τ 2 7 3

9 Real-Tme Syst (2009) 42: Fg. 3 Tmelne for T 1 under FPPS wth a smultaneous release of both tasks at tme zero. The numbers to the top rght corner of the boxes denote the response tmes of the respectve releases 3.1 Worst-case response tmes Ths secton presents theorems for the noton of crtcal nstant and to determne worst-case response tmes of tasks. Although these theorems are taken from Brl (2004), most of these results were already known; see for example Audsley et al. (1991), Joseph and Pandya (1986), Lu and Layland (1973). Auxlary lemmas on whch the proofs of these theorems and theorems n subsequent sectons are based are ncluded n the Appendx. Theorem 1 (Brl 2004, Theorem 4.1) In order to have a maxmal response tme for an executon k of task τ,.e. to have f k a k = WR, we may assume wthout loss of generalty that the phasng ϕ s such that ϕ j = a k for all j<. In other words, the phasng of the tasks release tmes s such that the release of the consdered executon of τ concdes wth the smultaneous release for all hgher prorty tasks. Ths latter pont n tme s called a crtcal nstant for task τ. Gven ths theorem, we conclude that tme 0 n Fg. 3 s a crtcal nstant for both task τ 1 and τ 2. From ths fgure, we therefore derve that the worst-case response tmes of tasks τ 1 and τ 2 are 2 and 5, respectvely. The next theorems can be used to determne the worst-case response tmes analytcally. Theorem 2 (Brl 2004, Theorem 4.2) 1 The worst-case response tme WR of a task τ s gven by the smallest x R + that satsfes the followng equaton, provded that x s at most T : x = C + x C j. (9) T j< j Theorem 3 (Brl 2004, Theorem 4.3) 1 The worst-case response tme WR of task τ can be found by the followng teratve procedure: WR (0) = C, (10) 1 The theorems n Brl (2004, Chap. 4) are based on worst-case computaton tmes of tasks. Because our basc model s based on fxed computaton tmes, we replaced WC and WC j n the orgnal equaton by C and C j, respectvely. We consder worst-case computaton tmes n Sect. 8.5.

10 72 Real-Tme Syst (2009) 42: WR (l+1) = C + j< (l) WR C j, l = 0, 1,... (11) The procedure s stopped when the same value s found for two successve teratons of l, or when the deadlne D s exceeded. 3.2 Worst-case occuped tmes In Fg. 3, taskτ 2 s preempted at tme 15 due to a release of task τ 1, and resumes ts executon at tme 17. The span of tme from a task τ s release tll the moment n tme that τ can start ts executon or resume ts executon after completon of a computaton tme C s termed occuped tme.theworst-case occuped tme (WO)of ataskτ s the largest occuped tme of any of ts jobs. In Brl (2004), t has been shown that the worst-case occuped tme can be descrbed n terms of the worst-case response tme by takng a lmt from above, 2.e. WO (C) = lm x C WR (x). (12) Consderng Fg. 3, we derve that worst-case occuped tmes WO 2 (0) and WO 2 (3) of task τ 2 are equal to 2 and 7, respectvely. Note that the worst-case occuped tme WO 2 (0) of task τ 2 s the longest possble span of tme from the release of τ 2 tll the moment n tme that τ 2 can start ts executon,.e. WO 2 (0) of τ 2 s the worstcase start tme WS 2 of τ 2. Hence, unlke worst-case response tmes, worst-case occuped tmes are also defned for a computaton tme equal to zero. Further note that WR 2 (C 2 ) = 5 < WO 2 (C 2 ) = 7. The next theorems can be used to determne the worst-case occuped tmes analytcally. Theorem 4 (Brl 2004, Theorem 4.4) 1 When the smallest postve soluton of (9) for a computaton tme C s at most D, the worst-case occuped tme WO of a task τ for a computaton tme C [0,C ] s gven by the smallest non-negatve x R that satsfes x = C + ( ) x + 1 C j. (13) T j< j Theorem 5 (Brl 2004, Theorem 4.5) 1 The worst-case occuped tme WO of task τ can be found by the followng teratve procedure. WO (0) = { j< C j for C = 0, WR for C > 0; (14) 2 A lmt taken from above s sometmes also denoted as lm x C + ; see Wessten (2003). When a functon f(x)s defned for a value x = x 0 and the lmt from above (or rght-hand lmt) of f(x)n x 0 s equal to f(x 0 ), f s termed rght (-hand) contnuous at x = x 0.

11 Real-Tme Syst (2009) 42: WO (l+1) = C + j< ( (l) ) WO + 1 C j, l = 0, 1,... (15) The procedure s stopped when the same value s found for two successve teratons of l. 3.3 Concludng remarks The proof of Theorem 4 derves (13) by startng from (12) and subsequently usng Lemma 16. Smlarlyto(12), we can express WR n terms of WO by takng a lmt from below, 3.e. WR (C) = lm WO (x). (16) x C The next two equatons express that WR (C) and WO (C) are left-contnuous and rght-contnuous, respectvely. WR (C) = lm WR (x), x C (17) WO (C) = lm WO (x). x C (18) Lemmas related to these latter three equatons can be found n the Appendx. 4 Exstng response tme analyss for FPDS refuted We frst recaptulate exstng response tme analyss under FPDS. Next, we show that the exstng analyss s pessmstc. We subsequently gve examples refutng the analyss,.e. examples that show that the exstng analyss s optmstc. 4.1 Recaptulaton of exstng worst-case response tme analyss for FPDS In ths secton, we recaptulate exstng worst-case response tme analyss for FPDS wth arbtrary phasng and deadlnes wthn perods as descrbed n Burns (1994), Burns and Wellngs (1997). We nclude a recaptulaton of the analyss for FPNS as presented n Tndell et al. (1994). The man reason for ncludng the latter s that t looks dfferent from the analyss for FPDS and s a bass for the analyss of controller area network (CAN). 3 A lmt taken from below s sometmes also denoted as lm x C ; see Wessten (2003). When a functon f(x)s defned for a value x = x 0 and the lmt from below (or left-hand lmt) of f(x)n x 0 s equal to f(x 0 ), f s termed left (-hand) contnuous at x = x 0.

12 74 Real-Tme Syst (2009) 42: Exstng analyss for FPDS The non-preemptve nature of subjobs may cause blockng of a task by at most one lower prorty task under FPDS. Moreover, a task can be blocked by at most one subjob of a lower prorty task. The maxmum blockng B D R + {0} of task τ by a lower prorty task s therefore equal to the longest computaton tme of any subjob of a task wth a prorty lower than task τ. Ths blockng tme s gven by { B D max j> max 1 k mj C j,k for <n, = (19) 0 for = n. Strctly spoken, ths blockng tme s a supremum (and not a maxmum) for all tasks, except for the lowest prorty task,.e. that value cannot be assumed for <n. The worst-case response tme WR D of a task τ under FPDS, arbtrary phasng, and deadlnes less than or equal to perods, as presented n Burns (1994) and Burns and Wellngs (1997), s gven by WR D = WR P (BD + C (F )) + (F ), (20) where WR P denotes the worst-case response tme of τ under FPPS. Accordng to Burns and Wellngs (1997), s an arbtrary small postve value needed to ensure that the fnal subjob has actually started. Hence, when task τ has consumed C (F ), the fnal subjob has (just) started Exstng analyss for FPNS In ths secton, we frst recaptulate the update of Joseph and Pandya (1986) gvenn Tndell et al. (1994) to take account of tasks beng non-preemptve. Next, we show that the update s very smlar to the analyss for FPDS as gven by (20). The non-preemptve nature of tasks may cause blockng of a task by at most one lower prorty task. The maxmum blockng B N R + {0} of task τ by a lower prorty task s equal to the longest computaton tme of a task wth a prorty lower than task τ,.e. B N = { max j> C j for <n, 0 for = n. Smlarly to B D, BN s a supremum for all tasks, except for the lowest prorty task,.e. that value cannot be assumed for <n. The worst-case response tme WR N s gven by (21) WR N = w + C, (22) where w s the smallest x R + that satsfes x = B N + x + τres C j. (23) T j< j

13 Real-Tme Syst (2009) 42: In ths latter equaton, τ res s the resoluton wth whch tme s measured. To calculate w, an teratve procedure based on recurrence relatonshps can be used. An approprate ntal value of ths procedure s w (0) = B N + j< C j. We now show that these results for FPNS are smlar to the exstng analyss for FPDS. To ths end, we substtute w = w τ res, x = x τ res, and τ res = n (22) and (23). Hence, the worst-case response tme WR N s gven by WR N = w + (C ), where w s the smallest x R + that satsfes x = B N + + j< Reusng the results for FPPS, we therefore get x C j. WR N = WR P (BN + ) + (C ). (24) Because we have F = C and B D = B N for FPNS, (24) for FPNS s smlar to (20) for FPDS. There s an aspect requrng further attenton, however. In partcular, (20)s based on an arbtrary small postve value whereas the analyss for FPNS s based on the resoluton τ res wth whch tme s measured. We wll return to ths ssue n Sect Exstng analyss s pessmstc Consder the set T 2 consstng of three tasks wth characterstcs as descrbed n Table 2. Based on (20) we derve WR D 2 = WRP 2 (BD 2 + C 2 (F 2 )) + (F 2 ) = WR P 2 (2 + 3 (2 )) + (2 ) = WR P 2 (3 + ) + (2 ) = (2 ) = 9. However, the exstng analyss does not take nto account that τ can only be blocked by a subjob of a lower prorty task f that subjob starts before the smultaneous release of τ and all tasks wth a hgher prorty than τ. Ths aspect can be taken nto account n the analyss by replacng B D n (20) by(b D ) +. The notaton x + stands for max{x,0}, whch s used to ndcate that the blockng tme can not become negatve Table 2 Task characterstcs of T 2 T D C τ τ τ

14 76 Real-Tme Syst (2009) 42: Fg. 4 Tmelne for T 2 under FPDS wth a release of tasks τ 1 and τ 2 at tme t = 1anda release of task τ 3 at tme t = 1 for the lowest prorty task. The worst-case response tme of τ 2 now becomes 7, as llustrated n Fg. 4.For 0, we therefore fnd a supremum (and not a maxmum) equal to 7 for the worst-case response tme of τ 2. As a result, the exstng analyss s pessmstc. 4.3 Exstng analyss s optmstc We wll gve three examples llustratng that the exstng analyss s optmstc. For all three examples, deadlnes are equal to perods,.e. D = T. The frst secton shows an obvous example,.e. an example wth a utlzaton factor U>1. The second secton shows an example wth U<1. The thrd secton shows an example wth U = 1. For all three examples, the task set conssts of just two tasks. For such task sets, the worst-case response tme analyss under FPDS presented n Burns (1994, 2001), Burns and Wellngs (1997) and n Brl et al. (2004) s very smlar. In partcular, the worst-case response tme WR D 2 of task τ 2 s determned by lookng at the response tme of the frst job of task τ 2 upon a smultaneous release wth task τ 1. However, the worst-case response tme of task τ 2 s not assumed for the frst job for all three examples An example wth U>1 An example refutng the worst-case response tme analyss s gven n Table 3. Note that the utlzaton factor U of ths set of tasks T 3 s gven by U = > 1. Hence, the task set s not schedulable. Based on (20), we derve WR D 2 = WRP 2 (B 2 + C 2 (F 2 )) + (F 2 ) = WR P 2 ( (3 )) + (3 ) = WR P 2 (1.5 + ) + (3 ) = (3 ) = 6.5. Ths value corresponds wth the response tme of the frst job of task τ 2 upon a smultaneous release wth task τ 1, as llustrated n Fg. 5. However, the same fgure also llustrates that the second job of τ 2 msses ts deadlne. Stated n other words, the exstng worst-case response tme analyss s optmstc.

15 Real-Tme Syst (2009) 42: Fg. 5 Tmelne for T 3 under FPDS wth a smultaneous release of both tasks at tme zero Table 3 Task characterstcs of T 3 T = D C τ τ Table 4 Task characterstcs of T 4 T = D C τ τ Fg. 6 Tmelne for T 4 under FPDS wth a smultaneous release of all tasks at tme zero An example wth U<1 Another example refutng the worst-case response tme analyss s gven n Table 4. Note that the utlzaton factor U of ths set of tasks T 4 s gven by U = < 1. Hence, the task set could be schedulable. Applyng (20) yelds WR D 2 = 6.1, whch corresponds wth the response tme of the frst job of task τ 2 upon a smultaneous release wth task τ 1 ; see Fg. 6. However, the same fgure also llustrates that the second job of task τ 2 msses ts deadlne An example wth U = 1 Consder task set T 5 gven n Table 5. The utlzaton factor U of ths set of tasks s gven by U = = 1. The task set s not schedulable by FPPS, as we showed n Sect. 3 that the task set s only schedulable when C 2 s at most 3. Fgure 7 shows a tmelne wth the executons of these two tasks under FPDS wth a smultaneous

16 78 Real-Tme Syst (2009) 42: Fg. 7 Tmelne for T 5 under FPDS wth a smultaneous release of all tasks at tme zero Table 5 Task characterstcs of T 5 T = D C τ τ Table 6 Task characterstcs of T 6 T C τ τ τ Fg. 8 Tmelne for T 6 under FPNS wth a smultaneous release of all tasks at tme zero. The numbers to the top rght corner of the boxes denote the response tmes of the respectve releases release at tme zero n an nterval of length 35,.e. equal to the hyperperod of the tasks. Applyng (20) yelds WR D 2 = 6.2, whch corresponds wth the response tme of the frst job of task τ 2 n Fg. 7. However, the response tme of the 5th job of task τ 2 s equal to 7, llustratng once agan that the exstng analyss s too optmstc. Nevertheless, the task set s schedulable under FPDS for ths phasng. Now, consder task set T 6 gven n Table 6, whch s smlar to task set T 5 gven n Table 5, except for the fact that rather than havng a second subjob for task τ 2 t has ataskτ 3. Fgure 8 shows a tmelne wth the executons of these three tasks under FPNS wth a smultaneous release at tme zero n an nterval of length 35,.e. equal to the hyperperod of the tasks. Applyng (20) yelds WR D 3 = 6.2, whch corresponds to the response tme of the frst job of task τ 3 n Fg. 8. However, the response tme of

17 Real-Tme Syst (2009) 42: the 5th job of task τ 3 s equal to 7, llustratng once agan that the exstng analyss s too optmstc. Nevertheless, the task set s schedulable under FPNS for ths phasng. 4.4 Concludng remark We have shown that we cannot restrct ourselves to the response tme of the frst job of a task when determnng the worst-case response tme of that task under FPDS. The reason for ths s that the fnal subjob of a task τ can defer the executon of hgher prorty tasks, whch can potentally gve rse to hgher nterference for subsequent jobs of task τ. Ths problem can therefore arse for all tasks, except for the hghest prorty task. González Harbour et al. (1991) dentfed the same nfluence of jobs of a task for relatve deadlnes at most equal to perods n the context of FPPS of perodc tasks wth varyng executon prorty. Consderng Fg. 7, we see that every job of task τ 2 n the nterval [0, 26.8) defers the executon of a job of task τ 1. Moreover, that deferred job of task τ 1 subsequently gves rse to addtonal nterference for the next job of task τ 2. Ths stuaton ends when the job of τ 2 s started at tme t = 28,.e. the 5th job of τ 2 does not defer the executon of a job of τ 1. Vewed n a dfferent way, we may state that the actve ntervals of the jobs of tasks τ 1 and τ 2 overlap n the nterval [0, 35). Note that ths overlappng starts at tme t = 0 and ends at tme t = 35, and we therefore term ths nterval [0, 35) a level-2 actve perod. Informally, a level- actve perod s a smallest nterval that only contans entre actve ntervals of jobs of task τ and jobs of tasks wth a hgher prorty than task τ. Hence, the actve nterval of every job of a task τ s contaned n a level- actve perod. To determne the worst-case response tme of a task τ, we therefore only have to consder level- actve perods. However, as llustrated by the examples shown n ths secton and mentoned above, we cannot restrct ourselves to the response tme of the frst job of a task τ when determnng the worst-case response tme of that task under FPDS. Instead, we have to consder the response tmes of all jobs n a level- actve perod. In a subsequent secton, we wll show that t suffces to consder only the response tmes of jobs n a level- actve perod that starts at a so-called ε-crtcal nstant. 5 Actve perod Ths secton presents a formal defnton of a level- actve perod, whch s based on the noton of pendng load, and theorems to determne the length of a level- actve perod. As mentoned before, a level- actve perod may contan multple jobs of τ. We therefore also defne the noton of a level-(, k) actve perod, and present a theorem to determne the length of such a perod. Informally, a level-(, k) actve perod s a smallest nterval that only contans (k + 1) entre actve ntervals of jobs 4 of task τ and entre actve ntervals of jobs of tasks wth a hgher prorty than task τ. 4 The jobs of task τ n a level-(, k) actve perod are numbered from 0 (zero) to k, gvng rse to (k + 1) jobs.

18 80 Real-Tme Syst (2009) 42: These notons and theorems form the bass for the worst-case analyss for FPDS n the next secton. We start wth the defnton of the noton level- actve perod n Sect Next, we provde examples of level- actve perods n Sect The length of a level- actve perod s the topc of Sect We refne the noton of level- actve perod to level-(, k) actve perod n Sect. 5.4, and conclude wth a theorem to determne ts length n Sect Level- actve perod The noton of level- actve perod s defned n terms of the noton of pendng load, whch on ts turn s defned n terms of the noton of actve job Actve job and pendng load Defnton 1 Ajobk of a task τ s actve at tme t f and only f t [a k,f k ), where a k and f k are the actvaton tme and the fnalzaton tme of that job, respectvely. As descrbed n Sect. 2.1, theactve nterval of job k of task τ s defned as the tme span between the actvaton tme of that job and ts completon,.e. [a k,f k ). We now defne the noton of pendng load n terms of actve job, and derve propertes for the pendng load. Defnton 2 The pendng load P τ (t) s the amount of processng at tme t that stll needs to be performed for the actve jobs of task τ that are released before tme t,.e. where σ τ P τ (t) = ( t ϕ T ) + C t 0 σ τ (t )dt, (25) { (t) = 1 ftaskτ s beng executed at tme t,.e.σ(t)=, 0 otherwse. The term ( t ϕ T ) + C n (25) s equal to the amount of processng that needs to be performed due to releases of task τ n [0,t).Theterm t 0 σ τ (t )dt s equal to the amount of processng that has been performed for τ. The rght-hand sde of (25) s therefore equal to the amount of processng at tme t due to releases of jobs of task τ before t that stll needs to be performed. We subsequently defne the notons of (cumulatve) pendng load P (t) and (processor) pendng load P(t). Defnton 3 The (cumulatve) pendng load P (t) s the amount of processng at tme t that stll needs to be performed for the actve jobs of tasks τ j wth j that are released before tme t,.e. P (t) = Pj τ (t) = ( ) t + ϕj t C j σ (t )dt, (26) T j j j 0

19 Real-Tme Syst (2009) 42: where σ (t) = j σ τ j (t) = { 1 fσ(t) {1,...,}, 0 otherwse. Defnton 4 The (processor) pendng load P(t)s the amount of processng at tme t that stll needs to be performed for the actve jobs of all tasks that are released before tme t,.e. P(t)= P n (t). (27) Corollary 1 The order n whch the tasks τ j wth j are executed s mmateral for the cumulatve pendng load P. For <n, the cumulatve pendng load P depends on blockng due to a lower prorty task. As an example, let P (t s ) = 0, then P (t) = C k for all t (t s,t s ) under FPDS f the followng three condtons hold: Ataskτ k wth s s released at tme t s. No other releases of τ j for j take place n [t s,t s ). A subjob of a lower prorty task s executng at tme t s and blocks task τ k durng [t s,t s ) due to the non-preemptve nature of the subjob. Because blockng due to a lower prorty task does not play a role for the (processor) pendng load, P(t)only depends on the actvatons of tasks. Corollary 2 The (processor) pendng load P(t) s ndependent of the schedulng algorthm, provded that the algorthm s non-dlng. Note that the notons of pendng load, cumulatve pendng load, and processor pendng load are ndependent of relatve deadlnes of tasks Defnton of a level- actve perod We now defne the noton of level- actve perod n terms of the pendng load P (t). Defnton 5 Alevel- actve perod s an nterval [t s,t e ) wth the followng three propertes: 1. P (t s ) = 0; 2. P (t e ) = 0; 3. P (t) > 0 for all t (t s,t e ). Let the blockng tme B (t s ) of a level- actve perod that starts at tme t s be defned as the length of the (optonally empty) ntal nterval durng whch the tasks τ j wth j are blocked by a subjob of a task wth a lower prorty. Note that B n (t s ) = 0 and 0 B (t s )<B D for <n.

20 82 Real-Tme Syst (2009) 42: Lemma 1 If a level- actve perod starts at tme t s and ends at tme t e, then the followng propertes hold: () Tasks τ j wth j are contnuously executng n [t s,t e ), except for an (optonally empty) ntal nterval [t s,t s + B (t s )) durng whch the tasks are blocked by a lower prorty task. () The length L (t s ) of that level- actve perod s at least B (t s ) + C k, where a task τ k wth k s released at tme t s. () The order n whch the tasks τ j wth j are executed s mmateral for the length L (t s ). Proof () Ths property follows mmedately from the non-preemptve nature of subjobs and the assumptons for fxed-prorty schedulng. () By defnton, P (t s ) = 0. Because the tasks τ j wth j are blocked n the (optonally empty) ntal nterval [t s,t s + B (t s )), and the level- actve perod contans at least the actve nterval of task τ k, the length L (t s ) of that level- actve perod s at least B (t s ) + C k. () Ths property follows mmedately from the defnton of a level- actve perod and Corollary 1. From ths defnton of the level- actve perod n terms of the pendng load P (t), we draw the followng concluson. Corollary 3 The level-n actve perod s ndependent of the schedulng algorthm, provded that the algorthm s non-dlng. Note that the noton of level- actve perod s ndependent of relatve deadlnes of tasks. Further note that a level- actve perod may, but need not, contan actvatons of task τ. In the sequel, we wll call a level- actve perod that contans an actvaton of task τ a proper level- actve perod. Smlarly, we call a level- actve perod that does not contan an actvaton of τ an mproper level- actve perod. Unless explctly stated otherwse, we use the phrase level- actve perod to denote a proper level- actve perod n the remander of ths document. 5.2 Examples We wll now consder two examples, one for FPPS based on the tmelne n Fg. 3 for T 1 and one for FPDS based on the tmelne n Fg. 7 for T 5. Consder Fg. 9, wth a tmelne for T 1 under FPPS, pendng loads P 1 (t), P2 τ (t), and P 2 (t), and level- actve perods. Note that P 1 (t) s equal to P1 τ (t) by defnton. From the graph for P 1 (t), we fnd that the nterval [0, 35) contans seven level-1 actve perods, correspondng wth the seven actvatons of task τ 1,.e.[0, 5), [5, 7), [10, 12), [15, 17), [20, 22), [25, 27), and [30, 32). The horzontal lne fragments n the graph for P2 τ (t) are caused by the fact that τ 2 s preempted by a job of task τ 1. From the graph for the pendng load P 2 (t), we fnd that the nterval [0, 35) contans eght level-2 actve perods,.e. [0, 5), [5, 7), [7, 10), [10, 12), [14, 19), [20, 25), [25, 27), and [28, 33). From these eght level-2 actve perods, [0, 5), [7, 10),

21 Real-Tme Syst (2009) 42: Fg. 9 Tmelne for T 1 under FPPS, pendng loads P 1 (t), P τ 2 (t), andp 2(t), and level- actve perods and level- busy perods. From the eght level-2 actve perods n the nterval [0, 35), fve are proper,.e. [0, 5), [7, 10), [14, 19), [20, 25), and[28, 33) contan actvatons of task τ 2. The other three are mproper,.e. [5, 7), [10, 12), and[25, 27) [14, 19), [20, 25), and [28, 33) are proper,.e. contan actvatons of task τ 2, and [5, 7), [10, 12), and [25, 27) are mproper. As mentoned before, the level-2 actve perod only depends on the actvatons of τ 1 and τ 2, and s ndependent of the schedulng algorthm. Consder Fg. 10, wth a tmelne for T 5 under FPDS, pendng loads P 1 (t), P τ 2 (t), and P 2 (t), and level- actve perods. From the graph for P 1 (t), we fnd that the nterval [0, 35) contans seven level-1 actve perods, correspondng wth the seven actvatons of task τ 1,.e.[0, 2), [5, 8.2), [10, 14.4), [15, 17.6), [20, 22.6), [25, 28.8), and [30, 32). The horzontal lne fragments n the graph for P 1 (t) are caused by the fact that τ 1 s blocked by a subjob of task τ 2. From the graph for the pendng load P 2 (t), we fnd that the nterval [0, 35) contans a sngle level-2 actve perod,.e. [0, 35).

22 84 Real-Tme Syst (2009) 42: Fg. 10 Tmelne for T 5 under FPDS, pendng loads P 1 (t), P2 τ (t), andp 2(t), and level- actve perods and level- busy perods 5.3 Length of a level- actve perod Ths secton presents three theorems for the length of a level- actve perod. A frst theorem presents a recursve equaton for the length of a level- actve perod. A next theorem states that under the followng assumpton a level- actve perod that starts wll also end. Assumpton 1 Ether U<1 or U 1 and the least common multple (lcm) of the perods of the tasks of T exsts. Hence, the assumpton s a suffcent condton to guarantee that a level- actve perod wll end when t starts. Because we assume ϕ 0 for all n, P (0) = 0for all n. We therefore conclude that, when Assumpton 1 holds, the tmelne conssts of a sequence of level- actve perods, optonally preceded by and separated by dle-

23 Real-Tme Syst (2009) 42: perods. A fnal theorem provdes an teratve procedure to determne the length of a level- actve perod A recursve equaton Theorem 6 The length L (t s ) of a level- actve perod that starts at tme t s s found for the smallest x R + that satsfes the followng equaton ( ) x ϕj (t s ) + C j. (28) x = B (t s ) + j Proof Because the level- actve perod starts at tme t s, P (t s ) = 0 by defnton. Now assume the level- actve perod under consderaton ends at tme t e. Hence, tme t e s the smallest t larger than t s for whch P (t) = 0, and the length L (t s ) of the actve perod becomes t e t s. We now derve (28) from P (t e ) = 0: P (t e ) = {(26)} j ( ) te ϕ + j C j te 0 σ (t)dt = P (t s ) + j = {P (t s ) = 0} j = 0. ( ) te (t s + ϕ j (t s )) + C j te ( ) te (t s + ϕ j (t s )) + C j t s σ (t)dt te t s σ (t)dt From Lemma 1, we derve that the lower prorty task s executng n [t s,t s + B (t s )), and only tasks τ j wth j are executng n [t s + B (t s ), t e ). Hence, we conclude that te σ (t)dt = t e (t s + B (t s )). t s Substtutng ths result n the former equaton, we get ( ) te (t s + ϕ j (t s )) + C j, t e (t s + B (t s )) = j and by subsequently substtutng t e = x + t s, we get (28). Because tme t e s the smallest t (larger than t s )forwhchp (t) = 0, x = t e t s s the smallest value n R + that satsfes (28), whch proves the theorem End of a level- actve perod We now present a theorem whch states that there exst postve solutons for the recursve equaton (28) fassumpton1 holds. To that end, we wll use Lemma 4.3 from Brl (2004) (see Lemma 15 n the Appendx), and frst prove two lemmas.

24 86 Real-Tme Syst (2009) 42: Lemma 2 There exsts a postve soluton for the recursve equaton (28) for the length of the level- actve perod f U < 1. Proof We wll prove that the condton U < 1 s suffcent by means of Lemma 4.3 of Brl (2004). Let f be defned as ( ) x ϕj (t s ) + C j. f(x)= B (t s ) + j We choose a = mn l C l 2, hence ( f(a)= f mn l ) C l = B (t s ) + ( C mnl l 2 j 2 ϕ j (t s ) ) + C j. By defnton, there s at least one task that s released at the start of the level- actve perod. Let task τ k wth k be released at tme t s,.e.ϕ k (t s ) = 0. We now get C mnl l 2 C l f(a) B (t s ) + C k = B (t s ) + C k > mn T k l 2 = a, hence f(a)>a. In order to choose an approprate b, we make the followng dervaton. f(x) B (t s ) + j x As U < 1, the relaton C j <B (t s ) + ( ) x + 1 C j = B (t s ) + xu + C j. T j j j x B (t s ) + xu + j C j holds for We now choose x B (t s ) + j C j 1 U. b = B (t s ) + j C j, 1 U and therefore get b>f(b). Now the condtons for Lemma 15 hold,.e. the functon f(x) s defned and strctly non-decreasng n an nterval [a,b] wth f(a)>a and f(b)<b. Hence, there exsts an x ( mn l 2, B (t s ) + j C ) j (1 U ) C l such that x = f(x).

25 Real-Tme Syst (2009) 42: Lemma 3 There exsts a postve soluton for the recursve equaton (28) for the length of the level-n actve perod f U = 1 and the least common multple of the perods of T exsts. Proof We frst observe that B n (t s ) = 0 for the level-n actve perod,.e. the lowest prorty task s never blocked. Next, we dstngush two complementary cases, a frst case wth ϕ (t s ) = 0 for all and a second case where ths does not hold. We prove the lemma by consderng both cases separately. For the frst case, we prove that for B n (t s ) = 0 and ϕ (t s ) = 0 for all the value x = lcm(t 1,...,T n ) s a soluton of (28). For these values of B n (t s ) and ϕ (t s ),(28) smplfes to x = x C j. T j n j Because lcm(t 1,...,T n ) C j = lcm(t 1,...,T n ) C j and j n = U = 1, we mmedately see that lcm(t 1,...,T n ) s a (postve) soluton. For the second case, we prove that the condton U = 1 and the least common multple of the perods of T exsts s suffcent by means of Lemma 15. Letf be defned as C j ( ) x ϕj (t s ) + C j. f(x)= j n We choose a = mn j n C j /2. Smlar to the proof of Lemma 2, we fnd f(a)>a.in order to choose an approprate b, we make the followng dervaton. f(x) x C j. T j n j We now consder two dsjunct cases for x = lcm(t 1,...,T n ).Iff(x)< j n T x j C j, we choose b = lcm(t 1,...,T n ), and therefore get b>f(b). Now the condtons for Lemma 15 hold,.e. the functon f(x) s defned and strctly non-decreasng n an nterval [a,b] wth f(a) > a and f(b) < b. Hence, there exsts an x C (mn j j n 2, lcm(t 1,...,T n )) such that x = f(x).iff(x)= j T x j C j, we found a (postve) soluton and we are also done. Theorem 7 If Assumpton 1 holds, a level- actve perod that s started at tme t s s guaranteed to end. Proof The theorem follows mmedately from Lemmas 2 and An teratve procedure The next theorem provdes an teratve procedure to determne the length of a level- actve perod.

26 88 Real-Tme Syst (2009) 42: Theorem 8 Let the level- actve perod start wth a release of a task τ k at tme t s. If Assumpton 1 holds, the length L (t s ) of that level- actve perod can be found by the followng teratve procedure: L (0) (t s ) = B (t s ) + C k, (29) L (l+1) (t s ) = B (t s ) + ( (l) ) L (t s ) ϕ j (t s ) + C j, l = 0, 1,... (30) T j j Proof From Lemmas 2 and 3, we know that there exsts a postve soluton of (28) when Assumpton 1 holds. To prove the theorem, we frst prove that the sequence s non-decreasng. Next, we prove that the procedure stops when the length L (t s ) s reached,.e. for the smallest soluton of (28). To that end, we show that all values n the sequence L (l) (t s ) are lower bounds on L (t s ). To show that the procedure termnates, we show that the sequence can only take a fnte number of values to reach that soluton. We prove that the sequence s non-decreasng, by nducton. To ths end,we start by notng that L (0) (t s ) = B (t s ) + C k > 0, and Next, f L (l+1) L (l+1) L (1) (t s ) = B (t s ) + ( (0) ) L (t s ) ϕ j (t s ) + C j T j j {ϕ s (t s ) = 0}B (t s ) + C k = L (0) (t s ). (t s ) L (l) (t s ), then we can conclude from (30) that also L (l+2) (t s ) (t s ), as fllng n a hgher value n the rght-hand sde of (30) gves a hgher or equal result. We next prove L (l) tem () we know L (0) on L (t s ), then (t s ) L (t s ), for all l = 0, 1,...,by nducton. From Lemma 1 (t s ) = B (t s ) + C k L (t s ). Next, f L (l) (t s ) s a lower bound ( (l) ) L (t s ) ϕ j (t s ) + C j j s a lower bound on the amount of processng that needs to be performed due to releases of task τ and ts hgher prorty tasks n the nterval of length L (l) (t s ), and hence L (l+1) (t s ) s also a lower bound on L (t s ). Fnally, we prove that the sequence can only take on a fnte number of values. To ths end, we note that L (l) (t s ) s bounded from below by B (t s ) + C k and from above by the soluton. Ths means that each factor ( L(l) (t s ) ϕ j (t s ) ) + n (30) can only take on a fnte number of values. Combnng ths for all hgher prorty tasks τ j, we can conclude that the rght-hand sde of (30) can only take on a fnte number of values.

27 Real-Tme Syst (2009) 42: Level-(, k) actve perod Smlar to a level- actve perod, a level-(, k) actve perod s defned n terms of the noton pendng load. For the defnton of a level-(, k) actve perod, we frst need to refne the noton of pendng load. We assume n ths secton that Assumpton 1 holds A refnement of pendng load Let a level- actve perod start at tme t s. As descrbed above, the length of that perod s gven by the smallest x>0satsfyng (28). Let the length of that perod be L (t s ). The number of jobs l (t s ) of task τ n that perod s now gven by L (t s ) ϕ (t s ) l (t s ) =. (31) We now refne our noton of pendng load P (t) by consderng only the frst k + 1 l (t s ) jobs of τ n the actve perod, where k N. Defnton 6 The pendng load P k (t) n a level- actve perod that started at tme t s <t and ends at tme t e t s the amount of processng at tme t that stll needs to be performed for at most the frst k + 1 l (t s ) jobs of τ and the jobs of tasks τ j wth j<that are released n [t s,t),.e. (( ) t (ts + ϕ (t s )) + P k (t) = mn,k+ 1) C T T + ( ) t (ts + ϕ j (t s )) + t C j σ (t )dt, (32) T j< j t s wth σ (t) as defned n Defnton 3. At the start of a level- actve perod and outsde level- actve perods, P k (t) s equal to zero Defnton of a level-(, k) actve perod Smlarly, we refne our noton of level- actve perod to level-(, k) actve perod. Defnton 7 Alevel-(, k) actve perod s an nterval [t s,t e ) wth the followng three propertes: 1. P k (t s ) = 0; 2. P k (t e ) = 0; 3. P k (t) > 0fort (t s,t e ) Length of a level-(, k) actve perod Theorem 9 Let the number of jobs of task τ nalevel- actve perod that starts at tme t s be gven by l (t s ). The length L k (t s ) of that level-(, k) actve perod wth

28 90 Real-Tme Syst (2009) 42: k<l (t s ) s the smallest x R + satsfes the followng equaton ( ) x ϕj (t s ) + C j. (33) x = B (t s ) + (k + 1)C + j< Proof The proof s smlar to the proof of Theorem 6. 6 Worst-case analyss for FPDS Ths secton provdes theorems for the notons of crtcal nstant and worst-case response tmes for tasks under FPDS, arbtrary phasng, and arbtrary relatve deadlnes, and theorems to determne the worst-case response tmes analytcally. We assume n ths secton that Assumpton 1 holds. Moreover, we consder an arbtrary level- actve perod wth a start at tme t s. As descrbed n Sect. 2.3, we wll use abbrevated representatons for the relatve notons usng a prme ( ) to denote the value of these notons relatve to tme t s,e.g.weusea k to denote a k(t s ). 6.1 A crtcal nstant Smlar to (1), the worst-case response tme WR D largest response tme under arbtrary phasng,.e. of a task τ under FPDS s the WR D = sup R k (ϕ). ϕ,k We can refne ths equaton by takng blockng of tasks and the noton of level- actve perod nto account. In partcular, we observe that all actve ntervals of jobs of task τ are contaned n level- actve perods. Assumng the start of an arbtrary level- actve perod at tme t s, the worst-case response tme WR D of task τ can therefore be descrbed as WR D = sup B,ϕ 1,...,ϕ max R 0 k<l (B,ϕ 1,...,ϕ ) k (B,ϕ 1,...,ϕ ), (34) where l s the number of jobs of task τ n that level- actve perod. We wll now frst present a lemma to determne the response tme of job k of task τ nalevel- actve perod. We subsequently present a theorem whch states that gven an nfntesmal tme ε>0, the maxmum response tme of task τ sassumedna level- actve perod whch starts at an ε-crtcal nstant. A next theorem refnes (34). Lemma 4 The response tme R k of job k of task τ n a level- actve perod that starts at tme t s wth 0 k<l and l the number of jobs of task τ n that level- actve perod s gven by R k (B,ϕ 1,...,ϕ ) = b k,m (B,ϕ 1,...,ϕ 1 ) + F (kt + ϕ ), (35)

29 Real-Tme Syst (2009) 42: where b k,m s the relatve begn tme of the fnal subjob of job k, gven by the smallest non-negatve x R satsfyng x = B + (k + 1)C F + j< ( x ϕ j + 1) + C j. (36) Proof We frst look at the relatve begn tme b k,m of the fnal subjob of that job k, and subsequently descrbe R k n terms of the relatve begn tme, the relatve actvaton tme a k and the computaton tme F of that fnal subjob. The fnal subjob of job k of task τ n the level- actve perod that starts at tme t s can begn at tme t s + b k,m when: The blockng subjob of the lower prorty task has executed B. All hgher prorty tasks that are released n [t s,t s + b k,m ] have a completon n that nterval. All earler jobs of task τ and all earler subjobs of job k that are released n [t s,t s + b k,m ] have a completon n that nterval. Note that the order n whch the subjobs n the nterval [t s,t s + b k,m ] are executed s rrelevant for the begn tme of the fnal subjob of job k of task τ. Stated n other words, the fnal subjob of job k of task τ can start for the smallest t s + x t s + max(b,a k ) for whch lm t t s +x P k (t) = F. We now derve ( (( t (ts lm P + ϕ k(t) = {(32)} lm mn ) ) +,k+ 1) C t t s +x t t s +x T + ( t (ts + ϕ j ) ) + ) t C j σ (t )dt T j< j t s (( = mn lm t t s +x + ( lm t t s +x j< t (ts + ϕ ) T ) +,k+ 1) C t (ts + ϕ j ) ) + ts +x C j σ (t )dt t s (( x ϕ = {Lemma 16} mn T + ) + 1),k+ 1 C + ( x ϕ + ts +x j + 1) C j σ (t )dt T j< j t s = {x max(b,ϕ + k T )} (k + 1) C + ( x ϕ ) + j +1 C j (x B T ) j< j = F.

30 92 Real-Tme Syst (2009) 42: The relatve begn tme b k,m (B,ϕ 1,...,ϕ 1 ) s therefore the smallest non-negatve x R satsfyng the followng equaton: x = B + (k + 1)C F + j< ( x ϕ j + 1) + C j. The relatve completon tme f k of job k of τ s now gven by the relatve begn tme b k,m plus the computaton tme F,.e.f k = b k,m + F. The response tme R k of the job k s gven by the relatve completon tme f k mnus the relatve actvaton tme a k,.e. R k (B,ϕ 1,...,ϕ ) = b k,m (B,ϕ 1,...,ϕ 1 ) + F (kt + ϕ ). Theorem 10 Gven an nfntesmal tme ε>0, the maxmum response tme of task τ under FPDS and arbtrary phasng s assumed when the level- actve perod s started at an ε-crtcal nstant,.e. when τ has a smultaneous release wth all hgher prorty tasks and a subjob of the lower prorty tasks wth computaton tme B D starts a tme ε before that smultaneous release. Proof Let R (B,ϕ 1,...,ϕ ) denote max 0 k<l (B,ϕ 1,...,ϕ ) R k (B,ϕ 1,...,ϕ ).We wll prove that R (B,ϕ 1,...,ϕ ) assumes a maxmum for ϕ j = 0 wth j and B = (BD ε) +. Hence, the maxmum s assumed when τ has a smultaneous release wth all hgher prorty tasks, and a subjob of a lower prorty task wth computaton tme B D starts an nfntesmal tme ε>0 before that smultaneous release, whch proves the theorem. Based on Theorem 7,.e. termnaton of a level- actve perod under Assumpton 1, we conclude that: Only a fnte number of jobs need to be consdered to determne the worst-case response tme of task τ. Every job of τ nalevel- actve perod has a fnte response tme. We wll now look at the value of the length L of the level- actve perod, the number l of jobs of task τ n the level- actve perod, and the response tme R k as a functon of the relatve phasng ϕ j wth j and the blockng tme B. Consder (28) forthe length L of a level- actve perod. The term x ϕ j n that equaton s a strctly non-ncreasng functon of ϕ j wth j. Because ϕ j 0, a maxmum of that term s assumed for ϕ j = 0. Moreover, the rght-hand sde of (28) s a strctly ncreasng functon of B, and the length L s therefore also a strctly ncreasng functon of B. The largest value of L s found for the largest value of B under consderaton,.e. for B = (BD ε) +. As a consequence, L assumes a maxmum for ϕ j = 0 for all j and B = (BD ε) +. Gven the behavor of L and (31), we conclude that the number of jobs l of task τ n the level- actve perod s a strctly non-ncreasng functon of ϕ j wth j and a strctly non-decreasng functon of B. As a consequence, l assumes a maxmum for ϕ j = 0 for all j and B = (BD ε) +.

31 Real-Tme Syst (2009) 42: From (35), we conclude that R k (B,ϕ 1,...,ϕ ) s a strctly decreasng functon of ϕ. Because ϕ 0, a maxmum s assumed for ϕ = 0. Now consder (36) for the relatve begn tme b k,m.theterm x ϕ j n that equaton s a strctly nonncreasng functon of ϕ j. Smlarly to ϕ, ϕ j 0, a maxmum of that term s therefore assumed for ϕ j = 0. Hence, b k,m (B, 0,...,0) domnates b k,m (B,ϕ 1,...,ϕ 1 ) for all values of B and all values of ϕ j wth j<. Moreover, the rght-hand sde of (36) s a strctly ncreasng functon of B, and b k,m (B, 0,...,0) s therefore also a strctly ncreasng functon of B. The largest value of b k,m (B, 0,...,0) s found for the largest value of B under consderaton,.e. for B = (BD ε) +.Asa consequence, R k (B,ϕ 1,...,ϕ ) also assumes a maxmum for ϕ j = 0 for all j and B = (BD ε) +. From the values of L, l and R k as a functon of the relatve phasng ϕ j wth j and the blockng tme B, we conclude that R (B,ϕ 1,...,ϕ ) s a strctly nonncreasng functon of ϕ 1,...,ϕ 1, a strctly decreasng functon of ϕ, and a strctly ncreasng functon of B. As a result, R (B,ϕ 1,...,ϕ ) assumes a maxmum for ϕ j = 0 wth j and B = (BD ε) +, whch proves the theorem. of task τ under FPDS and arb- Theorem 11 The worst-case response tme WR D trary phasng s gven by WR D = lm max ε 0 0 k<l ((BD ε)+,0,...,0) R k(( B D ε ) +, 0,...,0 ). (37) Proof Once agan, let R (B,ϕ 1,...,ϕ ) denote max R 0 k<l k ((BD ε) +, 0,...,0). ((BD ε)+,0,...,0) From the proof of Theorem 10, we derve that R (B, 0,...,0) domnates R (B,ϕ 1,...,ϕ ) for all values of B and all values of ϕ j wth j,.e. WR D = sup B,ϕ 1,...,ϕ R (B,ϕ 1,...,ϕ ) = sup R B (B, 0,...,0). Moreover, R (B,ϕ 1,...,ϕ ) s a strctly ncreasng,.e. monotonc, functon of B. Hence, WR D = sup R (( B (B, 0,...,0) = lm ε 0 R B D ε ) + ), 0,...,0, whch proves the theorem. In the sequel, we wll omt tralng zeros n the parameter lst, e.g. we wrte R k ((BD ε) + ) when ϕ j = 0forj. From the prevous two theorems, we draw the followng conclusons.

32 94 Real-Tme Syst (2009) 42: Corollary 4 The worst-case response tme WR D s a supremum (and not a maxmum) for all tasks, except for the lowest prorty task,.e. that value cannot be assumed for <n. Corollary 5 A crtcal nstant s a supremum for all tasks, except for the lowest prorty task,.e. that nstant cannot be assumed for <n. 6.2 Worst-case response tmes The next theorem descrbes WR D n terms of the worst-case response tme WR P and worst-case occuped tme WO P under FPPS. Frst, we present defntons and prove three lemmas for the worst-case length WL D of a level- actve perod, the maxmum number wl D jobs of task τ nalevel- actve perod, and the worst-case response tme WR D k of job k of task τ. Defnton 8 The worst-case length WL D of level- actve perod under FPDS s the largest length of that perod under arbtrary phasng,.e. WL D = sup L (B,ϕ 1,...,ϕ ). (38) B,ϕ 1,...,ϕ Defnton 9 The worst-case number wl D of jobs of task τ n a level- actve perod under FPDS s the largest number n that perod under arbtrary phasng,.e. wl D = sup l (B,ϕ 1,...,ϕ ). (39) B,ϕ 1,...,ϕ Defnton 10 The worst-case response tme WR D k of job k of task τ, wth 1 k< wl D, under FPDS s the largest response tme of job k of τ under arbtrary phasng,.e. WR D k = sup B,ϕ 1,...,ϕ R k (B,ϕ 1,...,ϕ ). (40) Lemma 5 The worst-case length WL D of a level- actve perod wth n under FPDS s gven by the smallest x R + that satsfes the followng equaton x = B D + j x C j. (41) Proof The term x ϕ j n (28) s a strctly non-ncreasng functon of ϕ j wth j. Because ϕ j 0, a maxmum of that term s assumed for ϕ j = 0. Now let L (B ) denote the length of a level- actve perod wth n for a smultaneous release of task τ wth all tasks wth a hgher prorty. Hence, L (B ) s the smallest x R+

33 Real-Tme Syst (2009) 42: satsfyng (28) wth ϕ j = 0,.e. the smallest x R+ satsfyng x = B + j x C j. (42) We wll now consder the cases <nand = n separately. { = n} The lowest prorty task s never blocked, therefore Bn D = 0, and we mmedately get (41) by substtutng B = 0n(42) for = n. { <n} The rght-hand sde of (42) s a strctly ncreasng functon of B, and L (B ) s therefore also a strctly ncreasng functon of B. The largest value for L (B ) s found for the largest value of B <BD. Hence, WLD s gven by WL D = lm B BD L (B ). (43) Gven Lemma 17, we can make the followng dervaton startng from ths equaton: ( WL D = {(42)} lm B B + L (B ) ) C j BD T j j = B D + L lm (B ) C j B j BD T j = {Lemma 17} B D + L lm (B ) C j B j BD T j = {(43)} B D + WL D C j. T j j Hence, the worst-case length WL D proves the lemma. s the smallest x R + satsfyng (41), whch Because B D s a supremum (and not a maxmum) for all tasks, except for the lowest prorty task, we draw the followng concluson from the prevous lemma. Corollary 6 The worst-case length WL D s a supremum (and not a maxmum) for all tasks, except for the lowest prorty task,.e. that value cannot be assumed for <n. Lemma 6 The maxmum number wl D of jobs of task τ n a level- actve perod wth n under FPDS s gven by WL wl D D =. (44) T Proof We frst derve (44) and subsequently prove that wl D s a maxmum.

34 96 Real-Tme Syst (2009) 42: As descrbed n the proof of Theorem 10, l (B ) s a strctly non-decreasng functon of the blockng tme B. Because BD s a supremum that cannot be assumed, the largest value for l (B ) s therefore found for the largest value of B <BD. Hence, wl D s gven by wl D = lm B BD l (B ). (45) Because L (B ) T s a strctly ncreasng functon of B, we can use Lemma 17 n the followng dervaton lm B BD l (B ) = lm B BD L (B ) T = {Lemma 17} = {(43)} WL D T lm B BD. L (B ) Equaton (44) mmedately follows from (45) and ths latter equaton. The proof that wl D s a maxmum conssts of two steps. We frst prove that l (B ) s left-contnuous n B D,.e. l (BD ) = lm B BD T l (B ), (46) and subsequently prove that l (B ) s constant n an nterval (BD suffcently small γ R +,.e. γ,b D ] for a B l (B ) = wld. B D γ<b BD To prove that l (B ) s left-contnuous n BD, we show that L (BD ) s defned and equal to WL D, and subsequently show that l (BD ) = wld. From Theorem 7, we know that L (B ) exsts f Assumpton 1 holds. Moreover, consderng Theorem 6 and Lemma 5, we conclude that WL D and L (BD ) are solutons of the same equaton,.e. L (BD ) = WLD. As a result, we get L l (BD ) = (B D) WL D = T T = wl D. To prove that l (B ) s constant n an nterval (BD γ R +, we use the defnton of a lmt: γ,b D ] for a suffcently small lm x X f(x)= Y ε>0 δ>0 x (X δ,x) f(x) Y <ε.

35 Real-Tme Syst (2009) 42: Because l (B ) s strctly non-decreasng and defned n BD,wehave 0 B BD l (B ) wld. Let ε (0, 1]. Now there exsts a δ (0,B D) such that 0 wld l (B )<ε 1for all B (BD δ,b D], hence wld l (B )>wld 1. Because wl D,l (B ) N, ths completes the proof. Note that unlke WL D, the value for wld can be assumed. Based on Lemma 6, we conclude that l ((BD γ) + ) = wl D for a suffcently small γ R +, and we can therefore exchange the order of the operators n (37),.e. Hence, WR D k s gven by WR D = max lm R 0 k<wl D k ((BD ε) + ). (47) ε 0 WR D k = lm ε 0 R k ((BD ε) + ). (48) Lemma 7 The worst-case response tme WR D k of job k wth 0 k<wld of a task τ under FPDS and arbtrary phasng s gven by { WR D k = WR P (BD + (k + 1)C F ) + F kt for <n, WO P n ((k + 1)C (49) n F n ) + F n kt n for = n, where WR P (BD + (k + 1)C F ) and WO P (BD + (k + 1)C F ) are the worstcase response tme and the worst-case occuped tme under FPPS of a task τ wth a computaton tme C = BD + (k + 1)C F, a perod T = kt + D F and a deadlne D = T. Proof Startng from (48), we derve WR D k = lm ε 0 R k ((BD ε) + ) = {(35)} lm(b k,m ε 0 ((B D ε) + ) + F kt ) = lm b k,m ε 0 ((B D ε) + ) + F kt, where b k,m ((B D ε) + ) denotes the relatve begn tme of the fnal subjob of job k of task τ wth 0 k<wl and ϕ j = 0forj as gven n (36). Hence, b k,m ((B D ε) + ) s the smallest x R + satsfyng x = ((B D ε) + ) + (k + 1)C F + ( ) x + 1 C j. T j< j

36 98 Real-Tme Syst (2009) 42: Now let task set T be dentcal to T except for the characterstcs of task τ,.e.τ has characterstcs C = (BD ε) + + (k + 1)C F, T = kt + D F, and D = T. Hence, task τ of T msses ts deadlne under FPPS and arbtrary phasng f and only f job k of task τ of T msses ts deadlne under FPDS, and arbtrary phasng and an amount of blockng (B D ε) +. Based on Theorem 4, we can now wrte ( (B b k,m D ε ) + ) ( (B = WO P D ε ) ) + + (k + 1)C F. For = n, we substtute Bn D = 0, and mmedately arrve at (49), whch proves the lemma for = n. For <n, we derve ( (B WR D k = lm ε 0 WOP D ε ) ) + + (k + 1)C F + F kt = {(16)} WR P ( ) B D + (k + 1)C F + F kt, whch proves the lemma for <n. Note that because the lowest prorty task s the only task that cannot be blocked,.e. Bn D = 0, the worst-case response tme analyss for FPDS s not unform for all tasks. Moreover, note that WR D k s a supremum (and not a maxmum) for all tasks, except for the lowest prorty task,.e. that value cannot be assumed for <n. Theorem 12 The worst-case response tme WR D of a task τ under FPDS and arbtrary phasng s gven by WR D = max WR D 0 k<wl D k. (50) Proof The theorem follows mmedately from (47) and (48), and requres Lemma An teratve procedure The next theorem provdes an teratve procedure to determne the worst-case response tme WR D for task τ under FPDS and arbtrary phasng. The procedure s stopped when the worst-case response tme WR D k of job k for task τ exceeds the deadlne D or when the level- actve perod s over. Ths latter condton s based on a property of WL D. Lemma 8 The worst-case length WL D k of a level-(, k) actve perod under FPDS s the smallest postve x R + satsfyng the followng equaton x = B D + (k + 1)C + x C j. (51) T j< j Proof The proof s smlar to the proof of Lemma 5.

37 Real-Tme Syst (2009) 42: Note that because B D s a supremum (and not a maxmum) for all tasks, except the lowest prorty task, WL D k s also supremum (and not a maxmum) for all tasks, except the lowest prorty task,.e. that value cannot be assumed for <n. Lemma 9 The worst-case length WL D k of a level-(, k) actve perod under FPDS s gven by WL D k = WRP (BD + (k + 1)C ), (52) where WR P (BD + (k + 1)C ) s the worst-case response tme under FPPS and arbtrary phasng of a task τ wth a computaton tme C = BD + (k + 1)C, a perod T = kt + D and a deadlne D = T. Proof The lemma follows from the smlarty between (9) and (51). The perod and deadlne of task τ have been chosen to be equal to the deadlne of job k + 1of task τ. Hence, when the teratve procedure to determne WR P (BD + (k + 1)C ) stops because the deadlne D s exceeded, the deadlne d,k+1 wll be exceeded as well. Lemma 10 Let k N be the smallest value for whch WR P (BD + (k + 1)C ) (k + 1)T. The worst-case length WL D of a level- actve perod s now gven by WR P (BD + (k + 1)C ). Proof To prove the lemma, we wll prove the followng equvalent relaton by means of a contradcton argument 0 k<wl D ( WL D k (k + 1)T k = wl D 1 ). We only consder k<wl D 1, because the proof for k = wl D 1 s smlar. Let WL D,k (k +1)T for 0 k<wl D 1. Usng Lemma 9, we derve WR P (BD + (k + 1)C ) (k + 1)T. Hence, task τ has a completon at or before (k + 1)T, and all hgher prorty tasks that are released n the nterval [0, WR P (BD + (k + 1)C )) have a completon n that nterval. Because task τ represents the executons of both the blockng lower prorty task as well as task τ, all executons of the correspondng jobs also have a completon n that nterval. Hence, the level- actve perod that started wth an ε-crtcal nstant ends at tme WR P (BD + (k + 1)C ). However, we now have that the length of the level- actve perod equals WL D,k, a value that s strctly smaller than WL D, whch s a contradcton. Therefore, our assumpton that WLD,k (k +1)T for 0 k<wl D 1 s wrong, whch proves the lemma. Theorem 13 The worst-case response tme WR D of a task τ can be found by the followng teratve procedure under Assumpton 1, usng (49): WR (0) = WR D,0, (53) WR (l+1) = max(wr (l), WR D,l+1 ), l = 0, 1,... (54)

38 100 Real-Tme Syst (2009) 42: The procedure s stopped when the worst-case response tme WR D k of job k of task τ exceeds the deadlne D or when the level- actve perod s over,.e. WR P (BD + (k + 1)C ) (k + 1)T. Proof Lemma 10 states that WR P (BD +(k +1)C ) (k +1)T s a proper termnaton condton to determne whether or not the level- actve perod s over before the release of job k + 1. Because of Theorem 7, the level- actve perods ends under Assumpton 1, and we therefore have to consder at most a fnte number wl D of jobs of task τ. As a result, the teratve procedure ends. We observe that the teratve procedure also stops when the deadlne D s exceeded, by the worst-case response tme WR D k of job k of τ.e. when the task set s not schedulable. Corollary 7 When Assumpton 1 holds, we can derve the schedulablty of a set of tasks T under FPDS and arbtrary phasng by checkng the schedulablty crteron WR D D usng Theorem 13. Corollary 8 To check the schedulablty crteron WR D D we do not need to determne the length WL D of the worst-case level- actve perod under FPDS frst. Instead, we can smply check whether or not the level- actve perod s over after every teraton. Fnally note that: WR D,k can be used as ntal value to calculate WRP (BD + (k + 1)C ) to determne whether or not the level- actve perod s over before the release of job k + 1. WR P (BD + (k + 1)C ) can be used as ntal value to calculate WR P (BD + (k + 2)C F ) to determne WR D,k+1. 7 Examples In ths secton, we wll llustrate the worst-case response tme analyss presented n Sect. 6 to determne the schedulablty of tasks and task sets under FPDS and arbtrary phasng of some examples of Sect. 4 usng the teratve procedure presented n Theorem Schedulablty of task τ 2 of T 2 The schedulablty of task τ 2 of task set T 2 s the topc of ths secton. The characterstcs of the tasks of T 2 can be found n Table 2 n Sect To determne the worst-case response tme WR D 2 for task τ 2, we frst derve B D 2 = 2 usng (19). Next, we determne WR (0) 2 usng Lemma 7,.e. WR (0) 2 = WR D 2,0 = WRP 2 (BD 2 + C 2 F 2 ) + F 2 = WR P 2 (3) + 2 = = 7. Because WR D 2,0 D 2 = 7 and WR P 2 (BD 2 + C 2) = WR P 2 (5) = 9 >T 2 = 7,.e. the level-2 actve perod s not over yet, we proceed wth the 2nd job.

39 Real-Tme Syst (2009) 42: For the 2nd job, we fnd WR D 2,1 = WRP 2 (BD 2 + 2C 2 F 2 ) + F 2 T 2 = WR P 2 (6) 5 = 10 5 = 5, and therefore WR (1) 2 = max(wr (0) 2, WRD 2,1 ) = max(7, 5) = 7. Now WRD 2,1 = 5 D 2 and WR P 2 (BD 2 + 2C 2) = WR P 2 (8) = 14 2T 2 = 14. Hence, we know that the level-2 actve perod s over, all jobs of task τ 2 meet ther deadlnes n that perod, and the worst-case response tme WR D 2 = Schedulablty of task τ 2 of T 4 We wll determne the schedulablty of task τ 2 of task set T 4 n ths secton. The characterstcs of the tasks of T 4 can be found n Table 4 n Sect We frst determne WR (0) 2 usng Lemma 7,.e. WR (0) 2 = WR D 2,0 = WOP 2 (C 2 F 2 ) + F 2 = WO P 2 (2) = = 6.1. Because WR D 2,0 D 2 = 7 and WR P 2 (BD 2 + C 2) = WR P 2 (4.1) = 8.1 >T 2 = 7, we proceed wth the 2nd job. For the 2nd job, we fnd WR D 2,1 = WOP 2 (2C 2 F 2 ) + F 2 T 2 = WO P 2 (6.1) 4.9 = = 7.2. Because WR D 2,1 >D 2 = 7, we conclude that task τ 2 s not schedulable. 7.3 Schedulablty of the task set T 5 In ths secton, we wll determne the schedulablty of the task set T 5. The characterstcs of the tasks of T 5 can be found n Table 5 n Sect To determne the worst-case response tme WR D 1 for task τ 1, we frst derve B D 1 = 3 usng (19). Next, we determne WR (0) 2 usng Lemma 7,.e. WR (0) 1 = WR D 1,0 = WRP 1 (BD 1 + C 1 F 1 ) + F 1 = = 5. Now WR D 1,0 = D 1 and WR D 1 (BD 1 + C 1) = 5 = T 1. Hence, we know that the level-1 actve perod s over, all jobs of task τ 1 meet ther deadlnes, and the worst-case response tme WR D 1 = 5. Next, we determne the worst-case response tme WR D 2 for task τ 2. To ths end, we frst determne WR (0) 2 usng Lemma 7,.e. WR (0) 2 = WR D 2,0 = WOP 2 (C 2 F 2 ) + F 2 = WO P 2 (1.2) + 3 = = 6.2. Because WR D 2,0 <D 2 = 7 and WR P 2 (BD 2 + C 2) = 8.2 >T 2 = 7, we proceed wth the 2nd job. For the 2nd job, we fnd WR D 2,1 = WOP 2 (2C 2 F 2 ) + F 2 T 2 = WO P 2 (5.4) 4 = = 5.4,

40 102 Real-Tme Syst (2009) 42: and therefore WR (1) 2 = max(wr (0) 2, WRD 2,1 ) = max(6.2, 5.4) = 6.2. Because WR D 2,1 <D 2 and WR P 2 (BD 2 + 2C 2) = 14.4 > 2T 2 = 14, we proceed wth the 3rd job. For the 3rd job, we fnd WR D 2,2 = WOP 2 (3C 2 F 2 ) + F 2 2T 2 = WO P 2 (9.6) 11 = = 6.6, and therefore WR (2) 2 = max(wr (1) 2, WRD 2,2 ) = max(6.2, 6.6) = 6.6. Because WR D 2,2 <D 2 and WR P 2 (BD 2 + 3C 2) = 22.6 > 3T 2 = 21, we proceed wth the 4th job. For the 4th job, we fnd WR D 2,3 = WOP 2 (4C 2 F 2 ) + F 2 3T 2 = WO P 2 (13.8) 18 = = 5.8, and therefore WR (3) 2 = max(wr (2) 2, WRD 2,3 ) = max(6.6, 5.8) = 6.6. Because WR D 2,3 <D 2 and WR P 2 (BD 2 + 4C 2) = 28.8 > 4T 2 = 28, we proceed wth the 5th job. For the 5th job, we fnd WR D 2,4 = WOP 2 (5C 2 F 2 ) + F 2 4T 2 = WO P 2 (18) 25 = = 7, and therefore WR (4) 2 = max(wr (3) 2, WRD 2,4 ) = max(6.6, 7) = 7. Now WRD 2,4 = D 2 and WR P 2 (BD 2 + 5C 2) = 35 = 5T 2. Hence we know that the level-2 actve perod s over, all jobs of task τ 2 meet ther deadlnes n that perod, and the worst-case response tme WR D 2 = 7. Because WR D D for all n, we conclude that T 5 s schedulable under FPDS and arbtrary phasng when deadlnes are equal to perods. 8 Dscusson Ths secton presents a theorem for the worst-case response tme of the hghest prorty task, compares the noton of level- actve perod wth smlar notons n the lterature, presents pessmstc varants for the worst-case response tme analyss of tasks, llustrates the revsed analyss for an advanced model for FPDS, and shows that our analyss s sustanable. 8.1 Worst-case response tme of hghest prorty task In Sect. 4.4, we concluded that the optmsm n the exstng analyss does not occur for the hghest prorty task. The next theorem provdes a formal bass for that concluson, by statng that the worst-case response tme of the hghest prorty task τ 1 can be found by only consderng the frst job of τ 1 n a level-1 actve perod started at an ε-crtcal nstant. Intutvely, a later job k of τ 1 wth k>1nalevel- actve perod can only have a longer response tme than the frst job when t s deferred by ts prevous job for a longer tme than the tme the frst job s blocked by a task wth a lower prorty than τ 1. Ths would requre that the utlzaton U τ 1 of τ 1 s larger than 1, however. Frst, we prove the followng lemma.

41 Real-Tme Syst (2009) 42: Lemma 11 The frst job of task τ 1 nalevel-1 actve perod has the largest response tme of all jobs of τ 1 n that perod. Proof The hghest prorty task τ 1 experences blockng of at most one subjob of a lower prorty task. If the frst job of τ 1 n a level-1 actve perod s blocked by an amount B, ts response tme R 1,0 (B) becomes R 1,0 (B) = B + C 1. Now, assume the level-1 actve perod contans l 1 > 1 jobs of task τ 1. The response tme R 1,k (B) of job k, wth 0 k<l 1, becomes R 1,k (B) = B + (k + 1)C 1 kt 1 = B + C 1 + k(c 1 T 1 ) = B + C 1 + k(u 1 1)T 1. Because we assume U 1, we must have U τ 1 < 1 when task τ 1 s blocked by a lower prorty task. Hence, we fnd R 1,k (B) < B + C 1 = R 1,0 (B), whch proves the lemma. Theorem 14 The worst-case response tme WR D 1 of the hghest prorty task τ 1 under FPDS s equal to WR D 1 = BD 1 + C 1. (55) Proof From equaton R 1,0 (B) = B + C 1, we conclude that R 1,0 (B) s a strctly ncreasng functon of B. Hence, we derve WR D 1 = sup R 1,0 (B) = lm (B + C 1 ) = B D B B B1 D 1 + C 1, whch proves the theorem. 8.2 A comparson wth exstng notons We wll now compare our noton of level- actve perod wth smlar notons n the lterature Level- busy perod n Lehoczky (1990) The noton of level- busy perod orgnates from Lehoczky (1990), where t has been ntroduced as an expedent to determne the worst-case response tmes of tasks for deadlnes larger than perods under FPPS and arbtrary phasng. The level- busy perod s defned as follows.

42 104 Real-Tme Syst (2009) 42: Defnton 11 Alevel- busy perod s a tme nterval [a,b] wthn whch jobs of prorty or hgher are processed throughout [a,b] but no jobs of level or hgher are processed n (a ε, a) or (b, b + ε) for suffcently small ε>0. Fgure 9 also shows the level-1 busy perods and level-2 busy perods for T 1.The level-1 busy perods n ths fgure only dffer from the level-1 actve perods by the ncluson of the end-ponts of the ntervals by the former. The dfference between level-2 busy perods and level-2 actve perods s more sgnfcant, however. Whereas the nterval [0, 12) s consttuted by four level-2 actve perods,.e. [0, 5), [5, 7), [7, 10), and [10, 12), the nterval s contaned n a sngle level-2 busy perod [0, 12]. Stated n other words, the level-2 busy perod unfes four adjacent level-2 actve perods. Smlarly, the nterval [20, 27) s consttuted by two level-2 actve perods,.e. [20, 25) and [25, 27), and the nterval s contaned n a sngle level-2 busy perod [20, 27]. Fgure 10 shows the level-1 busy perods and level-2 busy perods for T 1.Fromths fgure, we see that the level-2 busy perod never ends for U = 1, as also becomes mmedately clear from Defnton 11. Conversely, the level-2 actve perod that started at tme t = 0 ends at tme t = 35; see also Assumpton 1 and Theorem 7. We observe that the defnton of level- busy perod s ncluded n Klen et al. (1993) (on page D-4, referrng to Lehoczky (1990)), and the noton s used n Technque 5 Calculatng Response Tme wth Arbtrary Deadlnes and Blockng. As shown above, the busy perod never ends for U = 1. Notably, n Klen et al. (1993, pp. 4 35) t s only mentoned that we must be sure that the [...] utlzaton [...] s not greater than one. In Step 6 Decde f the busy perod s over the noton s used to determne whether or not the teratve procedure can be stopped. Notably, that decson s not based on the end of the busy perod, but on the end of the level- actve perod,.e. when the (worst-case) response tme WR P k of job k of task τ s less than or equal to T ;see also Theorem 13. There s another strkng dfference between the level- actve perod and the level- busy perod. A level- actve perod may start when a task wth a lower prorty s stll beng processed, as llustrated by the level-1 actve perod that starts at tme t = 5nFg.10. The correspondng level-1 busy perod does not start at tme t = 5, butattmet = 6.2 nstead. The fundamental dfference between both notons can be traced back to ther defntons; a busy perod s based on processng of jobs, whereas an actve perod s based on (pendng) load or actve jobs τ -Busy perod n González Harbour et al. (1991) In González Harbour et al. (1991), the noton of busy perod s slghtly modfed to accommodate the fact that a task τ may be composed of dstnct subtasks, each of whch may have ts own tmng requrements and fxed prorty. In the followng defnton, ρ denotes the mnmum prorty of the subtasks of task τ. Defnton 12 A τ -dle nstant s any tme t such that all work of prorty ρ or hgher started before t and all τ jobs also started before t have completed at or before t.

43 Real-Tme Syst (2009) 42: Defnton 13 A τ -busy perod s an nterval of tme [A,B] such that both A and B are τ -dle nstants and there s no tme t (A, B) such that t s a τ -dle nstant. Ths noton of τ -busy perod s smlar to our level- actve perod, wth as man dfference that a τ -busy perod s a closed nterval rather than a rght sem-open nterval. Although ths dfference may be vewed as phlosophcal, we prefer the usage of a rght sem-open nterval, whch we wll motvate by means of Fg. 10. Gven Defnton 12 and 13, tme t = 35 belongs to two τ 2 -busy perods,.e. [0, 35] and [35, 70]. Moreover, tme t = 35 s also a τ 2 -dle nstant. Hence, τ -busy perods can overlap, and when they overlap, the overlap s termed a τ -dle nstant. Ths s consdered to be counter-ntutve Level- busy perod n George et al. (1996) After a bref recaptulaton of the noton of level- busy perod of Lehoczky (1990) for FPPS, an nformal descrpton of a level- busy perod for FPNS under dscrete schedulng (Baruah et al. 1990b) s gven n George et al. (1996, Appendx A.2). Note that for dscrete schedulng, all task parameters are ntegers,.e. T, C, D Z + and ϕ Z + {0}, and preemptons are restrcted to nteger tme ponts. Unfortunately, there s an nconsstency n George et al. (1996). In Appendx A.2, the followng defnton s gven. Defnton 14 A level- busy perod s a processor busy perod n whch only nstances of tasks wth a prorty greater than or equal to that of τ execute. Accordngly, the nterval of tme that a lower prorty task blocks task τ and ts hgher prorty tasks s not ncluded n the level- busy perod n both the text of the proof of Lemma 6 n Sect and Fg. 6, whch s used to llustrate that proof. Conversely, that nterval s ncluded n the equaton to determne the length of the level- busy perod for the non-preemptve case, as descrbed n George et al. (1996, Appendx A.2). Note that George et al. (1996) does not reproduce the defnton of Lehoczky (1990) (see Defnton 11 above), but presents a new defnton. Surprsngly, the dfferences between these defntons are not dscussed. As an example, a (synchronous processor) busy perod n George et al. (1996) s descrbed as a rght sem-open nterval on page 6, whereas the level- busy perod n Lehoczky (1990) s a closed nterval. The noton of level- busy perod for FPNS n George et al. (1996) s smlar to our noton of level- actve perod under the assumpton that the equaton to determne the length of a level- busy perod for the non-preemptve case properly reflects the ntenton of the authors Level-π busy nterval n Lu (2000) In Lu (2000), an analyss method s descrbed to determne the schedulablty of tasks under FPPS whose relatve deadlnes are larger than ther respectve perods, usng

44 106 Real-Tme Syst (2009) 42: the term level-π busy nterval. A level-π busy nterval s defned as a left sem-open nterval (t 0,t],.e. the parttonng of the tmelne n Lu (2000) dffers from ours. Gven the descrpton n Lu (2000), our defnton of level- actve perod can be vewed as a slghtly modfed defnton of level-π busy nterval to accommodate our schedulng model for FPDS. 8.3 Pessmstc varants Gven (49) n Lemma 7, we observe that the worst-case response tme analyss s not unform for all tasks. The analyss can be made unform at the cost of potentally ntroducng pessmsm. Ths secton presents two lemmas wth pessmstc varants for the worst-case response tme analyss, one based on worst-case occuped tmes and one based on worst-case response tmes. For both varants, the teratve procedure presented n Theorem 13 can be used,.e. only the equatons for WR D k change, not the teratve procedure. We conclude ths secton wth a retrospect on the analyss for FPDS A unform analyss based on WO P Lemma 12 A pessmstc worst-case response tme ŴR D k of job k wth 0 k<wld of a task τ under FPDS and arbtrary phasng s gven by ŴR D k = WOP (BD + (k + 1)C F ) + F kt, (56) where WO P (BD + (k + 1)C F ) s the worst-case occuped tme under FPPS of ataskτ wth a computaton tme C = BD + (k + 1)C F, a perod T = kt + D F, and a deadlne D = T. Proof By defnton, WR P (C) WOP (C), hence WRD k ŴRD k. Because WRP 1 (C) = WO P 1 (C), ŴRD k s potentally pessmstc for 1 <<n. The pessmsm s llustrated by the set T 2 consstng of three tasks wth characterstcs as descrbed n Table 2 n Sect For the worst-case response tme ŴR D 2,0 of the frst job of task τ 2 we fnd ŴR D 2,0 = WOP 2 (BD 2 + C 2 F 2 ) + F 2 = WO P 2 ( ) + 2 = WO P 2 (3) + 2 = = 9. Because ŴR D 2,0 >D 2, T 2 s consdered unschedulable under FPDS based on Lemma 12. Conversely, applcaton of Lemma 7 yelds a value WR D 2 = 7 D 2. We observe that ŴR D 2,0 s equal to WR D 2 as determned n Sect. 4.2 by means of the exstng analyss as presented n Burns (1994) and Burns and Wellngs (1997). Ths equalty s not a concdence, for the followng two reasons. Frstly, remember

45 Real-Tme Syst (2009) 42: that because the characterstcs of the tasks of T 2 are ntegral multples of a value δ = 1 and = 0.2 δ, the value for WR D 2 does not change when s reduced to an arbtrary small postve value,.e. WR D 2 = lm ( WR P 2 (B2 D + C 2 (F 2 )) + (F 2 ) ). 0 Secondly, we can make the followng dervaton usng (12) ( WR P 2 (B2 D + C 2 (F 2 )) + (F 2 ) ) lm 0 = lm 0 ( WR P 2 (B D 2 + C 2 (F 2 )) ) + F 2 = WO P 2 (BD 2 + C 2 F 2 ) + F 2 = ŴR D 2,0 {(12)}. These two results show that ŴR D 2,0 = WR D 2 for T A unform analyss based on WR P We wll gve another pessmstc approach that s unform for all tasks, whch assumes a small postve value and s based on WR P. Lemma 13 A pessmstc worst-case response tme ŴR D k of job k wth 0 k<wld of a task τ under FPDS and arbtrary phasng s gven by ŴR D k = WRP (BD + (k + 1)C (F )) + (F ) kt (57) where: () WR P (BD + (k + 1)C (F )) s the worst-case response tme under FPPS of a task τ wth a computaton tme C = BD + (k + 1)C (F ), a perod T = kt + D (F ), and a deadlne D = T. () s a suffcently small postve number. Proof Because WR P 1 (C) = WOP 1 (C) = C, ŴRD 1,0 = ŴRD 1,0 = WRD 1. Hence, ths approach s not pessmstc for = 1. We wll now prove that WR P (C + ) WO P (C) for 1 < n. The potental addtonal pessmsm ntroduced by (57) now mmedately follows from Lemma 12,.e.ŴR D k ŴRD k. By defnton, task τ can start executng an addtonal amount of computaton tme after havng executed an amount C at tme WO P (C). Because executon of that addtonal computaton tme takes at least an amount of tme, we mmedately get WR P (C + ) WOP (C) +, whch proves the theorem. Based on (12), we frst conclude that both lemmas are smlar for an arbtrary small postve value of,.e. lm 0 ŴR D k = ŴRD k. The addtonal pessmsm potentally ntroduced by Lemma 13 s llustrated by the set T 7 consstng of three tasks wth

46 108 Real-Tme Syst (2009) 42: Table 7 Task characterstcs of T 7 T = D C τ τ τ characterstcs as descrbed n Table 7. For ths example, the task characterstcs are ntegral multples of δ = 0.5. For = 0.6 >δ, we fnd ŴR D 2,0 = 12, whch s larger than τ 2 s deadlne. Conversely, the worst-case response tme ŴR D 2 of task τ 2 determned by means of Theorem 13 usng Lemma 12 yelds ŴR D 2 = WRD 2 = 9 D 2. For = 0.4 <δ, we fnd ŴR D 2,0 = 9. For ths value of, ŴRD 2,0 = ŴRD 2 = WRD 2 = 9 D 2, and reducng the value of wll not change the value found for ŴR D 2,0. The next lemma provdes a suffcent condton to guarantee that Lemma 13 ntroduces no addtonal pessmsm compared to Lemma 12. Lemma 14 If the greatest common dvsor (gcd R+ ) of the perods and computaton tmes of the tasks exsts, and s equal to δ, then <δs a suffcent condton to guarantee that Lemma 13 ntroduces no addtonal pessmsm compared to Lemma 12. Proof To prove the lemma, t suffces to prove <δ WR P (BD + (k + 1)C (F )) = WO P (BD + (k + 1)C F ). From Theorem 2, we derve that WR P (BD + (k + 1)C (F )) sgvenbythe smallest x R + that satsfes the followng equaton, provded that x s at most kt + D (F ), x = B D + (k + 1)C (F ) + x C j. T j< j By substtutng x = x +, we get x = B D + (k + 1)C F + j< x + C j. When the greatest common dvsor (gcd R+ ) of the perods and computaton tmes of the tasks exsts and s equal to δ, all task parameters are ntegral multples of δ (by defnton), and x wll also be an ntegral multple of δ. Letx = n x δ and = n Tj δ for an arbtrary j<, where n x,n Tj N +. Now we get x + nx + δ =. n Tj

47 Real-Tme Syst (2009) 42: BasedonLemma20, we conclude 0 < δ < 1 nx + δ n Tj nx = + 1. n Tj Hence, f the gcd R+ exsts and s equal to δ>, the smallest x R + satsfyng the recursve equaton gven above s a soluton for both WR P (BD + (k + 1)C (F )) and WO P (BD + (k + 1)C F ), whch proves the lemma. We fnally observe that the analyss presented n Lemma 13 s smlar to the revsed schedulablty analyss for CAN presented n Davs et al. (2007). The latter analyss s an evolutonary mprovement of the analyss gven by Tndell et al. (1994, 1995), and Tndell and Burns (1994). A fxed value for s used n Davs et al. (2007), correspondng to the transmsson tme for a sngle bt on CAN A retrospect Usng our notaton, the worst-case response tme of a task τ under FPDS, arbtrary phasng, and deadlnes less than or equal to perods, as descrbed n Lu (2000) can be gven by WR P (BD + C ). As observed n Burns et al. (1993), ths analyss s pessmstc, because a task τ cannot be preempted whle executng ts last subjob,.e. F. The orgnal mprovement of the worst-case response tme of a task τ under FPDS as presented n Burns et al. (1993) was not based on B D as gven n (19), but on the maxmum length of deferred preempton. We nterpret ths latter phrase as a blockng tme B gven by 5 B = max max C j,k. (58) 2 j n 1 k m j Though pessmstc, ths orgnal mprovement s correct,.e. not optmstc. The problem wth the analyss n Burns (1994), Burns and Wellngs (1997) s caused by the fact that the non-preemptve behavor of the fnal subjob of task τ tself s not taken nto account for >1, as llustrated by Fg. 7 n Sect for task τ 2. As descrbed n Davs et al. (2007) n the context of schedulablty analyss for CAN, ths problem can therefore be resolved at the cost of potentally ntroducng addtonal pessmsm by usng B D, whch s gven by B D = max(b D,F ). (59) Conversely, the problem wth the analyss n Burns (1994), Burns and Wellngs (1997) does not occur when B D = B D,.e. when BD F. 8.4 An advanced model for FPDS The model for FPDS descrbed n Sect. 2.2 assumes that each job of a task τ conssts of a sequence of m non-preemptable subjobs. In ths secton, we wll llustrate by means of an example how our analytcal results can be appled n a context where 5 From Lemma 11, we conclude that we may gnore the hghest prorty task n (58).

48 110 Real-Tme Syst (2009) 42: Fg. 11 An example of a DAG of subjobs, representng the flow graph of task τ ataskτ conssts of a (rooted and connected) drected acyclc graph (DAG) of m non-preemptable subjobs. Consder Fg. 11, wth a DAG of subjobs representng the flow graph of task τ. The nodes of ths graph represent the subjobs and the edges represent the successor relatonshps of subjobs. The graph has a sngle root node, wth a computaton tme of C,1, and two leaf nodes, wth computaton tmes C,7 and C,9, respectvely. Durng the executon of a job, a sngle path from the root node to a leaf s traversed. Hence, a job wll ether execute the subjobs wth computaton tmes C,2 and C,3 or the subjob wth computaton tme C,4. Smlarly, a job wll ether execute C,6 and C,7 or C,8 and C,9. The structure of task τ plays a role durng the analyss of the task tself, and for a lower prorty task. The analyss of tasks wth a hgher prorty than τ s smlar to the case where a job conssts of a sequence of subjobs. For the analyss of a task wth a lower prorty than τ, we need to determne the longest computaton tme C of τ for all possble paths through the graph. For our example, ths s equal to C = C,1 + max(c,2 + C,3,C,4 ) + C,5 + max(c,6 + C,7,C,8 + C,9 ). For the analyss of task τ tself, every leaf node of the DAG gves rse to a case that needs to be examned ndvdually. For our example, we therefore get two cases, a frst case for the leaf node C,7,.e. C = C,1 + max(c,2 + C,3,C,4 ) + C,5 + C,6 + C,7, F = C,7, and a second case for the leaf node C,9,.e. C = C,1 + max(c,2 + C,3,C,4 ) + C,5 + C,8 + C,9, F = C,9. The worst-case response tme WR D of task τ s the maxmum of the worst-case response tmes of these two cases. Note that f C F C F and F F, then t suffces to consder the frst case only. Smlarly, f C F C F and F F, then t suffces to consder only the second case. As an alternatve, we can also take a pessmstc approach, and determne WR D based on Ĉ = max(c F,C F ) + max(f,f ), F = max(f,f ).

49 Real-Tme Syst (2009) 42: Table 8 Task characterstcs of T 8 T = D C τ τ τ We wll now llustrate the analyss for τ wth a numercal example. Consder the set T 8 n Table 8. Assume a structure of each job of τ 2 as llustrated n Fg. 11, and let the computaton tmes of the subjobs of task τ 2 be gven by C 2,1 = 1, C 2,2 = 3, C 2,3 = 4, C 2,4 = 6, C 2,5 = 1, C 2,6 = 3, C 2,7 = 2, C 2,8 = 1, C 2,9 = 5. We now fnd C 2 = 1 + max(3 + 4, 6) = 14, F 2 = 2, C 2 = 1 + max(3 + 4, 6) = 15, and F 2 = 5. Because C 2 F 2 = 12 >C 2 F 2 = 10 and F 2 = 2 < F 2 = 5, we have to determne the worst-case response tmes for both cases. Usng the analyss presented n Sect. 6, we fnd 21 for the frst case and 20 for the second case. The worst-case response tme of τ 2 s therefore assumed for the frst case,.e. WR D 2 = 21. For the pessmstc approach, we fnd Ĉ 2 = max(12, 10) + max(2, 5) = 17, F 2 = 5, and derve a worst-case response tme for task τ 2 equal to A note on sustanablty Ths secton brefly dscusses the mpact of assumng worst-case rather than fxed computaton tmes and mnmum nter-arrval tmes (or worst-case perods) rather than fxed perods. Stated dfferently, we consder the sustanablty (Baruah and Burns 2006) of our schedulablty analyss for FPDS. As descrbed n Baruah and Burns (2006), a schedulablty test s sustanable f any task system deemed schedulable by the test remans so f t behaves better than mandated by ts system specfcatons,.e. sustanablty requres that schedulablty be preserved n stuatons n whch t should be easer to ensure schedulablty. The concept of sustanablty s defned as Defnton 15 (From Baruah and Burns 2006) A schedulablty test for a schedulng polcy s sustanable f any system deemed schedulable by the schedulablty test remans schedulable when the parameters of one or more ndvdual job[s] are changed n any, some, or all of the followng ways: () decreased executon requrements; () later arrval tmes; () smaller jtter; and (v) larger relatve deadlnes. That paper also proves that response tme analyss for FPPS s sustanable. For FPDS, we have to adapt Defnton 15 to our model,.e. Defnton 15 s based on a model that only consders jobs of tasks and does not explctly consder subjobs of tasks. Defnton 16 A schedulablty test for our real-tme schedulng model for FPDS s sustanable f any system deemed schedulable by the test remans schedulable when the parameters are changed n any, some, or all of the followng ways: () decreased

50 112 Real-Tme Syst (2009) 42: executon requrements of subjobs; () later arrval tmes of jobs; () larger relatve deadlnes of jobs. Wth such an adaptaton, the schedulablty analyss for our model for FPDS s sustanable, as expressed by the followng theorem, for whch we merely provde a sketch of a proof. Theorem 15 Based on Defnton 16, the schedulablty analyss for our model of FPDS as expressed by the schedulablty test (3) Theorem 12 and Lemmas 7, 12, and 13 s sustanable. Sketch of proof Sustanablty of our schedulablty analyss for FPDS mmedately follows from (3) and the fact that: the maxmum number wl D of jobs of task τ nalevel- actve perod, and the (pessmstc) worst-case response tmes WR D k n (49), ŴRD k n (56), and ŴRD k n (57) are strctly non-ncreasng for decreasng computaton tmes of subjobs and ncreasng perods of tasks. Based on Theorem 15, we conclude that we can replace computaton tmes by worst-case computaton tmes and perods by mnmum nter-arrval tmes (or worstcase perods) n our real-tme schedulng models n Sect. 2. To llustrate the sgnfcance of our adaptaton let s consder an example showng that Defnton 15 s not suffcent for our model. In partcular, we show that a schedulable task system under FPDS becomes unschedulable when the computaton tme C of a task remans the same, but ts dstrbuton to the task ts subjobs changes. As an example, the task characterstcs of T 9 n Table 9 only dffer of those of T 5 n Table 5 n Sect n the dstrbuton of the computaton tme C 2 = 4.2ofτ 2 to ts subjobs. Unlke T 5, T 9 s not schedulable,.e. the second job of task τ 2 n Fg. 12 msses ts deadlne upon a smultaneous release of both tasks. Table 9 Task characterstcs of T 9 T = D C τ τ Fg. 12 Tmelne for T 9 under FPDS wth a smultaneous release of all tasks at tme zero and a deadlne mss of task τ 2 at tme t = 14

51 Real-Tme Syst (2009) 42: Conclusons In ths paper, we revsted exstng worst-case response tme analyss of hard real-tme tasks under FPDS, arbtrary phasng and relatve deadlnes at most equal to perods. We showed by means of a number of examples that exstng analyss s pessmstc and/or optmstc, both for FPDS as well as for FPNS, beng a specal case of FPDS. From these examples, we concluded that the worst-case response tme of a task s not necessarly assumed for the frst job of a task when released at a crtcal nstant. The reason for ths s that the fnal subjob of a task can defer the executon of hgher prorty tasks, whch can potentally gve rse to hgher nterference for subsequent jobs of that task. Ths problem can therefore arse for all tasks, except for the hghest prorty task. We observed that González Harbour et al. (1991) dentfed the same nfluence of jobs of a task for relatve deadlnes at most equal to perods n the context of FPPS of perodc tasks wth varyng executon prorty. We provded revsed worst-case response tme analyss, resolvng the problems wth exstng approaches. The analyss s based on known concepts of crtcal nstant and busy perod for FPPS, for whch we gave slghtly modfed defntons to accommodate for our schedulng model for FPDS. To prevent confuson wth exstng defntons of busy perod, we used the term actve perod for our defnton n ths document. We gave a formal defnton of actve perod, dscussed condtons for ts termnaton, and presented a suffcent condton wth a formal proof. We showed that the crtcal nstant, longest actve perod, and worst-case response tme for a task are suprema rather than maxma for all tasks, except for the lowest prorty task. Hence, that nstant, perod, and response tme cannot be assumed for any task, except for the lowest prorty task. These anomales for the lowest prorty task are caused by the fact that only the lowest prorty task cannot be blocked. We expressed worst-case response tmes under FPDS n terms of worst-case response tmes and worst-case occuped tme under FPPS, and presented an teratve procedure to determne worst-case response tmes under FPDS. We brefly compared the noton of level- actve perod wth smlar notons n the lterature. We concluded that the notons of τ -busy perod n González Harbour et al. (1991), level- busy perod n George et al. (1996), and level-π busy nterval n Lu (2000) are smlar to our noton of level- actve perod. There are strkng dfferences wth the noton of busy perod n Lehoczky (1990), however. In partcular, the level-n busy perod never ends for a utlzaton factor U = 1. Moreover, we observed that although Klen et al. (1993) refers to the noton of busy perod from Lehoczky (1990) n ther descrpton of a method to determne worst-case response tmes of tasks under FPPS, arbtrary phasng and deadlnes larger than perods, ther termnaton condton s actually based on the noton of actve perod rather than busy perod. We also presented unform, but pessmstc varants of our worst-case response tme analyss, and showed that the evolutonary mprovement of the analyss for CAN as presented n Davs et al. (2007) corresponds to one of these varants. We llustrated our analyss for an advanced model for FPDS, where tasks are structured as flow graphs of subjobs rather than sequences. Fnally, we showed that our analyss for FPDS s sustanable and therefore also applcable for sporadc task systems.

52 114 Real-Tme Syst (2009) 42: Acknowledgements We thank Alan Burns and Robert I. Davs from the Unversty of York for dscussons, and the IST funded ARTIST 2 Network of Excellence on Embedded Systems Desgn for makng those dscussons possble. We also thank Mke J. Holendersk and the anonymous referees of the Real Tme Systems Journal for there comments on earler versons of ths paper. Open Access Ths artcle s dstrbuted under the terms of the Creatve Commons Attrbuton Noncommercal Lcense whch permts any noncommercal use, dstrbuton, and reproducton n any medum, provded the orgnal author(s) and source are credted. Appendx: Auxlary defntons and lemmas Ths appendx presents auxlary defntons for greatest common dvsor and least common multple for both postve ratonal numbers and postve real numbers. Moreover, t presents auxlary lemmas for a strctly ncreasng functon f(x)and an auxlary lemma for the celng functon and the floor functon. Defnton 17 The least common multple for postve ratonal numbers (lcm Q+ )s defned as lcm Q+ (r 1,...,r l ) = mn{r Q + r = n 1 r 1 = =n l r l wth n 1,...,n l N + }, (60) where l N and l 2, and r 1,...,r l Q +. Defnton 18 The greatest common dvsor for postve ratonal numbers (gcd Q+ )s defned as gcd Q+ (r 1,...,r l ) = max{r Q + n 1 r = r 1,...,n l r = r l wth n 1,...,n l N + }, (61) where l N and l 2, and r 1,...,r l Q +. Defnton 19 The least common multple for postve real numbers (lcm R+ )sdefned as lcm R+ (r 1,...,r l ) = mn{r R + r = n 1 r 1 = =n l r l wth n 1,...,n l N + }, (62) where l N and l 2, and r 1,...,r l R +. Defnton 20 The greatest common dvsor for postve real numbers (gcd R+ )s defned as gcd R+ (r 1,...,r l ) = max{r R + n 1 r = r 1,...,n l r = r l wth n 1,...,n l N + }, (63) where l N and l 2, and r 1,...,r l R +. Unlke gcd Q+ and lcm Q+, the greatest common dvsor for postve real numbers gcd R+ and the least common multple for postve real numbers lcm R+ need not exst.

53 Real-Tme Syst (2009) 42: Lemma 15 (Brl 2004, Lemma 4.3) Let f(x)be defned and strctly non-decreasng n an nterval [a,b] wth f(a)>aand f(b)<b. Then there exsts a value c (a, b) such that f(c)= c. Proof See Brl (2004). Lemma 16 (Brl (2004, Lemma 4.5)) When lm x X f(x) s defned, and f(x) s strctly ncreasng n an nterval (X, X + γ) for suffcently small γ R +, then the followng equaton holds: lm f(x) = lm f(x) + 1. (64) x X x X Proof See Brl (2004). Lemma 17 When lm x X f(x)s defned, and f(x) s strctly ncreasng n an nterval (X γ,x) for a suffcently small γ R +, then the followng equaton holds: lm f(x) = lm f(x). (65) x X x X Proof The proof uses the defnton of lmt: lm x X f(x)= Y ε>0 δ>0 x (X δ,x) f(x) Y <ε. We frst prove the relaton X γ<x<x f(x)<y, and subsequently prove the lemma. The proof of the relaton s based on a contradcton argument. Because lm x X f(x) s defned, we may wrte lm x X f(x) = Y. Assume f(x 1 ) Y for an x 1 (X γ,x). Choose an x 2 (x 1,X). Because f(x) s strctly ncreasng n (X γ,x), f(x 2 )>f(x 1 ) Y. Now choose ε = f(x 2 ) Y, then x (x2,x)f(x)>f(x 2 )>Y and hence f(x) Y > f(x 2 ) Y =ε, whch contradcts the fact that lm x X f(x)= Y. For the proof of the lemma, we consder two man cases: Y Z and Y/ Z. Let Y Z. Accordng to the relaton proved above, 0 <Y f(x)for all x (X γ,x). Let ε (0, 1]. Now there exsts a δ 1 (0,γ) such that 0 <Y f(x)<ε 1 for all x (X δ 1,X), hence Y>f(x)>Y 1,.e. f(x) = Y = Y. So, lm f(x) = lm Y = Y = lm f(x). x X x X x X

54 116 Real-Tme Syst (2009) 42: Next, let Y/ Z. Letε (0,Y Y ]. Now there exsts a δ 2 (0,γ)such that for all x (X δ 2,X) 0 <Y f(x)<ε Y Y, hence Y>f(x)>Y ε Y,.e. f(x) = Y. For ths second man case we therefore also fnd lm f(x) = lm x X Y = Y = lm f(x), x X x X whch proves the lemma. The proofs of the followng two lemmas are smlar to the proofs of the prevous two lemmas. Lemma 18 When lm x X f(x)s defned, and f(x) s strctly ncreasng n an nterval (X γ,x) for a suffcently small γ R +, then the followng equaton holds: lm f(x) = lm f(x) 1. (66) x X x X Lemma 19 When lm x X f(x)s defned, and f(x)s strctly ncreasng n an nterval (X, X + γ)for suffcently small γ R +, then the followng equaton holds: lm f(x) = lm f(x). (67) x X x X Lemma 20 For n Z, m Z\{0}, and ε R, the followng relaton holds: n + ε n 0 <ε<1 = + 1. (68) m m Proof The proof s based on propertes of celng and floor functons. Let n + ε = l, m wth l Z, hence m(l 1)<n+ ε ml. For 0 <ε<1, we get n + ε/ Z and n<n+ ε<n+ 1. Moreover, because m(l 1), ml Z, we derve 0 <ε<1 (m(l 1)<n+ ε ml m(l 1) n<ml).

55 Real-Tme Syst (2009) 42: Wth m 0, we derve from m(l 1) n<mlthat n m = l 1, whch proves the lemma. References Audsley NC, Burns A, Rchardson MF, Wellngs AJ (1991) Hard real-tme schedulng: The deadlne monotonc approach. In: Proc of the 8th IEEE workshop on real-tme operatng systems and software (RTOSS), May 1991, pp Baeten JCM, Mddelburg CA (2002) Process algebra wth tmng. Sprnger, Berln Baruah S (2005) The lmted-preempton unprocessor schedulng of sporadc systems. In: Proc of the 17th Euromcro conference on real-tme systems (ECRTS), July 2005, pp Baruah S, Burns A (2006) Sustanable schedulablty analyss. In: Proc of the 27th IEEE real-tme systems symposum (RTSS), December 2006, pp Baruah SK, Mok AK, Roser LE (1990a) Preemptvely schedulng hard-real-tme sporadc tasks on one processor. In: Proc of the 11th IEEE real-tme systems symposum (RTSS), December 1990, pp Baruah SK, Roser LE, Howell RR (1990b) Algorthms and complexty concernng the preemptve schedulng of perodc, real-tme tasks on one processor. Real-Tme Syst 2(4): Brl RJ (2004) Real-tme schedulng for meda processng usng condtonally guaranteed budgets. PhD thess, Technsche Unverstet Endhoven (TU/e), The Netherlands pdf Brl RJ (2006) Exstng worst-case response tme analyss of real-tme tasks under fxed-prorty schedulng wth deferred preempton s too optmstc. Techncal report CS 06-05, Department of Mathematcs and Computer Scence, Technsche Unverstet Endhoven (TU/e), The Netherlands, February 2006 Brl RJ, Verhaegh WFJ, Lukken JJ (2004) Exact worst-case response tmes of real-tme tasks under fxedprorty schedulng wth deferred preempton. In: Proc of the work-n-progress (WP) sesson of the 16th Euromcro conference on real-tme systems (ECRTS), Techncal report from the Unversty of Nebraska-Lncoln, Department of Computer Scence and Engneerng (TR-UNL-CSE ), June 2004, pp Brl RJ, Lukken JJ, Davs RI, Burns A (2006) Message response tme analyss for deal controller area network (CAN) refuted. In: Proc of the 5th nternatonal workshop on real tme networks (RTN), July 2006, pp Brl RJ, Lukken JJ, Verhaegh WFJ (2007) Worst-case response tme analyss of real-tme tasks under fxed-prorty schedulng wth deferred preempton revsted. In: Proc of the 19th Euromcro conference on real-tme systems (ECRTS), July 2007, pp Burns A (1994) Preemptve prorty based schedulng: An approprate engneerng approach. In: Son S (ed) Advances n real-tme systems. Prentce-Hall, Englewood Clffs, pp Burns A (2001) Defnng new non-preemptve dspatchng and lockng polces for Ada. In: Proc of the 6th Ada-Europe nternatonal conference, May Lecture notes n computer scence, vol Sprnger, Berln, pp Burns A, Wellngs AJ (1997) Restrcted taskng models. In: Proc of the 8th nternatonal real-tme Ada workshop, pp Burns A, Ncholson M, Tndell K, Zhang N (1993) Allocatng and schedulng hard real-tme tasks on a pont-to-pont dstrbuted system. In: Proc of the 1st workshop on parallel and dstrbuted real-tme systems, Aprl 1993, pp Buttazzo GC (2005) Hard real-tme computng systems predctable schedulng algorthms and applcatons, 2nd edn. Sprnger, Berln Davs RI, Burns A, Brl RJ, Lukken JJ (2007) Controller area network (CAN) schedulablty analyss: refuted, revsted and revsed. Real-Tme Syst 35(3): George L, Rverre N, Spur M (1996) Preemptve and non-preemptve real-tme un-processor schedulng. Techncal report 2966, Insttut Natonal de Recherche en Informatque et en Automatque (INRIA), France, September 1996

56 118 Real-Tme Syst (2009) 42: González Harbour M, Klen MH, Lehoczky JP (1991) Fxed-prorty schedulng wth varyng executon prorty. In: Proc of the 12th IEEE real-tme systems symposum (RTSS), December 1991, pp Gopalakrshnan R, Parulkar GM (1996) Brngng real-tme schedulng theory and practce closer for multmeda computng. In: Proc of the ACM Sgmetrcs conference on measurement & modelng of computer systems, May 1996, pp 1 12 Hermant J-F, Leboucher L, Rverre N (1996) Real-tme fxed and dynamc prorty drven schedulng algorthms: theory and practce. Techncal report 3081, Insttut Natonal de Recherche en Informatque et en Automatque (INRIA), France, December 1996 Hooman J (1991) Specfcaton and compostonal verfcaton of real-tme systems. PhD thess, Technsche Unverstet Endhoven (TU/e), The Netherlands Joseph M, Pandya P (1986) Fndng response tmes n a real-tme system. Comput J 29(5): Klen MH, Ralya T, Pollak B, Obenza R, González Harbour M (1993) A practtoner s handbook for realtme analyss: gude to rate monotonc analyss for real-tme systems. Kluwer Academc, Dordrecht Koymans R (1990) Specfyng real-tme propertes wth metrc temporal logc. Real-Tme Syst 2(4): Lee S, Lee C-G, Lee M, Mn SL, Km C-S (1998) Lmted preemptble schedulng to embrace cache memory n real-tme systems. In: Proc of the ACM Sgplan workshop on languages, complers and tools for embedded systems (LCTES), June Lecture notes n computer scence, vol Sprnger, Berln, pp Lehoczky JP (1990) Fxed prorty schedulng of perodc task sets wth arbtrary deadlnes. In: Proc of the 11th IEEE real-tme systems symposum (RTSS), December 1990, pp Lu JWS (2000) Real-tme systems. Prentce-Hall, Englewood Clffs Lu CL, Layland JW (1973) Schedulng algorthms for multprogrammng n a real-tme envronment. J ACM 20(1):46 61 Mok AK-L (1983) Fundamental desgn problems of dstrbuted systems for the hard-real-tme envronment. PhD thess, Massachusetts Insttute of Technology. pdf/mit-lcs-tr-297.pdf Mok AK, Poon W-C (2005) Non-preemptve robustness under reduced system load. In: Proc of the 26th IEEE real-tme systems symposum (RTSS), December 2005, pp Regehr J (2002) Schedulng tasks wth mxed preempton relatons for robustness to tmng faults. In: Proc of the 23rd IEEE real-tme systems symposum (RTSS), December 2002, pp Sha L, Rajkumar R, Lehoczky JP (1990) Prorty nhertance protocols: an approach to real-tme synchronsaton. IEEE Trans Comput 39(9): Smonson J, Patel JH (1995) Use of preferred preempton ponts n cache-based real-tme systems. In: Proc of the IEEE nternatonal computer performance and dependablty symposum (IPDS), Aprl 1995, pp Spur M (1996) Analyss of deadlne scheduled real-tme systems. Techncal report 2772, Insttut Natonal de Recherche en Informatque et en Automatque (INRIA), France, January 1996 Tndell K, Burns A (1994) Guaranteeng message latences on controller area network (CAN). In: Proc of the 1st nternatonal CAN conference, September 1994, pp 1 11 Tndell K, Hansson H, Wellngs AJ (1994) Analysng real-tme communcatons: Controller area network (CAN). In: Proc of the 15th IEEE real-tme systems symposum (RTSS), December 1994, pp Tndell K, Burns A, Wellngs AJ (1995) Calculatng controller area network (CAN) message response tmes. Control Eng Pract 3(8): Wang Y, Saksena M (1999) Schedulng fxed-prorty tasks wth preempton threshold. In: Proc of the 6th nternatonal conference on real-tme computng systems and applcatons (RTCSA), December 1999, pp Wessten EW (2003) CRC Concse Encyclopeda of Mathematcs. Chapman & Hall/CRC, London

57 Real-Tme Syst (2009) 42: Render J. Brl receved a B.Sc. and an M.Sc. (both wth honors) from the Unversty of Twente, and a Ph.D. from the Technsche Unverstet Endhoven, the Netherlands. He started hs professonal career n January 1984 at the Delft Unversty of Technology. From May 1985 tll August 2004, he has been wth Phlps, and worked n both Phlps Research as well as Phlps Busness Unts. He worked on varous topcs, ncludng fault tolerance, formal specfcatons, software archtecture analyss, and dynamc resource management, and n dfferent applcaton domans, e.g. hgh-volume electroncs consumer products and (low volume) professonal systems. In September 2004, he made a transfer back to the academc world,.e. to the System Archtecture and Networkng (SAN) group of the Mathematcs and Computer Scence department of the Technsche Unverstet Endhoven. Hs man research nterests are currently n the area of reservaton-based resource management for networked embedded systems wth real-tme constrants. Johan J. Lukken s head of the System Archtecture and Networkng Research group at Endhoven Unversty of Technology snce He receved M.Sc. and Ph.D. from Gronngen Unversty n the Netherlands. In 1991 he joned Endhoven Unversty after a two years leave at the Calforna Insttute of Technology. Hs research nterests nclude the desgn and performance analyss of parallel and dstrbuted systems. Untl 2000 he was nvolved n large-scale smulatons n physcs and chemstry. Snce 2000, hs research focus has shfted to the applcaton doman of networked resource-constraned embedded systems. Contrbutons of the SAN group are n the area of component-based mddleware for resource-constraned devces, dstrbuted coordnaton, Qualty of Servce n networked systems and schedulablty analyss n real-tme systems. Wm F.J. Verhaegh receved the mathematcal engneerng degree wth honors n 1990 from the Technsche Unverstet Endhoven, the Netherlands. Snce then, he s wth the Phlps Research Laboratores n Endhoven, the Netherlands. From 1990 untl 1998, he has been a member of the department Dgtal VLSI, where he has been workng on hgh-level synthess of DSP systems for vdeo applcatons, wth the emphass on schedulng problems and technques. Based on ths work, he receved a Ph.D. degree n 1995 from the Technsche Unverstet Endhoven. Snce 1998, he s workng on varous optmzaton aspects of multmeda systems, networks, and applcatons. On the one hand, ths concerns applcaton-level resource management and schedulng, for optmzaton of qualty of servce of multmeda systems. On the other hand, ths concerns adaptve algorthms and machne learnng algorthms for user nteracton ssues, such as content flterng and automatc playlst generaton.