A Review of Fixed Priority and EDF Scheduling for Hard Real-Time Uniprocessor Systems

Transcription

1 A Revew of Fxed Prorty and EDF Schedulng for Hard Real-Te Unprocessor Systes Robert I. Davs Real-Te Systes Research Group, Departent of Coputer Scence, Unversty of York, York, UK. ABSTRACT Ths paper was wrtten to accopany a talk at the ETR Suer School n Toulouse 213. It provdes a background and prer to schedulng and schedulablty analyss for hard real-te sngle processor systes. The paper focuses on the two an schedulng algorths used: Fxed Prorty and Earlest Deadlne Frst. The paper has two broad as: Frstly to provde a gude to the fundaental results for these realte schedulng algorths. Secondly to provde a bref revew of extensons aed at () ltng the effects and overheads of pre-epton and () accountng for the effects of preeptons; specfcally cache related pre-epton delays. The paper also brefly looks back at success stores n real-te schedulng, and forwards at the current hot topcs n ths research area. Categores and Subject Descrptors C.3 [Specal-Purpose and Applcaton-Based Systes]: Realte and ebedded systes. General Ters Algorths, Perforance, Theory, Verfcaton. Keywords Real-te; schedulng; schedulablty analyss; fxed prorty; EDF; sngle processor; unprocessor; survey; lted preepton; cache-related pre-epton delays. 1 INTRODUCTION Ths paper was wrtten to accopany a talk at the ETR Suer School n Toulouse 213. It as to provde a bref gude and revew of schedulng and schedulablty analyss for hard real-te unprocessor systes. Ths s a broad feld whch has been the subject of extensve research, startng n the 197 s and 198 s, gatherng pace n the 199 s through to the present day. In ths paper, we necessarly cannot gve full coverage of the feld; nstead, we seek to provde a prer coverng the fundaental results, copleented by a bref revew hghlghtng two nterestng topcs or thees that buld upon these foundatons. For the nterested reader, we provde EWL 13, August 26 27, 213, Toulouse, FRANCE. Copyrght retaned by the authors. references to further readng n the for of books, surveys, and the orgnal sources. We hope that readers wll be curous to fnd out exactly how soe of the technques entoned, but not covered n detal, actually work and so go on to read the cted lterature. 2 ADDITIONAL READING Burns and Wellngs Book Real-Te Systes and Prograng Languages: Ada 95, Real Te Java and Real Te Posx [19] provdes a useful ntroducton to real-te schedulng. For an excellent descrpton of early work, the nterested reader s also referred to two surveys: Fxed Prorty Schedulng An Hstorcal Perspectve [6], and Real-Te Schedulng Theory A Hstorcal Perspectve [62], both publshed n Real-Te Systes whch s the an journal coverng ths feld. An up-to-date source of further readng can be found on the IEEE Techncal Cottee on Real-Te Systes (TCRTS) webste va the lst of senal papers recoended for educaton (see 3 BACKGROUND Real-te systes are charactersed not only by the need for functonal correctness, but also the need for teporal or tng correctness. Real-te systes contnually ontor and respond to stul fro the envronent and the physcal systes that they control. In order for such systes to behave correctly, they ust not only execute the correct coputatons, but also do so wthn predefned te constrants. These te constrants are typcally expressed n ters of end-to-end deadlnes on the elapsed te between a stul and the correspondng response. Applcatons n real-te systes ay be classfed as hard real-te, where falure to eet a deadlne consttutes a falure of the applcaton; or soft realte, where latency n excess of the deadlne leads only to a degraded qualty of servce. Today, hard real-te systes are found n any dverse applcaton areas ncludng; autootve electroncs, avoncs, space systes, edcal systes, household autoaton, and robotcs. Many real-te systes are coposed of tasks. Tasks are software coponents (progras) that execute n response to soe stulus and output a response. Often the stulus s perodc (e.g. te-drven) thus a perodc task ght execute once every 5s. Tasks can also be event-drven fro soe stulus that does not necessarly occur wth a fxed perod, but nstead has soe nu nter-arrval te, for exaple an nterrupt that occurs fro a sensor trggered on each cycle or rotaton of an engne. Such tasks are referred to as sporadc. Each nstance or executon of a task s referred to as a job of that task. Thus each task can gve rse to a potentally unbounded sequence of jobs.

2 Te constrants n a real-te syste typcally put a lt on the axu pertted response te fro the arrval of the event (or ter tck) that subsequently leads to the release of a job of a task by the operatng syste, untl the job copletes ts executon and outputs ts response. Ths te lt s the relatve deadlne (or just deadlne) of the task. The arrval te of the trggerng event or ter tck plus the relatve deadlne of the task gves the absolute deadlne of the job. More coplex behavours ay be pleented such that the te constrant s fro the stulus for one task, through to the response of a dfferent task. Ths te nterval s referred to as the end-to-end response te and t ust eet the correspondng end-to-end deadlne f the syste s to operate correctly. In a sngle processor syste, the processor can only run a sngle task at any gven te, hence there s an portant choce to be ade as to whch job to run when there s ore than one job that s ready to execute. The schedulng polcy eployed deternes ths choce, whch s then pleented by the scheduler, whch s part of the real-te operatng syste. Schedulng polces ay be classfed accordng to whether they ake schedulng decsons onlne (dependng on the current state of the actve jobs) or offlne (usng a precoputed table). The latter approach s referred to as statc cyclc schedulng. Statc cyclc schedulng was used n early real-te systes, and s stll used today n very sple systes, and also n any safety crtcal systes (e.g. arcraft flght control systes). Statc cyclc schedulng lacks flexblty and has a nuber of other dsadvantages steng fro the use of pre-coputed tables. For these reasons n has been supplanted n any real-te systes by onlne schedulng polces. In ths paper, we focus soley on onlne schedulng and n partcular two schedulng polces that both use prortes to deterne whch job to execute. 1. Fxed Prorty (FP) schedulng 2. Earlest Deadlne Frst (EDF) schedulng Wth fxed prorty schedulng, each task s assgned a statc prorty whch s then nherted by all jobs of that task. By contrast, wth EDF schedulng the prorty of each job s derved fro ts absolute deadlne, such that the job wth the earlest absolute deadlne has the hghest prorty. Fxed prorty schedulng has fxed task prortes, whereas EDF has fxed job prortes. Other schedulng algorths such as Shortest Reanng Processng Te Frst (SRPT) [4], [64] and Least Laxty Frst (LLF) [6] have dynac job prortes that change as each job executes. Schedulng polces are classfed as pre-eptve, f they allow an already runnng job to be suspended so that another (hgher prorty) job ay execute, or non-pre-eptve f they do not. Thus there are pre-eptve and non-pre-eptve varants of both fxed prorty and EDF schedulng. In hard real-te systes, tng correctness (eetng all deadlnes) s an essental aspect of correct syste behavour, hence t s portant to be able to ensure before the syste s deployed that all deadlnes wll be et durng operaton 1 1 In practce often what s requred s an assurance that there s only a very sall probablty that deadlnes wll be ssed. Probablstc analyss whch seeks to address ths ore refned vew s the topc of another talk at ETR 213. assung that the syste behaves as expected (.e. no faults, correct functonal behavour etc.). Schedulablty analyss s the nae gven to a atheatcal approach used to deterne offlne f deadlnes can be guaranteed to be et at runte. Schedulablty analyss requres nforaton about: () The possble scenaros and patterns of arrvals of the jobs of each task. () The executon behavour of the tasks, n partcular, an upper bound on the longest te a job of each task could take to execute non-pre-eptvely, referred to as the worst-case executon te (WCET), and any nteractons between the tasks (e.g. utually exclusve access to shared resources). () The schedulng polcy used (e.g. fxed prorty) and any odfcatons to ths (e.g. when jobs ay teporarly execute at a hgher prorty, due to nonpre-eptve behavour, or when accessng shared resources etc.). In ths paper, we wll assue that an upper bound on the WCET of each task s known. We note that WCET analyss s an actve feld of research, and the proble of deternng an accurate estate of the WCET s a dffcult one, partcularly for processors wth hardware acceleraton features such as ppelnes and caches [71]. We now ntroduce soe ternology that s used n research focused on schedulng and schedulablty analyss. We defne what s eant by schedulable and feasble task sets, optal schedulng algorths and optal prorty assgnent polces, and dfferent types of schedulablty test, ncludng exact ones. We say that a task set s schedulable wth respect to soe schedulng algorth, f all vald sequences of jobs that ay be generated by the task set can be scheduled by the schedulng algorth wthout any ssed deadlnes. Further, we say a task set s feasble f there exsts soe schedulng algorth that can schedule all possble sequences of jobs that ay be generated by the task set wthout ssng any deadlnes. A schedulng algorth s sad to be optal wth respect to a task odel f t can schedule all of the feasble task sets that coply wth the odel. A schedulablty test s sad to be suffcent, wth respect to a schedulng algorth, f all of the task sets that are deeed schedulable accordng to the test are n fact schedulable under the schedulng algorth. Slarly, a schedulablty test s sad to be necessary, f all of the task sets that are deeed unschedulable accordng to the test are n fact unschedulable under the schedulng algorth. A schedulablty test that s both suffcent and necessary s referred to as exact. Thus our a n schedulablty analyss research s to fnd effcent, exact schedulablty tests; however, there s typcally a tradeoff between the coputatonal coplexty of the tests and ther effectveness. Hence soetes we need to use effcent tests that only gve suffcent results rather than exact tests that take too long to copute. In fxed prorty schedulng, a prorty assgnent polcy P s sad to be optal wth respect to soe class of task sets (e.g. those wth arbtrary deadlnes), and soe class of fxed prorty schedulng algorth (e.g. pre-eptve) f there are no task sets n the class that are schedulable under the schedulng

3 algorth usng any other prorty orderng polcy that are not also schedulable usng the prorty assgnent deterned by polcy P. Usng an optal prorty assgnent polcy we can schedule any task set that s schedulable usng any other prorty assgnent polcy; hence n analyss of fxed prorty systes, we always a to fnd and use optal prorty assgnent polces. The rest of the paper s organsed as follows. In Secton 4, we outlne the fundaental research nto schedulablty analyss for sngle processor systes usng fxed prorty and EDF schedulng. In Secton 5 we provde a foral taskng odel and descrbe general fors of schedulablty analyss for both fxed prorty and EDF schedulng of sporadc task sets. These are the buldng blocks that researchers workng n ths area need to be aware of. In Secton 6, we provde a gude to research along two thees that buld upon the fundaental schedulablty analyss research: ltng pre-epton, and accountng for ts costs. Secton 7 concludes the paper wth a bref look at success stores for real-te schedulablty analyss and hot topcs of future research. 4 FIXED PRIORITY AND EDF SCHEDULING In ths secton, we provde an overvew of the fundaental work on schedulablty analyss for sngle processor systes usng fxed prorty and EDF schedulng. 4.1 Pre-eptve schedulng In 1973, Lu and Layland [56] consdered fxed prorty preeptve schedulng of synchronous 2 task sets coprsng ndependent perodc tasks, wth bounded worst-case executon tes, and deadlnes equal to ther perods (so called plctdeadlnes). Lu and Layland showed that rate onotonc prorty orderng, where tasks are assgned prortes n order of ther perods, wth a shorter perod plyng a hgher prorty, s the optal prorty assgnent polcy for such task sets. They further proved that that usng rate onotonc prorty orderng, fxed prorty pre-eptve schedulng can schedule any plct-deadlne task set that has a total utlsaton U ln( 2). 693, where the utlsaton of a task s ts worst-case executon te dvded by ts perod, and the total utlsaton of the task set s sply the su of the ndvdual task utlsatons. Lu and Layland [56] also showed that that EDF can schedule any task set that has a total utlsaton U 1. In 1974, Dertouzos [41] showed that pre-eptve EDF s an optal sngle processor schedulng algorth, n the sense that f a vald schedule exsts for a task set, then the schedule produced by EDF wll also eet all deadlnes. Least Laxty Frst [6] s another such optal algorth. Research nto real-te schedulng durng the 198 s and early 199 s focused on lftng any of the restrctons of the Lu and Layland task odel. Task arrvals were pertted to be sporadc, wth known nal nter-arrval tes, (stll slghtly confusngly referred to n the lterature as perods), and task deadlnes were pertted to be less than or equal to ther perods (so called constraned deadlnes) or less than, equal to, or greater than ther perods (so called arbtrary deadlnes). In 1982, Leung and Whtehead [53] showed that deadlne onotonc prorty orderng, where tasks are assgned prortes n order of ther deadlnes, wth a shorter deadlne plyng a hgher prorty, s the optal prorty orderng for constraned-deadlne task sets. Exact schedulablty tests for pre-eptve fxed prorty schedulng of constraned-deadlne task sets were ntroduced by Joseph and Pandya n 1986 [48], Lehoczky et al. n 1989 [55], and Audsley et al. n 1993 [5]. In 199, Lehoczky [54] showed that deadlne onotonc prorty orderng s not optal for task sets wth arbtrary deadlnes; however, an optal prorty orderng can be deterned n ths case, n at ost n ( n 1) / 2 task schedulablty tests usng Audsley s optal prorty assgnent (OPA) algorth 3 [4], [7]. Exact schedulablty tests for task sets wth arbtrary deadlnes were developed by Lehoczky [54] n 199, and Tndell et al. [67] n The release jtter proble was frst address by Audsley et al. [5] n 1993 for constraned deadlne task sets, and later by Tndell et al. [67] for the arbtrary deadlne case. Here, jobs of a task ay follow a perodc or sporadc arrval pattern, but do not necessarly becoe ready to execute edately, nstead ther release ay be delayed by up to a axu te referred to as the Release Jtter of the task. Ths s a typcal behavour n real systes where there ay be soe latency n respondng to the event that trggers task executon. Zuhly and Burns [75] showed that for constraned deadlne task sets wth release jtter, Deadlne nus Jtter onotonc prorty orderng s optal. Exact schedulablty tests for constraned and arbtrarydeadlne task sets scheduled usng pre-eptve EDF were ntroduced by Baruah et al. [1], [11] n 199. Subsequently, exact tests for EDF have been developed by George and Herant [43] and Zhang et al. [73], [74] that are ore effcent n practce. 4.2 Resource sharng In pre-eptve systes, echanss are requred to ensure that jobs access shared resources n utual excluson, otherwse data corrupton, or erroneous behavour of hardware devces could ensue. If sple seaphores are used to seralse resource access, then unbounded prorty nverson can occur. Prorty nverson s the nae gven to the stuaton where a hgh prorty job atteptng to lock a resource has to wat not only for the low prorty job that currently holds the resource, but also any jobs of nteredate prorty that ay pre-ept t. A nuber of concurrency control protocols have been developed whch avod ths proble. These nclude the Prorty Inhertance Protocol (PIP) [63] and the Prorty Celng Protocol (PCP) [63] orgnally developed for fxed prorty schedulng and the Stack Resource Polcy (SRP) [8] ntroduced by Baker n 1991, whch s applcable to both fxed prorty and EDF schedulng. SRP s the ost effectve protocol and has been adopted by the OSEK and Autosar operatng syste standards. The Stack Resource Polcy [8] assocates a pre-epton level wth each resource. In EDF (FP) schedulng, ths pre-epton 2 A task set s referred to as synchronous f all of the tasks can share a coon release te. 3 Ths algorth s optal n the sense that t fnds a schedulable prorty orderng whenever such an orderng exsts.

4 level corresponds to the shortest relatve deadlne (hghest prorty) of any task that accesses the resource. At run-te, when a job locks a resource, ts current prorty s saved (on a stack), and t contnues to execute at the hgher of ts prevous prorty and the pre-epton level of the resource. On unlockng the resource the job returns to ts prevous prorty (obtaned fro the stack). Thus SRP allows resource lockng to be properly nested. The Stack Resource Polcy [8] has a nuber of portant propertes: o Once a job starts executng t never has to wat to access a resource. Ths eans that wth SRP, resource accesses add no addtonal context swtches, and sngle stack executon s possble savng eory. o No job j can be delayed fro executng by lower preepton level jobs for longer than the axu sngle resource access te of any such lower prorty job, where the resource s shared wth other jobs of the sae or a hgher pre-epton level than job j. o The syste s deadlock free. These propertes strctly bound the blockng te for whch hgher prorty jobs can be delayed by the executon of lower prorty jobs. Baker [8] ntally provded a suffcent schedulablty test for EDF and SRP. Later, n 1996 Spur [65] odfed the exact test for EDF schedulng to account for resource lockng under SRP. Exact tests for fxed prorty schedulng wth ether PCP or SRP where ntroduced by Audsley et al. n 1993 [5] for constraned deadlne task sets, and Tndell et al. [67] n 1994 for arbtrary deadlne task sets. 4.3 Non-pre-eptve schedulng In 198, K and Naghbdadeh [5], and n 1991, Jeffay et al. [47], gave exact schedulablty tests for plct-deadlne task sets under non-pre-eptve EDF schedulng. These tests were extended by George et al. [44] n 1996, to the general case of sporadc task sets wth arbtrary deadlnes. Whle pre-eptve EDF s an optal sngle processor schedulng algorth, n the non-pre-eptve case no workconservng algorth s optal. (A work-conservng schedulng algorth s one that does not dle the processor when there are tasks ready to be executed). Ths s because n general, wth non-pre-eptve schedulng, t s necessary to nsert dle te to acheve a feasble schedule. The nterested reader s referred to [44] for exaples of ths behavour. In 1995, Howell and Venkatrao [46] showed that for nonconcrete strctly perodc task sets, where the tes at whch each task ay be frst released are unknown, the proble of deternng a feasble non-pre-eptve schedule s NP hard. Further, they showed that for sporadc task sets, no optal onlne nserted dle te algorth can exst. In other words, clarvoyance s needed to deterne a feasble non-preeptve schedule. Whle no work-conservng algorth s optal n the strong sense that t can schedule any task set for whch a feasble nonpre-eptve schedule exsts; n 1995, George et al. [45] showed that non-pre-eptve EDF s optal n the weak sense that t can schedule any task set for whch a feasble work-conservng, non-pre-eptve schedule exsts. For fxed prorty non-pre-eptve schedulng of arbtrarydeadlne task sets, George et al. [44] derved an exact schedulablty test based on the approach of Tndell et al. [67] for the pre-eptve case. George et al. showed that unlke n the pre-eptve case, deadlne onotonc prorty orderng s not optal for constraned-deadlne task sets scheduled nonpre-eptvely. Further, they showed that Audsley s optal prorty assgnent algorth [7] s applcable, and can be used to deterne an optal prorty orderng n ths case. Subsequent research by Brl et al. [17] has refned exact analyss of non-pre-eptve fxed prorty schedulng, correctng ssues of both pesss and opts, and extendng the schedulablty tests to co-operatve schedulng where each task s ade up of a nuber of non-pre-eptve regons. There are varous approaches that allow lted preepton and thus represent a coprose between fully preeptve and fully non-pre-eptve schedulng. We return to these n Secton 6. 5 SCHEDULABILITY TESTS In ths secton we frst descrbe the sporadc task odel, whch s coonly used n real-te schedulng research, and fors the bass for the schedulablty analyss that we then recaptulate. We also provde detals of the defacto standard notaton soetes referred to as Burns Standard Notaton [32], whch s used n ths area. 5.1 Task odel and notaton We consder the schedulng of a set of sporadc tasks (or task se on a sngle processor. Each task set coprses a statc set of n tasks ( 1.. n ), where n s a postve nteger. Wthout loss of generalty, we assue that the ndex of task also represents the task prorty used n fxed prorty schedulng, hence 1 has the hghest fxed-prorty, and n the lowest. We use the notaton hp () (and lp () ) to ean the set of tasks wth prortes hgher than (lower than), and the notaton hep () (and lep () ) to ean the set of tasks wth prortes hgher than or equal to (lower than or equal to). Each task s characterzed by ts bounded worst-case executon te C, nu nter-arrval te or perod T, relatve deadlne D, and release jtter J. Each task therefore gves rse to a potentally unbounded sequence of jobs, each of whch has an executon te upper bounded by C, an arrval te at least T after the arrval of the prevous job of the sae task, and an absolute deadlne that s D after ts arrval. Due to release jtter, release of each job ay be delayed for a varable aount of te fro to J after ts arrval. In an plct-deadlne task set, all tasks have D T. In a constraned-deadlne task set, all tasks have D T, whle n an arbtrary-deadlne task set, task deadlnes are ndependent of ther perods. The set of arbtrary-deadlne task sets s therefore a superset of the set of constraned-deadlne task sets, whch s tself a superset of the set of plct deadlne task sets. The worst-case response te R of a task s gven by the longest possble te fro the arrval of the task untl t copletes executon. Thus task s schedulable f and only f R D, and a task set s schedulable f and only f R D. The utlzaton U, of a task s gven by ts executon te dvded by ts perod ( U = C / T ). The total utlzaton U, of a task set s the su of the utlzatons of all of ts tasks.

5 The followng assuptons are ade about the behavour of the tasks: o The arrval tes of the tasks are ndependent and unknown a pror, hence the tasks ay share a coon release te. o The tasks ay access utually exclusve shared resources accordng to the Stack Resource Polcy [8]. Thus each task ay be subject to a axu aount of blockng B that equates to the longest te that a job of a task of lower prorty (pre-epton level) ay lock a resource that s shared wth tasks of the sae or hgher prorty (pre-epton level) than task. o The tasks are otherwse ndependent, and do not voluntarly suspend theselves. A job s sad to be ready f t has outstandng coputaton awatng executon by the processor. Under pre-eptve EDF schedulng, at any gven te the ready job wth the earlest absolute deadlne s selected for executon by the processor. In contrast, under pre-eptve fxed prorty schedulng, at any gven te the earlest released unfnshed job of the hghest prorty ready task s selected for executon. Slarly, under non-pre-eptve EDF, at any gven te when there s no currently executng job then the ready job wth the earlest absolute deadlne s selected for executon by the processor. Under non-pre-eptve fxed prorty schedulng, at any gven te when there s no currently executng job then the earlest released unfnshed job of the hghest prorty ready task s selected for executon. 5.2 Analyss for pre-eptve FP schedulng We now gve a suary of Response Te Analyss [5] used to provde an exact schedulablty test for fxed prorty preeptve schedulng of constraned-deadlne task sets. We then recaptulate the soewhat ore coplex response te analyss for arbtrary-deadlne task sets [67]. Frst, we ntroduce the concepts of a crtcal nstant, and busy perods, whch are fundaental to response te analyss. The crtcal nstant for a task s defned as a pont n te when the arrval of a job of that task wll have the longest possble response te, gven any vald pattern of arrvals and executon behavour of the tasks n the task set. The ter prorty level- busy perod refers to a perod of te [ t 1, t 2) durng whch tasks, of prorty or hgher, that were released at the start of the busy perod at t 1, or durng the busy perod but strctly before ts end at t 2, are ether executng or ready to execute. Under fxed prorty pre-eptve schedulng, the crtcal nstant for a task occurs wthn the longest prorty level- busy perod, whch s assocated wth an arrval sequence such that the frst jobs of task and all hgher prorty tasks are released sultaneously after ther axu pertted release jtter, and subsequent jobs of the sae tasks are released as soon as possble thereafter. For tasks wth arbtrary deadlnes and large release jtter, ths eans that ore than one job of the task ay be released at the sae te. In ths case, we assue that jobs of the sae task are nevertheless released n order of ther arrval, and that a later arrvng job cannot overtake an earler arrval and so be released frst. Further, we assue that later arrvng jobs of the sae task effectvely have lower prorty, and are executed after earler arrvals. Fnally, we assue that just pror to the ntal sultaneous release, a lower prorty task locks the resource resultng n the axu blockng te B. For a constraned deadlne task, the worst-case response te R corresponds to ts release jtter plus the length of the prorty level- busy perod descrbed above. The busy perod coprses three coponents, () the blockng te B, () the executon te of the task tself C, and () so called nterference, equal to the te for whch task s prevented fro executng by all hgher prorty tasks. The length of the busy perod w, can be coputed usng the followng fxed pont teraton [5], wth the suaton ter gvng the nterference due to the set of hgher prorty tasks hp(). w J 1 j w B C C j (1) jhp( ) T j Iteraton starts wth an ntal value w, typcally 1 w B C, and ends when ether w w n whch case 1 the worst-case response te R s gven by w J, or 1 when w D J n whch case the task s unschedulable. The fxed pont teraton s guaranteed to converge provded that the overall task set utlsaton s less than or equal to 1. We note that teraton ay be speeded up usng the technques descrbed n [33]. Equaton (1) provdes an exact schedulablty test for the fxed prorty pre-eptve schedulng of constraned-deadlne task sets wth any fxed prorty orderng. For fxed prorty pre-eptvely scheduled systes, where task deadlnes are arbtrary, executon of one job of a task ay not necessarly be copleted before the next job s released. Hence a nuber of jobs of task ay be present wthn a prorty level- busy perod, wth earler jobs delayng the executon of later ones. In ths case, the crtcal nstant and hence the worst-case response te for task occurs for soe job released wthn the prorty level- busy perod correspondng to the sequence of releases prevously descrbed, but not necessarly the frst job. The length L of ths longest prorty level- busy perod can be found va the followng fxed pont teraton: L J 1 j L B C j (2) jhep ( ) T j Iteraton starts wth an ntal value guaranteed to be no larger than the nu soluton, for exaple L C, and ends 1 when L L. The nuber of jobs Q of task n the busy perod s gven by: L J Q (3) T In general t s necessary to copute the response tes of all jobs of a task wthn the longest prorty level- busy perod n order to deterne the task s worst-case response te. The copleton te W, q of the qth job (where q = s the frst job) of task, wth respect to the start of the busy perod, s gven by the followng fxed pont teraton:

6 w 1, q J j w, q B ( q 1) C C j (4) jhp( ) T j Iteraton starts wth an ntal value w,q, typcally 1 w, q B ( q 1) C, and ends when ether w, q w, q n 1 whch case,, 1 W q w, q or when w, q qt D J n whch case job q, and hence task s unschedulable. To fnd the worst-case response te of task, copleton tes W, q need to be calculated for jobs q,1,2,3,... Q 1. The worst-case response te of task s then gven by: p R ax q,1,2... Q 1( W, q qt J ) (5) Task s schedulable provded that R D. Equatons (4) and (5) gve an exact schedulablty test for the pre-eptve fxed prorty schedulng of arbtrary-deadlne task sets wth any specfed prorty orderng. 5.3 Analyss for non-pre-eptve FP schedulng We now recaptulate Response Te Analyss used to provde an exact schedulablty test for fxed prorty non-pre-eptve schedulng of arbtrary deadlne task sets. Ths analyss has slartes to the pre-eptve case, but also soe key dfferences. George et al. [44] and Brl et al. [17] showed that for fxed prorty non-pre-eptve schedulng, the longest response te of a job of task occurs for soe job of that task wthn a prorty level- busy perod startng wth the sultaneous release of jobs of task and all hgher prorty tasks that have been subject to the axu release jtter, wth subsequent jobs of those tasks released as soon as possble. Further, the nu possble aount of te pror to the ntal sultaneous release, a lower prorty task k s released, and ths task has the longest executon te of any such lower prorty task. (Note that n the followng, we assue dscrete te and so = 1). The worst-case length of the prorty level- busy perod s agan gven by the nu soluton to (2), wth B defned to be the longest te that task and any hgher prorty tasks can be blocked fro executng by the non-pre-eptve executon of a lower prorty task: B ax ( C 1) (6) klp( ) k Note n (6) the axu of an epty set s assued to be zero. The nuber of jobs Q of task n the actve perod s gven by: L J Q (7) T The start te W, q of the qth job (where q = s the frst job) of task easured wth respected to the ntal sultaneous release s gven by the nu soluton to the followng fxed pont teraton: w 1, q w, q B qc 1 C j (8) jhp( ) T j Iteraton starts wth an ntal value w,q, typcally 1 w, q B qc, and ends when ether w, q w, q n whch 1 case, 1 W q w, q, or when: w, q C qt D J n whch case job q, and hence task s unschedulable. To fnd the worst-case response te, the start tes W, q need to be calculated for jobs q,1,2,3,... Q 1. The worst-case response te of task s then gven by: P q,1,2... Q 1 q R ax NP ( W, C qt J ) Task s schedulable provded that R D. (9) Equatons (8) and (9) gve an exact schedulablty test for nonpre-eptve fxed prorty schedulng of arbtrary-deadlne task sets wth any specfed prorty orderng. The exact analyss for both pre-eptve and non-pre-eptve fxed prorty schedulng s pseudo-polynoal n coplexty. In practce, the technques descrbed above based on fxed pont teraton are fast enough to be used to deterne task set schedulablty, and even to perfor senstvty analyss by varyng task paraeters and deternng f the task set reans schedulable. In cases where a sgnfcant trade-off between accuracy and speed s requred, then Davs and Burns [35] provde response te upper bounds for both pre-eptve and non-pre-eptve cases that are lnear n coplexty and can be coputed n O(n) te for all n tasks n a task set. 5.4 Prorty assgnent The schedulablty tests gven above for both fxed prorty pre-eptve and non-pre-eptve schedulng are agnostc as far as the prorty assgnent polcy s concerned; they work wth any gven assgnent. However, prorty assgnent can have a very large effect on schedulablty as shown by Davs et al. [3], hence t s portant whenever possble to use an optal prorty assgnent polcy. Audsley s Optal Prorty Assgnent (OPA) algorth [4], [7], reproduced below, provdes an optal prorty assgnent for arbtrary-deadlne sporadc task sets wth respect to both pre-eptve and non-pre-eptve fxed prorty schedulng. Ths greedy approach requres at ost n(n+1)/2 task schedulablty tests to deterne a schedulable prorty orderng whenever such an orderng exsts. Ths copares favourably wth the n! tests that would otherwse be requred, assung a brute force approach that checks every possble cobnaton of prortes. for each prorty level k, lowest frst { for each unassgned task { f( s schedulable at prorty k wth all other unassgned tasks assued to have hgher prortes) { assgn to prorty k break (contnue outer loop) return unschedulable return schedulable Algorth 1: Optal Prorty Assgnent Algorth Whle t provdes an optal prorty assgnent, the OPA algorth has one sgnfcant drawback: t does not ake any choce about prorty orderng other than to ensure schedulablty. Thus usng the OPA algorth the prorty assgnent obtaned can leave the task set of the edge of unschedulablty, wth any sall ncrease n the executon te of a task resultng n a deadlne ss. Davs and Burns addressed

7 ths ssue wth the Robust Prorty Assgnent algorth (RPA) [34]. Ths algorth s based on Audsley s OPA algorth; however, t akes use of a scalng factor whch s used to characterse addtonal nterference. Ths addtonal nterference ay take a very general for provded that t s onotoncally ncreasng wth the prorty level, and the te nterval consdered. For exaple the scalng factor ght characterse an ncrease n the executon te or one, or all of the tasks, or of soe addtonal nterference at the hghest prorty level. Usng an approprate choce of addtonal nterference, the RPA algorth provdes a prorty assgnent that s not only optal, but also as robust as possble to ncreases n nterference. The RPA algorth s shown below. for each prorty level k, lowest frst { for each unassgned task { bnary search for the largest value of for whch task s schedulable at prorty k f no tasks are schedulable at prorty k { return unschedulable else { assgn the schedulable task that tolerates the ax at prorty k to prorty k return schedulable Algorth 2: Robust Prorty Assgnent Algorth For ore nforaton on prorty assgnent, the nterested reader s drected to the presentaton that accopaned the keynote talk Gettng ones prortes rght at RTNS 212 [26]. 5.5 Analyss for pre-eptve EDF We now recaptulate schedulablty analyss for arbtrarydeadlne task sets under pre-eptve EDF. In the analyss of EDF schedulng, we use the concept of processor deand. bound. Ths s defned for an nterval of length t, as the axu possble processor deand fro jobs wth both ther release tes and ther deadlnes n the nterval. A necessary condton for schedulablty s that for any nterval of length t, the processor deand bound ust not exceed the length of the nterval. Otherwse the processor could not possbly coplete all of the jobs that are released n the nterval and also need to fnsh wthn the nterval. Exact tests for EDF were frst gven by Baruah et al [1], [11]. Below, we provde extended versons, ncludng blockng and release jtter due to Spur [65] and Zhang and Burns [74]. The processor deand bound functon h( s gven by: h A task set s schedulable under pre-eptve EDF f and only f U 1 and the processor LOAD s 1 where the processor LOAD s gven by: n t J D t ( ) ax, 1C (1) T 1 h( b( LOAD ax (11) t t where resource access s accounted for by the blockng ter: b( ax C, D J t, D J t (12) a k a In whch C a, k s the axu aount of te that a task a ay spend accessng a resource also used by soe task k wth a hgher pre-epton level. (Note that the pre-epton level corresponds to D J ). k k If LOAD 1 for all values of t n the nterval (, L ], where L s defned as follows, then the task set s schedulable. L ax(( D1 T1 J ax ( b( d )) ddax 1 1U Note nor correcton ade to (13) w.r.t. publshed verson. a ),...( Dn, ( T J k k Tn, J D ) U n, ), ) (13) Note the length L of the nterval that needs to be exaned s also lted to the length of the longest prorty level-n busy perod, whch ay be coputed usng (2) and should be used nstead of (13) n the case that U = 1. The only values of t that need to be checked n the nterval (, L ] are those where the LOAD can change,.e. t kt D J for nteger values of k. However, there can stll be a very large nuber of such values that need to be checked. Zhang and Burns [73], [74] used the fact that h( + b( s onotoncally non-decreasng wth respect to t as the bass for a hghly effcent exact schedulablty test for pre-eptve EDF, called Quck Processor-deand Analyss (QPA). The QPA algorth s reproduced below. In practce, ths algorth typcally checks a far sall set of values n order to ascertan schedulablty than the classcal ethod. QPA schedulablty test for pre-eptve EDF: An arbtrary deadlne sporadc task set s schedulable under pre-eptve EDF schedulng f and only f U 1 and the result of the followng teratve algorth s h( b( DJn, where DJ n s the sallest deadlne nus jtter of any of the tasks, and d s an absolute task deadlne,.e. d kt D J for nteger values of k. 1 t ax{ d d L 2 whle( h( b( t h( b( DJn) { 3 f( h( b( t ) 4 { t h( b( 5 else 6 { t ax{ d d t 7 8 f( h( b( DJn ) task set s schedulable 9 else task set s unschedulable Algorth 3: QPA for EDF 5.6 Analyss for non-pre-eptve EDF George et al. [44] extended the schedulablty test for preeptve EDF to the non-pre-eptve case va the use of a revsed blockng factor b(. b( ax : D J t ( C 1) (14) Note n (14) the axu of an epty set s assued to be zero. Effectvely, non-pre-eptve EDF schedulng ay be consdered a specal case of resource sharng n pre-eptve EDF and thus the QPA schedulablty analyss (Algorth 3) ay also be used n ths case [74].

8 5.7 Coparsons between FP and EDF Fxed prorty and EDF schedulng are n soe respects copetng technques. A theoretcal coparson between the effectveness of fxed prorty and EDF schedulng has been derved by consderng the speedup factor: the axu processor speed necessary so that fxed prorty schedulng can schedule any task set that s schedulable on a processor of unt speed under EDF. Ths gves a easure of the worst-case suboptalty of fxed prorty schedulng. Lu and Layland s utlsaton bounds [56] effectvely show that the speedup factor requred to guarantee that pre-eptve fxed prorty schedulng can schedule any plct-deadlne task set schedulable by EDF s 1/ln(2) Davs et al. [37] showed that the exact speedup factor for the constraned deadlne case s 1/ (where s the atheatcal constant defned by the transcendental equaton ln( 1/), hence, ). Further, for the arbtrary deadlne case [38], the speedup factor s lower bounded by 1 / and upper bounded by 2. These upper and lower bounds also hold for a coparson between non-pre-eptve fxed prorty and non-pre-eptve EDF schedulng [39]. It reans an nterestng open proble what the exact speedup factors are n these cases. The average case sub-optalty of pre-eptve fxed prorty schedulng has also been nvestgated by Lehoczky et al. [55] who used task set average breakdown utlsaton as a etrc. The breakdown utlsaton s obtaned by scalng task executon tes untl the task set s only just schedulable and then recordng the utlsaton. For large plct-deadlne task sets they found that the breakdown utlsaton for s approx. 88%, correspondng to a penalty of approxately 12% n processng capacty copared to pre-eptve EDF. Bn and Buttazzo [16] later showed that breakdown utlsaton suffers fro a bas whch tends to penalse fxed prorty schedulng by favourng task sets where the utlsaton of ndvdual tasks s slar. They ntroduced an optalty degree etrc whch gves a farer coparson and shows that the average penalty for usng fxed prorty-pre-eptve schedulng s sgnfcantly saller. In the paper Rate Monotonc vs. EDF: Judgent Day, Buttazzo [23] provded a coprehensve coparson and revew of the advantages and dsadvantages of fxed prorty versus EDF schedulng polces; lookng at factors such as pleentaton coplexty, runte overhead, nuber of preeptons, robustness durng overload, jtter on task executon, and responsveness when handlng aperodc (soft real-te) tasks. Whle EDF has any advantages, t s notable that to ths day, the overwhelng ajorty of coercal Real-Te Operatng Systes pleent fxed prorty rather than EDF schedulng. Ths s done for reasons of splcty and lower overheads: n partcular, the nuber of prorty levels s often lted allowng a bt-ap representaton of the ready queue to be used, resultng n O(1) queung and de-queung operatons. 6 BUILDING ON THE FUNDAMENTALS In the prevous secton, we recaptulated schedulablty analyss for fxed prorty and EDF, pre-eptve and non-preeptve schedulng. In the ajorty of theoretcal work on real-te schedulng, the overheads of pre-epton are assued to be neglgble. Indeed ths assupton s ade n the analyss presented so far. In real systes; however, the overheads of pre-epton can be large and have a sgnfcant pact on syste schedulablty. There are two an threads of research that address ths proble. Frstly, work that seeks to nse the nuber of pre-eptons va the use of fxed or floatng non-pre-eptve regons (e.g. co-operatve schedulng), or va pre-epton thresholds, and non-pre-epton groups, where tasks copete for the processor at a dfferent prorty to the one at whch they actually execute. Secondly, work that seeks to account for and reduce the cost of pre-eptons, n partcular cache related preepton delays, va the ntegraton of CRPD analyss nto schedulablty tests, and va anpulatng task layouts and cache parttonng to prove schedulablty. 6.1 Ltng pre-epton Pre-eptve and non-pre-eptve schedulng can be consdered as two extrees. It ay be argued that fully preeptve schedulng allows too any pre-eptons, whch can have a hgh cost f pertted at any arbtrary pont. By contrast, non-pre-eptve schedulng perts no pre-eptons, and can have a severe effect on schedulablty when soe tasks have uch shorter deadlnes than others. Indeed, wth just two tasks one of whch has an executon te that s longer than the other s deadlne, then non-pre-eptve schedulng s sply not vable. Lted pre-epton schedulng s the general ter gven to a collecton of dfferent schedulng polces aed at reducng the nuber of pre-eptons, and hence ther overall cost, whle ensurng that the addtonal blockng ntroduced does not ake the syste unschedulable. The frst approach s deferred pre-epton. Two dfferent odels of schedulng wth deferred pre-epton have been developed n the lterature. In both cases, non-pre-eptve regons are used to lt pre-epton. Ths s partcularly effectve towards the end of a task s executon where a fnal non-pre-eptve regon ay prevent any further context swtches and consequent pre-epton related overheads before the task copletes. In the fxed odel, ntroduced by Burns n 1994 [2] for fxed prorty schedulng, the locaton of each non-pre-eptve regon s statcally deterned pror to executon. Pre-epton s then only pertted at pre-defned locatons n the code of each task, referred to as pre-epton ponts. Ths ethod s also referred to as co-operatve schedulng, as the tasks cooperate, provdng re-schedulng / pre-epton ponts to prove schedulablty. Exact schedulablty analyss for cooperatve fxed prorty schedulng was derved by Brl et al. n 29 [17]. Snce then, Bertogna et al. have ntegrated preepton costs nto analyss of the fxed odel, consderng both fxed [13] and varable [14] pre-epton costs, for both EDF and fxed schedulng. Bertogna et al. [15], also derved a ethod of coputng the optal length of the fnal non-preeptve regon of each task n order to axze schedulablty for a gven prorty assgnent. Fnally, n 212, Davs and Bertogna [29] ntroduced an optal algorth that s able to fnd a schedulable cobnaton of prorty assgnent and fnal-non-pre-eptve regon lengths whenever such a schedulable cobnaton exsts. Ths algorth proves upon the schedulablty that can be obtaned by ether fully pre-eptve or fully non-pre-eptve fxed prorty schedulng.

9 In the floatng odel of non-pre-eptve regons, ntroduced by Baruah n 25 [12] for EDF schedulng, an upper bound s gven on the length of the longest non-pre-eptve regon of each task. However, the locaton of each non-pre-eptve regon s not known a pror and ay vary at run-te, for exaple under the control of the operatng syste. Analyss for the floatng odel under fxed prorty schedulng was gven by Yao et al. [72]. Alternatve approaches to ltng pre-epton nclude Preepton Thresholds (FPTS) [69], [61] and Non-pre-epton Groups [28] n whch each task has a base prorty at whch t copetes for the processor, and a threshold prorty at whch t executes, thus ltng pre-epton to those tasks wth a base prorty hgher than the threshold. In [69], [61] Saksena and Wang attepted to derve an ntegrated approach to prorty and pre-epton threshold assgnent, but dd not succeed n fndng an optal algorth wth less than exponental coplexty. Research by Brl et al. [18] n 212 cobnes the deas of deferred pre-epton and pre-epton thresholds, generalsng both nto a sngle schee wth pre-epton thresholds between a set of sub-jobs whch execute non-preeptvely. For further nforaton on lted pre-epton schedulng the reader s referred to the survey n [22]. 6.2 Accountng for pre-epton costs There s a runte overhead when the processor swtches fro the executon of one job to another. There are varous coponents that ake up ths overhead. These nclude: o The cost of scheduler operatons (e.g. run-queue anpulaton, ter nterrupts etc.). o The cost of the context swtch fro one task to another (savng and restorng regsters etc.). o For processors wth cache, the cache related preepton delay (CRPD). Wth careful RTOS desgn, the frst two coponents can often be tghtly bounded and ether ncluded n the executon te of each task, or odelled as vrtual hgh prorty tasks. Further, the addtonal blockng effects caused by the operatng syste can also be odelled and ncluded n the analyss see for exaple the work of Whtha et al. [7]. Cache related preepton delays can however be a substantal proporton of a tasks executon te, and are hghly dependent on both the pre-eptng task and the pre-epted task. Cache s fast local eory, whch s used to store nstructons (and data) provng access tes and hence brdgng the large gap that exsts between the speed of an eory and the speed of hgh perforance processors. Meory requests that can be servced by the cache can be sgnfcantly faster (1 to 1 tes faster n soe cases) than accesses that have to go to an eory. However, cache s typcally uch saller than an eory and so not all of the nstructons and data needed by all tasks can be held n cache at the sae te. Analyss of cache related pre-epton delays uses the concept of useful cache blocks (UCBs) and evctng cache blocks (ECBs) based on the work by Lee et al. [52]. The ECBs for a task are the cache blocks that ay be loaded nto cache by the task durng ts executon. Whereas, UCBs are blocks that are not only loaded nto cache by the task, but are certan to be reused before potentally beng evcted by the task, not countng evctons fro pre-eptng tasks. If a UCB s evcted by a pre-eptng task, then an addtonal delay ay be ntroduced f the UCB needs to be re-loaded when t otherwse would not have been. Ths s tered a cache related preepton delay. Dependng on the approach used, CRPD analyss cobnes nforaton about the UCBs belongng to the pre-epted task(s) wth the ECBs of the pre-eptng task(s). Usng ths nforaton, the total nuber of cache blocks that are evcted, whch ust then be reloaded after pre-epton, can be calculated and cobned wth the cost of reloadng a block to gve the CRPD. A nuber of approaches have been developed for calculatng the CRPD under pre-eptve fxed prorty schedulng. These nclude the UCB-Only approach of Lee et al. [52], whch consders only the pre-epted task(s), and the ECB-Only approach of Busquets et al. [21] whch consders only the preeptng task. Approaches that consder both pre-epted and pre-eptng tasks nclude the UCB-Unon approach of Tan and Mooney [66] and the ECB-Unon approach of Alteyer et al. [1] and ther ult-set extensons [3]. Untl recently, there has been sgnfcantly less work n developng CRPD analyss for pre-eptve EDF schedulng. In 27, Ju et al. [49] consdered the ntersecton of the preepted task s UCBs wth the pre-eptng task s ECBs; however, ths ethod has sgnfcant pesss as each par of tasks s consdered separately. In 213, Lunnss et al. [58] adapted the state-of-the-art approaches [1], [3] for calculatng CRPD for fxed prorty schedulng to work wth EDF. Cache related pre-epton delays can have a sgnfcant effect on schedulablty. They can vary substantally dependng on how the code of tasks s lad out n eory and hence how t s apped to the cache, thus provdng scope for optsng schedulablty va anpulatng the poston of tasks n eory [59]. One way of elnatng CRPDs s to partton the use of cache so that each task has ts own cache partton, hence ensurng that the ECBs of a pre-eptng task do not overlap wth the UCBs of any pre-epted task. However, such parttonng coes at a prce. The saller cache partton allocated to each task ncreases the non-pre-eptve WCET of the task. Recent work by Alteyer at el. [2] has nvestgated ths trade-off concludng that n any cases, even optal cache parttonng s less effectve than sharng the cache and accountng for CRPD va ntegrated analyss [1], [3]. 7 SUMMARY AND CONCLUSIONS In ths paper, we have provded a gude to the fundaental schedulablty analyss results for pre-eptve and non-preeptve, fxed prorty and EDF, schedulng of sporadc task sets. We have also brefly covered extensons to ths work that relax the rather unrealstc assupton that schedulng and preepton costs are neglgble. In real systes, partcularly those runnng on processors wth cache, the costs of pre-epton can be relatvely hgh when copared wth the non-pre-eptve WCETs of the tasks. Thus technques for ltng pre-epton are needed, as well as analyss that accounts for the cost of pre-epton overheads. 7.1 Lookng back: success stores Despte the advent of ultcore processors and substantal research nto ultprocessor schedulng, research nto realte schedulng on sngle core systes stll thrves. One of the hardest challenges n ths area s to progress fro sple

10 odels of the syste and tasks to ore coplex odels that capture the portant behavours of real systes and applcatons. A good exaple of where ths has happened and a ajor success story for research nto schedulablty analyss s Controller Area Network. Ths s a sple broadcast network used to connect Electronc Control Unts n autootve systes. (CAN s used n the vast ajorty of cars anufactured today, wth sales of CAN enabled croprocessors approachng 1 bllon per year). Fro the perspectve of analysng essage response tes, the broadcast behavour of CAN eans that ts analyss resebles that of a sngle processor wth fxed prorty non-pre-eptve schedulng. Precse schedulablty analyss has been derved for Controller Area Networks [36] that akes certan assuptons n ters of the behavour of the nodes connected to the bus (perfect prorty queues); however, n practce these assuptons ay not always hold. Recent work has sort to address partcular features of the hardware such as nonabortable transt requests [51], and the fact that FIFO queung polces are often used [3], [31]. These extensons ake the analyss ore flexble and better suted to coercal use. 7.2 Lookng ahead: research drectons Wthn the thee of sngle processor real-te schedulng, hot research topcs today nclude: () () () The ntegraton of schedulablty analyss, and WCET analyss, and the developent of ntegrated technques such as lted pre-epton to prove schedulablty (see Secton 6). Schedulng of Mxed Crtcalty Systes based on fxed prorty [68], [9] and EDF [42]. Probablstc approaches to schedulablty analyss [57] and WCET analyss [25], [24], and ther ntegraton [27]. 8 ACKNOWLEDGEMENTS Ths work was funded n part by the EPSRC project MCC (EP/K11626/1). The author would lke to thank Wll Lunnss for hs help wth the background ateral on CRPD analyss. 9 REFERENCES [1] S. Alteyer, R.I. Davs, C. Maza Cache related Preepton Delay aware response te analyss for fxed prorty pre-eptve systes. In proceedngs IEEE Real- Te Systes Syposu (RTSS), pages , 211. [2] S. Alteyer, Roeland Doua, W. Lunnss, R.I. Davs, Evaluaton of Cache Parttonng for Hard Real-Te Systes. Under subsson to IEEE Real-Te Systes Syposu (RTSS) 213. [3] S. Alteyer, R.I. Davs, C. Maza Iproved cache related pre-epton delay aware response te analyss for fxed prorty pre-eptve systes. Real-Te Systes, Volue 48, Issue 5, pages , Sept 212 [4] N.C. Audsley, "Optal prorty assgnent and feasblty of statc prorty tasks wth arbtrary start tes", Techncal Report YCS 164, Dept. Coputer Scence, Unversty of York, UK, [5] N.C. Audsley, A. Burns, M. Rchardson, A.J. Wellngs, Applyng new Schedulng Theory to Statc Prorty Preeptve Schedulng. Software Engneerng Journal, 8(5), pages , [6] N.C. Audsley, A. Burns, R.I. Davs, K. W. Tndell and A. J. Wellngs, Fxed Prorty schedulng an Hstorcal perspectve. Real-Te Systes 8(3), pages [7] N.C. Audsley On prorty assgnent n fxed prorty schedulng, Inforaton Processng Letters, 79(1): 39-44, May 21. [8] T.P. Baker Stack-based Schedulng of Real-Te Processes. Real-Te Systes Journal (3)1, pages [9] S.K. Baruah, A. Burns, R.I. Davs Response Te Analyss for Mxed Crtcalty Systes. In proceedngs 32nd IEEE Real-Te Systes Syposu (RTSS'11), pages 34-43, Nov 29th - Dec 2nd, 211. [1] S.K. Baruah, A.K. Mok, L.E. Roser, Preeptvely Schedulng Hard-Real-Te Sporadc Tasks on One Processor. In proceedngs IEEE Real-Te Systes Syposu (RTSS), pages182-19, 199. [11] S.K. Baruah, L.E. Roser, R.R. Howell, Algorths and Coplexty Concernng the Preeptve Schedulng of Perodc Real-Te Tasks on one Processor. Real-Te Systes, 2(4), pages , 199. [12] S.K. Baruah. The lted-preepton unprocessor schedulng of sporadc task systes. In Proceedngs Eurocro Conference on Real-Te Systes (ECRTS), pages , 25. [13] M. Bertogna, G. Buttazzo, M. Marnon, G. Yao, F. Esposto, M. Caccao. "Preepton ponts placeent for sporadc task sets", In proceedngs Eurocro Conference on Real-Te Systes (ECRTS), 21. [14] M. Bertogna, O. Xhan, M. Marnon, F. Esposto, G. Buttazzo. "Optal Selecton of Preepton Ponts to Mnze Preepton Overhead", In proceedngs Eurocro Conference on Real-Te Systes (ECRTS), 211. [15] M. Bertogna, G. Buttazzo, G. Yao. "Iprovng Feasblty of Fxed Prorty Tasks usng Non-Preeptve Regons", In proceedngs Real-Te Systes Syposu (RTSS), 211. [16] E. Bn, G.C. Buttazzo, Measurng the Perforance of Schedulablty Tests, Real-Te Systes 3 (1-2), pages , 25. [17] R.J. Brl, J.J Lukken, W.F. Verhaegh, Worst-case response te analyss of real-te tasks under fxedprorty schedulng wth deferred pre-epton. Real- Te Systes. 42, 1-3, pages , 29. [18] R.J. Brl, M.M.H.P. van den Heuvel, U. Keskn, J.J. Lukken, Generalzed fxed-prorty schedulng wth lted preeptons, In proceedngs Eurocro Conference on Real-Te Systes (ECRTS), pages 29-22, 212. [19] A. Burns, A.J. Wellngs, Real-Te Systes and Prograng Languages: Ada 95, Real Te Java and Real Te Posx, Addson Wesley, 21.