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-Tme Unprocessor Systems Robert I. Davs Real-Tme Systems Research Group, Department of Computer Scence, Unversty of York, York, UK. ABSTRACT Ths paper was wrtten to accompany a talk at the ETR Summer School n Toulouse 213. It provdes a background and prmer to schedulng and schedulablty analyss for hard real-tme sngle processor systems. The paper focuses on the two man schedulng algorthms used: Fxed Prorty and Earlest Deadlne Frst. The paper has two broad ams: Frstly to provde a gude to the fundamental results for these realtme schedulng algorthms. Secondly to provde a bref revew of extensons amed at () lmtng the effects and overheads of pre-empton and () accountng for the effects of preemptons; specfcally cache related pre-empton delays. The paper also brefly looks back at success stores n real-tme schedulng, and forwards at the current hot topcs n ths research area. Categores and Subject Descrptors C.3 [Specal-Purpose and Applcaton-Based Systems]: Realtme and embedded systems. General Terms Algorthms, Performance, Theory, Verfcaton. Keywords Real-tme; schedulng; schedulablty analyss; fxed prorty; EDF; sngle processor; unprocessor; survey; lmted preempton; cache-related pre-empton delays. 1 INTRODUCTION Ths paper was wrtten to accompany a talk at the ETR Summer School n Toulouse 213. It ams to provde a bref gude and revew of schedulng and schedulablty analyss for hard real-tme unprocessor systems. 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 prmer coverng the fundamental results, complemented by a bref revew hghlghtng two nterestng topcs or themes that buld upon these foundatons. For the nterested reader, we provde references to further readng n the form of books, surveys, and the orgnal sources. We hope that readers wll be curous to fnd out exactly how some of the technques mentoned, but not covered n detal, actually work and so go on to read the cted lterature. 2 ADDITIONAL READING Burns and Wellngs Book Real-Tme Systems and Programng Languages: Ada 95, Real Tme Java and Real Tme Posx [19] provdes a useful ntroducton to real-tme 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-Tme Schedulng Theory A Hstorcal Perspectve [62], both publshed n Real-Tme Systems whch s the man journal coverng ths feld. An up-to-date source of further readng can be found on the IEEE Techncal Commttee on Real-Tme Systems (TCRTS) webste va the lst of semnal papers recommended for educaton (see 3 BACKGROUND Real-tme systems are charactersed not only by the need for functonal correctness, but also the need for temporal or tmng correctness. Real-tme systems contnually montor and respond to stmul from the envronment and the physcal systems that they control. In order for such systems to behave correctly, they must not only execute the correct computatons, but also do so wthn predefned tme constrants. These tme constrants are typcally expressed n terms of end-to-end deadlnes on the elapsed tme between a stmul and the correspondng response. Applcatons n real-tme systems may be classfed as hard real-tme, where falure to meet a deadlne consttutes a falure of the applcaton; or soft realtme, where latency n excess of the deadlne leads only to a degraded qualty of servce. Today, hard real-tme systems are found n many dverse applcaton areas ncludng; automotve electroncs, avoncs, space systems, medcal systems, household automaton, and robotcs. Many real-tme systems are composed of tasks. Tasks are software components (programs) that execute n response to some stmulus and output a response. Often the stmulus s perodc (e.g. tme-drven) thus a perodc task mght execute once every 5ms. Tasks can also be event-drven from some stmulus that does not necessarly occur wth a fxed perod, but nstead has some mnmum nter-arrval tme, for example an nterrupt that occurs from 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 Tme constrants n a real-tme system typcally put a lmt on the maxmum permtted response tme from the arrval of the event (or tmer tck) that subsequently leads to the release of a job of a task by the operatng system, untl the job completes ts executon and outputs ts response. Ths tme lmt s the relatve deadlne (or just deadlne) of the task. The arrval tme of the trggerng event or tmer tck plus the relatve deadlne of the task gves the absolute deadlne of the job. More complex behavours may be mplemented such that the tme constrant s from the stmulus for one task, through to the response of a dfferent task. Ths tme nterval s referred to as the end-to-end response tme and t must meet the correspondng end-to-end deadlne f the system s to operate correctly. In a sngle processor system, the processor can only run a sngle task at any gven tme, hence there s an mportant choce to be made as to whch job to run when there s more than one job that s ready to execute. The schedulng polcy employed determnes ths choce, whch s then mplemented by the scheduler, whch s part of the real-tme operatng system. Schedulng polces may be classfed accordng to whether they make schedulng decsons onlne (dependng on the current state of the actve jobs) or offlne (usng a precomputed table). The latter approach s referred to as statc cyclc schedulng. Statc cyclc schedulng was used n early real-tme systems, and s stll used today n very smple systems, and also n many safety crtcal systems (e.g. arcraft flght control systems). Statc cyclc schedulng lacks flexblty and has a number of other dsadvantages stemmng from the use of pre-computed tables. For these reasons n has been supplanted n many real-tme systems by onlne schedulng polces. In ths paper, we focus soley on onlne schedulng and n partcular two schedulng polces that both use prortes to determne 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 from 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 algorthms such as Shortest Remanng Processng Tme Frst (SRPT) [4], [64] and Least Laxty Frst (LLF) [6] have dynamc job prortes that change as each job executes. Schedulng polces are classfed as pre-emptve, f they allow an already runnng job to be suspended so that another (hgher prorty) job may execute, or non-pre-emptve f they do not. Thus there are pre-emptve and non-pre-emptve varants of both fxed prorty and EDF schedulng. In hard real-tme systems, tmng correctness (meetng all deadlnes) s an essental aspect of correct system behavour, hence t s mportant to be able to ensure before the system s deployed that all deadlnes wll be met durng operaton 1 1 In practce often what s requred s an assurance that there s only a very small probablty that deadlnes wll be mssed. Probablstc analyss whch seeks to address ths more refned vew s the topc of another talk at ETR 213. assumng that the system behaves as expected (.e. no faults, correct functonal behavour etc.). Schedulablty analyss s the name gven to a mathematcal approach used to determne offlne f deadlnes can be guaranteed to be met at runtme. Schedulablty analyss requres nformaton 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 tme a job of each task could take to execute non-pre-emptvely, referred to as the worst-case executon tme (WCET), and any nteractons between the tasks (e.g. mutually exclusve access to shared resources). () The schedulng polcy used (e.g. fxed prorty) and any modfcatons to ths (e.g. when jobs may temporarly execute at a hgher prorty, due to nonpre-emptve behavour, or when accessng shared resources etc.). In ths paper, we wll assume 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 problem of determnng an accurate estmate of the WCET s a dffcult one, partcularly for processors wth hardware acceleraton features such as ppelnes and caches [71]. We now ntroduce some termnology that s used n research focused on schedulng and schedulablty analyss. We defne what s meant by schedulable and feasble task sets, optmal schedulng algorthms and optmal prorty assgnment polces, and dfferent types of schedulablty test, ncludng exact ones. We say that a task set s schedulable wth respect to some schedulng algorthm, f all vald sequences of jobs that may be generated by the task set can be scheduled by the schedulng algorthm wthout any mssed deadlnes. Further, we say a task set s feasble f there exsts some schedulng algorthm that can schedule all possble sequences of jobs that may be generated by the task set wthout mssng any deadlnes. A schedulng algorthm s sad to be optmal wth respect to a task model f t can schedule all of the feasble task sets that comply wth the model. A schedulablty test s sad to be suffcent, wth respect to a schedulng algorthm, f all of the task sets that are deemed schedulable accordng to the test are n fact schedulable under the schedulng algorthm. Smlarly, a schedulablty test s sad to be necessary, f all of the task sets that are deemed unschedulable accordng to the test are n fact unschedulable under the schedulng algorthm. A schedulablty test that s both suffcent and necessary s referred to as exact. Thus our am n schedulablty analyss research s to fnd effcent, exact schedulablty tests; however, there s typcally a tradeoff between the computatonal complexty of the tests and ther effectveness. Hence sometmes we need to use effcent tests that only gve suffcent results rather than exact tests that take too long to compute. In fxed prorty schedulng, a prorty assgnment polcy P s sad to be optmal wth respect to some class of task sets (e.g. those wth arbtrary deadlnes), and some class of fxed prorty schedulng algorthm (e.g. pre-emptve) f there are no task sets n the class that are schedulable under the schedulng

3 algorthm usng any other prorty orderng polcy that are not also schedulable usng the prorty assgnment determned by polcy P. Usng an optmal prorty assgnment polcy we can schedule any task set that s schedulable usng any other prorty assgnment polcy; hence n analyss of fxed prorty systems, we always am to fnd and use optmal prorty assgnment polces. The rest of the paper s organsed as follows. In Secton 4, we outlne the fundamental research nto schedulablty analyss for sngle processor systems usng fxed prorty and EDF schedulng. In Secton 5 we provde a formal taskng model and descrbe general forms 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 themes that buld upon the fundamental schedulablty analyss research: lmtng pre-empton, and accountng for ts costs. Secton 7 concludes the paper wth a bref look at success stores for real-tme schedulablty analyss and hot topcs of future research. 4 FIXED PRIORITY AND EDF SCHEDULING In ths secton, we provde an overvew of the fundamental work on schedulablty analyss for sngle processor systems usng fxed prorty and EDF schedulng. 4.1 Pre-emptve schedulng In 1973, Lu and Layland [56] consdered fxed prorty preemptve schedulng of synchronous 2 task sets comprsng ndependent perodc tasks, wth bounded worst-case executon tmes, and deadlnes equal to ther perods (so called mplctdeadlnes). Lu and Layland showed that rate monotonc prorty orderng, where tasks are assgned prortes n order of ther perods, wth a shorter perod mplyng a hgher prorty, s the optmal prorty assgnment polcy for such task sets. They further proved that that usng rate monotonc prorty orderng, fxed prorty pre-emptve schedulng can schedule any mplct-deadlne task set that has a total utlsaton U ln( 2). 693, where the utlsaton of a task s ts worst-case executon tme dvded by ts perod, and the total utlsaton of the task set s smply the sum 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-emptve EDF s an optmal sngle processor schedulng algorthm, n the sense that f a vald schedule exsts for a task set, then the schedule produced by EDF wll also meet all deadlnes. Least Laxty Frst [6] s another such optmal algorthm. Research nto real-tme schedulng durng the 198 s and early 199 s focused on lftng many of the restrctons of the Lu and Layland task model. Task arrvals were permtted to be sporadc, wth known mnmal nter-arrval tmes, (stll slghtly confusngly referred to n the lterature as perods), and task deadlnes were permtted to be less than or equal to ther perods (so called constraned deadlnes) or less than, equal to, or greater than ther perods (so called arbtrary deadlnes). In 1982, Leung and Whtehead [53] showed that deadlne monotonc prorty orderng, where tasks are assgned prortes n order of ther deadlnes, wth a shorter deadlne mplyng a hgher prorty, s the optmal prorty orderng for constraned-deadlne task sets. Exact schedulablty tests for pre-emptve 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 monotonc prorty orderng s not optmal for task sets wth arbtrary deadlnes; however, an optmal prorty orderng can be determned n ths case, n at most n ( n +1) / 2 task schedulablty tests usng Audsley s optmal prorty assgnment (OPA) algorthm 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 problem 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 may follow a perodc or sporadc arrval pattern, but do not necessarly become ready to execute mmedately, nstead ther release may be delayed by up to a maxmum tme referred to as the Release Jtter of the task. Ths s a typcal behavour n real systems where there may be some 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 mnus Jtter monotonc prorty orderng s optmal. Exact schedulablty tests for constraned and arbtrarydeadlne task sets scheduled usng pre-emptve EDF were ntroduced by Baruah et al. [1], [11] n 199. Subsequently, exact tests for EDF have been developed by George and Hermant [43] and Zhang et al. [73], [74] that are more effcent n practce. 4.2 Resource sharng In pre-emptve systems, mechansms are requred to ensure that jobs access shared resources n mutual excluson, otherwse data corrupton, or erroneous behavour of hardware devces could ensue. If smple semaphores are used to seralse resource access, then unbounded prorty nverson can occur. Prorty nverson s the name gven to the stuaton where a hgh prorty job attemptng to lock a resource has to wat not only for the low prorty job that currently holds the resource, but also any jobs of ntermedate prorty that may pre-empt t. A number of concurrency control protocols have been developed whch avod ths problem. 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 most effectve protocol and has been adopted by the OSEK and Autosar operatng system standards. The Stack Resource Polcy [8] assocates a pre-empton level wth each resource. In EDF (FP) schedulng, ths pre-empton 2 A task set s referred to as synchronous f all of the tasks can share a common release tme. 3 Ths algorthm s optmal 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-tme, 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-empton level of the resource. On unlockng the resource the job returns to ts prevous prorty (obtaned from the stack). Thus SRP allows resource lockng to be properly nested. The Stack Resource Polcy [8] has a number of mportant propertes: o Once a job starts executng t never has to wat to access a resource. Ths means that wth SRP, resource accesses add no addtonal context swtches, and sngle stack executon s possble savng memory. o No job j can be delayed from executng by lower preempton level jobs for longer than the maxmum sngle resource access tme of any such lower prorty job, where the resource s shared wth other jobs of the same or a hgher pre-empton level than job j. o The system s deadlock free. These propertes strctly bound the blockng tme 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] modfed 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-emptve schedulng In 198, Km and Naghbdadeh [5], and n 1991, Jeffay et al. [47], gave exact schedulablty tests for mplct-deadlne task sets under non-pre-emptve 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-emptve EDF s an optmal sngle processor schedulng algorthm, n the non-pre-emptve case no workconservng algorthm s optmal. (A work-conservng schedulng algorthm s one that does not dle the processor when there are tasks ready to be executed). Ths s because n general, wth non-pre-emptve schedulng, t s necessary to nsert dle tme to acheve a feasble schedule. The nterested reader s referred to [44] for examples of ths behavour. In 1995, Howell and Venkatrao [46] showed that for nonconcrete strctly perodc task sets, where the tmes at whch each task may be frst released are unknown, the problem of determnng a feasble non-pre-emptve schedule s NP hard. Further, they showed that for sporadc task sets, no optmal onlne nserted dle tme algorthm can exst. In other words, clarvoyance s needed to determne a feasble non-preemptve schedule. Whle no work-conservng algorthm s optmal n the strong sense that t can schedule any task set for whch a feasble nonpre-emptve schedule exsts; n 1995, George et al. [45] showed that non-pre-emptve EDF s optmal n the weak sense that t can schedule any task set for whch a feasble work-conservng, non-pre-emptve schedule exsts. For fxed prorty non-pre-emptve 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-emptve case. George et al. showed that unlke n the pre-emptve case, deadlne monotonc prorty orderng s not optmal for constraned-deadlne task sets scheduled nonpre-emptvely. Further, they showed that Audsley s optmal prorty assgnment algorthm [7] s applcable, and can be used to determne an optmal prorty orderng n ths case. Subsequent research by Brl et al. [17] has refned exact analyss of non-pre-emptve fxed prorty schedulng, correctng ssues of both pessmsm and optmsm, and extendng the schedulablty tests to co-operatve schedulng where each task s made up of a number of non-pre-emptve regons. There are varous approaches that allow lmted preempton and thus represent a compromse between fully preemptve and fully non-pre-emptve schedulng. We return to these n Secton 6. 5 SCHEDULABILITY TESTS In ths secton we frst descrbe the sporadc task model, whch s commonly used n real-tme schedulng research, and forms the bass for the schedulablty analyss that we then recaptulate. We also provde detals of the defacto standard notaton sometmes referred to as Burns Standard Notaton [32], whch s used n ths area. 5.1 Task model and notaton We consder the schedulng of a set of sporadc tasks (or task se on a sngle processor. Each task set comprses a statc set of n tasks ( τ 1.. τ n ), where n s a postve nteger. Wthout loss of generalty, we assume that the ndex of task τ also represents the task prorty used n fxed prorty schedulng, hence τ 1 has the hghest fxed-prorty, and τ n the lowest. We use the notaton hp () (and lp () ) to mean the set of tasks wth prortes hgher than (lower than), and the notaton hep () (and lep () ) to mean the set of tasks wth prortes hgher than or equal to (lower than or equal to). Each task τ s characterzed by ts bounded worst-case executon tme C, mnmum nter-arrval tme or perod T, relatve deadlne D, and release jtter J. Each task τ therefore gves rse to a potentally unbounded sequence of jobs, each of whch has an executon tme upper bounded by C, an arrval tme at least T after the arrval of the prevous job of the same task, and an absolute deadlne that s D after ts arrval. Due to release jtter, release of each job may be delayed for a varable amount of tme from to J after ts arrval. In an mplct-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 mplct deadlne task sets. The worst-case response tme R of a task τ s gven by the longest possble tme from the arrval of the task untl t completes executon. Thus task τ 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 tme dvded by ts perod ( U = C / T ). The total utlzaton U, of a task set s the sum of the utlzatons of all of ts tasks.

5 The followng assumptons are made about the behavour of the tasks: o The arrval tmes of the tasks are ndependent and unknown a pror, hence the tasks may share a common release tme. o The tasks may access mutually exclusve shared resources accordng to the Stack Resource Polcy [8]. Thus each task τ may be subject to a maxmum amount of blockng B that equates to the longest tme that a job of a task of lower prorty (pre-empton level) may lock a resource that s shared wth tasks of the same or hgher prorty (pre-empton level) than task τ. o The tasks are otherwse ndependent, and do not voluntarly suspend themselves. A job s sad to be ready f t has outstandng computaton awatng executon by the processor. Under pre-emptve EDF schedulng, at any gven tme the ready job wth the earlest absolute deadlne s selected for executon by the processor. In contrast, under pre-emptve fxed prorty schedulng, at any gven tme the earlest released unfnshed job of the hghest prorty ready task s selected for executon. Smlarly, under non-pre-emptve EDF, at any gven tme 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-emptve fxed prorty schedulng, at any gven tme 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-emptve FP schedulng We now gve a summary of Response Tme Analyss [5] used to provde an exact schedulablty test for fxed prorty preemptve schedulng of constraned-deadlne task sets. We then recaptulate the somewhat more complex response tme analyss for arbtrary-deadlne task sets [67]. Frst, we ntroduce the concepts of a crtcal nstant, and busy perods, whch are fundamental to response tme analyss. The crtcal nstant for a task τ s defned as a pont n tme when the arrval of a job of that task wll have the longest possble response tme, gven any vald pattern of arrvals and executon behavour of the tasks n the task set. The term prorty level- busy perod refers to a perod of tme [ 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-emptve schedulng, the crtcal nstant for a task τ occurs wthn the longest prorty level- busy perod, whch s assocated wth an arrval sequence such that the frst jobs of task τ and all hgher prorty tasks are released smultaneously after ther maxmum permtted release jtter, and subsequent jobs of the same tasks are released as soon as possble thereafter. For tasks wth arbtrary deadlnes and large release jtter, ths means that more than one job of the task may be released at the same tme. In ths case, we assume that jobs of the same 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 assume that later arrvng jobs of the same task effectvely have lower prorty, and are executed after earler arrvals. Fnally, we assume that just pror to the ntal smultaneous release, a lower prorty task locks the resource resultng n the maxmum blockng tme B. For a constraned deadlne task τ, the worst-case response tme R corresponds to ts release jtter plus the length of the prorty level- busy perod descrbed above. The busy perod comprses three components, () the blockng tme B, () the executon tme of the task tself C, and () so called nterference, equal to the tme for whch task τ s prevented from executng by all hgher prorty tasks. The length of the busy perod w, can be computed usng the followng fxed pont teraton [5], wth the summaton term gvng the nterference due to the set of hgher prorty tasks hp(). m w + J m+ 1 j w = B + C + C j (1) j hp( ) T j Iteraton starts wth an ntal value w, typcally m+1 m w = B + C, and ends when ether w = w n whch case m+1 the worst-case response tme R s gven by w + J, or m+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 may be speeded up usng the technques descrbed n [33]. Equaton (1) provdes an exact schedulablty test for the fxed prorty pre-emptve schedulng of constraned-deadlne task sets wth any fxed prorty orderng. For fxed prorty pre-emptvely scheduled systems, where task deadlnes are arbtrary, executon of one job of a task may not necessarly be completed before the next job s released. Hence a number of jobs of task τ may 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 tme for task τ occurs for some 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: m L + J m+ 1 j L = B + C j (2) j hep( ) T j Iteraton starts wth an ntal value guaranteed to be no larger than the mnmum soluton, for example L = C, and ends m+1 m when L = L. The number of jobs Q of task τ n the busy perod s gven by: L + J Q = (3) T In general t s necessary to compute the response tmes of all jobs of a task τ wthn the longest prorty level- busy perod n order to determne the task s worst-case response tme. The completon tme W, q of the qth job (where q = s the frst job) of task τ, wth respect to the start of the busy perod, s gven by the followng fxed pont teraton:

6 m w + + 1, q J m j w, q = B + ( q + 1) C + C j (4) j hp( ) T j Iteraton starts wth an ntal value w,q, typcally m+ 1 m w, q = B + ( q + 1) C, and ends when ether w, q = w, q n + 1 whch case,, = m m+1 W q w, q or when w, q qt > D J n whch case job q, and hence task τ s unschedulable. To fnd the worst-case response tme of task τ, completon tmes W, q need to be calculated for jobs q =,1,2,3,... Q 1. The worst-case response tme of task τ s then gven by: p R = max q=,1,2... Q 1( W, q qt + J ) (5) Task τ s schedulable provded that R D. Equatons (4) and (5) gve an exact schedulablty test for the pre-emptve fxed prorty schedulng of arbtrary-deadlne task sets wth any specfed prorty orderng. 5.3 Analyss for non-pre-emptve FP schedulng We now recaptulate Response Tme Analyss used to provde an exact schedulablty test for fxed prorty non-pre-emptve schedulng of arbtrary deadlne task sets. Ths analyss has smlartes to the pre-emptve case, but also some key dfferences. George et al. [44] and Brl et al. [17] showed that for fxed prorty non-pre-emptve schedulng, the longest response tme of a job of task τ occurs for some job of that task wthn a prorty level- busy perod startng wth the smultaneous release of jobs of task τ and all hgher prorty tasks that have been subject to the maxmum release jtter, wth subsequent jobs of those tasks released as soon as possble. Further, the mnmum possble amount of tme pror to the ntal smultaneous release, a lower prorty task τ k s released, and ths task has the longest executon tme of any such lower prorty task. (Note that n the followng, we assume dscrete tme and so = 1). The worst-case length of the prorty level- busy perod s agan gven by the mnmum soluton to (2), wth B defned to be the longest tme that task τ and any hgher prorty tasks can be blocked from executng by the non-pre-emptve executon of a lower prorty task: B = max ( C 1) (6) k lp( ) k Note n (6) the maxmum of an empty set s assumed to be zero. The number of jobs Q of task τ n the actve perod s gven by: L + J Q = (7) T The start tme W, q of the qth job (where q = s the frst job) of task τ measured wth respected to the ntal smultaneous release s gven by the mnmum soluton to the followng fxed pont teraton: m + w m 1, q w, q = B + qc C j (8) j hp( ) T j Iteraton starts wth an ntal value w,q, typcally m+ 1 m w, q = B + qc, and ends when ether w, q = w, q n whch + 1 case, = m m+1 W q w, q, or when: w, q + C qt > D J n whch case job q, and hence task τ s unschedulable. To fnd the worst-case response tme, the start tmes W, q need to be calculated for jobs q =,1,2,3,... Q 1. The worst-case response tme of task τ s then gven by: P R = max NP ( W, ) q,1,2... Q 1 q + C qt + J (9) = Task τ s schedulable provded that R D. Equatons (8) and (9) gve an exact schedulablty test for nonpre-emptve fxed prorty schedulng of arbtrary-deadlne task sets wth any specfed prorty orderng. The exact analyss for both pre-emptve and non-pre-emptve fxed prorty schedulng s pseudo-polynomal n complexty. In practce, the technques descrbed above based on fxed pont teraton are fast enough to be used to determne task set schedulablty, and even to perform senstvty analyss by varyng task parameters and determnng f the task set remans schedulable. In cases where a sgnfcant trade-off between accuracy and speed s requred, then Davs and Burns [35] provde response tme upper bounds for both pre-emptve and non-pre-emptve cases that are lnear n complexty and can be computed n O(n) tme for all n tasks n a task set. 5.4 Prorty assgnment The schedulablty tests gven above for both fxed prorty pre-emptve and non-pre-emptve schedulng are agnostc as far as the prorty assgnment polcy s concerned; they work wth any gven assgnment. However, prorty assgnment can have a very large effect on schedulablty as shown by Davs et al. [3], hence t s mportant whenever possble to use an optmal prorty assgnment polcy. Audsley s Optmal Prorty Assgnment (OPA) algorthm [4], [7], reproduced below, provdes an optmal prorty assgnment for arbtrary-deadlne sporadc task sets wth respect to both pre-emptve and non-pre-emptve fxed prorty schedulng. Ths greedy approach requres at most n(n+1)/2 task schedulablty tests to determne a schedulable prorty orderng whenever such an orderng exsts. Ths compares favourably wth the n! tests that would otherwse be requred, assumng a brute force approach that checks every possble combnaton of prortes. for each prorty level k, lowest frst { for each unassgned task τ { f(τ s schedulable at prorty k wth all other unassgned tasks assumed to have hgher prortes) { assgn τ to prorty k break (contnue outer loop) return unschedulable return schedulable Algorthm 1: Optmal Prorty Assgnment Algorthm Whle t provdes an optmal prorty assgnment, the OPA algorthm has one sgnfcant drawback: t does not make any choce about prorty orderng other than to ensure schedulablty. Thus usng the OPA algorthm the prorty assgnment obtaned can leave the task set of the edge of unschedulablty, wth any small ncrease n the executon tme of a task resultng n a deadlne mss. Davs and Burns addressed

7 ths ssue wth the Robust Prorty Assgnment algorthm (RPA) [34]. Ths algorthm s based on Audsley s OPA algorthm; however, t makes use of a scalng factor α whch s used to characterse addtonal nterference. Ths addtonal nterference may take a very general form provded that t s monotoncally ncreasng wth the prorty level, and the tme nterval consdered. For example the scalng factor α mght characterse an ncrease n the executon tme or one, or all of the tasks, or of some addtonal nterference at the hghest prorty level. Usng an approprate choce of addtonal nterference, the RPA algorthm provdes a prorty assgnment that s not only optmal, but also as robust as possble to ncreases n nterference. The RPA algorthm s shown below. for each prorty level k, lowest frst { for each unassgned task τ { bnary search for the largest value of α for whch task τ s schedulable at prorty k f no tasks are schedulable at prorty k { return unschedulable else { assgn the schedulable task that tolerates the max α at prorty k to prorty k return schedulable Algorthm 2: Robust Prorty Assgnment Algorthm For more nformaton on prorty assgnment, the nterested reader s drected to the presentaton that accompaned the keynote talk Gettng ones prortes rght at RTNS 212 [26]. 5.5 Analyss for pre-emptve EDF We now recaptulate schedulablty analyss for arbtrarydeadlne task sets under pre-emptve EDF. In the analyss of EDF schedulng, we use the concept of processor demand. bound. Ths s defned for an nterval of length t, as the maxmum possble processor demand from jobs wth both ther release tmes and ther deadlnes n the nterval. A necessary condton for schedulablty s that for any nterval of length t, the processor demand bound must not exceed the length of the nterval. Otherwse the processor could not possbly complete 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 demand bound functon h( s gven by: n t J D h t + ( ) = max, + C 1 (1) = T 1 A task set s schedulable under pre-emptve EDF f and only f U 1 and the processor LOAD s 1 where the processor LOAD s gven by: h( LOAD = max (11) t t where resource access s accounted for by the blockng term: ( C D J > t D J b( = max,, (12) a k a In whch C a, k s the maxmum amount of tme that a task τ a may spend accessng a resource also used by some task τ k wth a hgher pre-empton level. (Note that the pre-empton 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. ( T + J D ) U L = max(( D1 T1 J1),...( Dn, Tn, J n, ), ) 1 U (13) Note the length L of the nterval that needs to be examned s also lmted to the length of the longest prorty level-n busy perod, whch may be computed 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 number of such values that need to be checked. Zhang and Burns [73], [74] used the fact that h( s monotoncally non-decreasng wth respect to t as the bass for a hghly effcent exact schedulablty test for pre-emptve EDF, called Quck Processor-demand Analyss (QPA). The QPA algorthm s reproduced below. In practce, ths algorthm typcally checks a far small set of values n order to ascertan schedulablty than the classcal method. QPA schedulablty test for pre-emptve EDF: An arbtrary deadlne sporadc task set s schedulable under pre-emptve EDF schedulng f and only f U 1 and the result of the followng teratve algorthm s h( DJ mn, where DJ mn s the smallest deadlne mnus 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 = max{ d d L 2 whle( h ( t h( > DJ mn ) { 3 f( h ( < t ) 4 { t = h( 5 else 6 { t = max{ d d < t 7 8 f( h( DJ mn ) task set s schedulable 9 else task set s unschedulable Algorthm 3: QPA for EDF 5.6 Analyss for non-pre-emptve EDF George et al. [44] extended the schedulablty test for preemptve EDF to the non-pre-emptve case va the use of a revsed blockng factor b(. b( = max ( C : D J > t a 1) k k (14) Note n (14) the maxmum of an empty set s assumed to be zero. Effectvely, non-pre-emptve EDF schedulng may be consdered a specal case of resource sharng n pre-emptve EDF and thus the QPA schedulablty analyss (Algorthm 3) may also be used n ths case [74].

8 5.7 Comparsons between FP and EDF Fxed prorty and EDF schedulng are n some respects competng technques. A theoretcal comparson between the effectveness of fxed prorty and EDF schedulng has been derved by consderng the speedup factor: the maxmum 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 measure of the worst-case suboptmalty of fxed prorty schedulng. Lu and Layland s utlsaton bounds [56] effectvely show that the speedup factor requred to guarantee that pre-emptve fxed prorty schedulng can schedule any mplct-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/ Ω (where Ω s the mathematcal constant defned by the transcendental equaton ln( 1/ Ω) = Ω, hence, Ω ). Further, for the arbtrary deadlne case [38], the speedup factor s lower bounded by 1 / Ω and upper bounded by 2. These upper and lower bounds also hold for a comparson between non-pre-emptve fxed prorty and non-pre-emptve EDF schedulng [39]. It remans an nterestng open problem what the exact speedup factors are n these cases. The average case sub-optmalty of pre-emptve fxed prorty schedulng has also been nvestgated by Lehoczky et al. [55] who used task set average breakdown utlsaton as a metrc. The breakdown utlsaton s obtaned by scalng task executon tmes untl the task set s only just schedulable and then recordng the utlsaton. For large mplct-deadlne task sets they found that the breakdown utlsaton for s approx. 88%, correspondng to a penalty of approxmately 12% n processng capacty compared to pre-emptve EDF. Bn and Buttazzo [16] later showed that breakdown utlsaton suffers from a bas whch tends to penalse fxed prorty schedulng by favourng task sets where the utlsaton of ndvdual tasks s smlar. They ntroduced an optmalty degree metrc whch gves a farer comparson and shows that the average penalty for usng fxed prorty-pre-emptve schedulng s sgnfcantly smaller. In the paper Rate Monotonc vs. EDF: Judgment Day, Buttazzo [23] provded a comprehensve comparson and revew of the advantages and dsadvantages of fxed prorty versus EDF schedulng polces; lookng at factors such as mplementaton complexty, runtme overhead, number of preemptons, robustness durng overload, jtter on task executon, and responsveness when handlng aperodc (soft real-tme) tasks. Whle EDF has many advantages, t s notable that to ths day, the overwhelmng majorty of commercal Real-Tme Operatng Systems mplement fxed prorty rather than EDF schedulng. Ths s done for reasons of smplcty and lower overheads: n partcular, the number of prorty levels s often lmted allowng a bt-map 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-emptve and non-preemptve schedulng. In the majorty of theoretcal work on real-tme schedulng, the overheads of pre-empton are assumed to be neglgble. Indeed ths assumpton s made n the analyss presented so far. In real systems; however, the overheads of pre-empton can be large and have a sgnfcant mpact on system schedulablty. There are two man threads of research that address ths problem. Frstly, work that seeks to mnmse the number of pre-emptons va the use of fxed or floatng non-pre-emptve regons (e.g. co-operatve schedulng), or va pre-empton thresholds, and non-pre-empton groups, where tasks compete 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-emptons, n partcular cache related preempton delays, va the ntegraton of CRPD analyss nto schedulablty tests, and va manpulatng task layouts and cache parttonng to mprove schedulablty. 6.1 Lmtng pre-empton Pre-emptve and non-pre-emptve schedulng can be consdered as two extremes. It may be argued that fully preemptve schedulng allows too many pre-emptons, whch can have a hgh cost f permtted at any arbtrary pont. By contrast, non-pre-emptve schedulng permts no pre-emptons, and can have a severe effect on schedulablty when some tasks have much shorter deadlnes than others. Indeed, wth just two tasks one of whch has an executon tme that s longer than the other s deadlne, then non-pre-emptve schedulng s smply not vable. Lmted pre-empton schedulng s the general term gven to a collecton of dfferent schedulng polces amed at reducng the number of pre-emptons, and hence ther overall cost, whle ensurng that the addtonal blockng ntroduced does not make the system unschedulable. The frst approach s deferred pre-empton. Two dfferent models of schedulng wth deferred pre-empton have been developed n the lterature. In both cases, non-pre-emptve regons are used to lmt pre-empton. Ths s partcularly effectve towards the end of a task s executon where a fnal non-pre-emptve regon may prevent any further context swtches and consequent pre-empton related overheads before the task completes. In the fxed model, ntroduced by Burns n 1994 [2] for fxed prorty schedulng, the locaton of each non-pre-emptve regon s statcally determned pror to executon. Pre-empton s then only permtted at pre-defned locatons n the code of each task, referred to as pre-empton ponts. Ths method s also referred to as co-operatve schedulng, as the tasks cooperate, provdng re-schedulng / pre-empton ponts to mprove 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 preempton costs nto analyss of the fxed model, consderng both fxed [13] and varable [14] pre-empton costs, for both EDF and fxed schedulng. Bertogna et al. [15], also derved a method of computng the optmal length of the fnal non-preemptve regon of each task n order to maxmze schedulablty for a gven prorty assgnment. Fnally, n 212, Davs and Bertogna [29] ntroduced an optmal algorthm that s able to fnd a schedulable combnaton of prorty assgnment and fnal-non-pre-emptve regon lengths whenever such a schedulable combnaton exsts. Ths algorthm mproves upon the schedulablty that can be obtaned by ether fully pre-emptve or fully non-pre-emptve fxed prorty schedulng.

9 In the floatng model of non-pre-emptve regons, ntroduced by Baruah n 25 [12] for EDF schedulng, an upper bound s gven on the length of the longest non-pre-emptve regon of each task. However, the locaton of each non-pre-emptve regon s not known a pror and may vary at run-tme, for example under the control of the operatng system. Analyss for the floatng model under fxed prorty schedulng was gven by Yao et al. [72]. Alternatve approaches to lmtng pre-empton nclude Preempton Thresholds (FPTS) [69], [61] and Non-pre-empton Groups [28] n whch each task has a base prorty at whch t competes for the processor, and a threshold prorty at whch t executes, thus lmtng pre-empton to those tasks wth a base prorty hgher than the threshold. In [69], [61] Saksena and Wang attempted to derve an ntegrated approach to prorty and pre-empton threshold assgnment, but dd not succeed n fndng an optmal algorthm wth less than exponental complexty. Research by Brl et al. [18] n 212 combnes the deas of deferred pre-empton and pre-empton thresholds, generalsng both nto a sngle scheme wth pre-empton thresholds between a set of sub-jobs whch execute non-preemptvely. For further nformaton on lmted pre-empton schedulng the reader s referred to the survey n [22]. 6.2 Accountng for pre-empton costs There s a runtme overhead when the processor swtches from the executon of one job to another. There are varous components that make up ths overhead. These nclude: o The cost of scheduler operatons (e.g. run-queue manpulaton, tmer nterrupts etc.). o The cost of the context swtch from one task to another (savng and restorng regsters etc.). o For processors wth cache, the cache related preempton delay (CRPD). Wth careful RTOS desgn, the frst two components can often be tghtly bounded and ether ncluded n the executon tme of each task, or modelled as vrtual hgh prorty tasks. Further, the addtonal blockng effects caused by the operatng system can also be modelled and ncluded n the analyss see for example the work of Whtham et al. [7]. Cache related preempton delays can however be a substantal proporton of a tasks executon tme, and are hghly dependent on both the pre-emptng task and the pre-empted task. Cache s fast local memory, whch s used to store nstructons (and data) mprovng access tmes and hence brdgng the large gap that exsts between the speed of man memory and the speed of hgh performance processors. Memory requests that can be servced by the cache can be sgnfcantly faster (1 to 1 tmes faster n some cases) than accesses that have to go to man memory. However, cache s typcally much smaller than man memory and so not all of the nstructons and data needed by all tasks can be held n cache at the same tme. Analyss of cache related pre-empton 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 may 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 from pre-emptng tasks. If a UCB s evcted by a pre-emptng task, then an addtonal delay may be ntroduced f the UCB needs to be re-loaded when t otherwse would not have been. Ths s termed a cache related preempton delay. Dependng on the approach used, CRPD analyss combnes nformaton about the UCBs belongng to the pre-empted task(s) wth the ECBs of the pre-emptng task(s). Usng ths nformaton, the total number of cache blocks that are evcted, whch must then be reloaded after pre-empton, can be calculated and combned wth the cost of reloadng a block to gve the CRPD. A number of approaches have been developed for calculatng the CRPD under pre-emptve fxed prorty schedulng. These nclude the UCB-Only approach of Lee et al. [52], whch consders only the pre-empted task(s), and the ECB-Only approach of Busquets et al. [21] whch consders only the preemptng task. Approaches that consder both pre-empted and pre-emptng tasks nclude the UCB-Unon approach of Tan and Mooney [66] and the ECB-Unon approach of Altmeyer et al. [1] and ther mult-set extensons [3]. Untl recently, there has been sgnfcantly less work n developng CRPD analyss for pre-emptve EDF schedulng. In 27, Ju et al. [49] consdered the ntersecton of the preempted task s UCBs wth the pre-emptng task s ECBs; however, ths method has sgnfcant pessmsm 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-empton delays can have a sgnfcant effect on schedulablty. They can vary substantally dependng on how the code of tasks s lad out n memory and hence how t s mapped to the cache, thus provdng scope for optmsng schedulablty va manpulatng the poston of tasks n memory [59]. One way of elmnatng 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-emptng task do not overlap wth the UCBs of any pre-empted task. However, such parttonng comes at a prce. The smaller cache partton allocated to each task ncreases the non-pre-emptve WCET of the task. Recent work by Altmeyer at el. [2] has nvestgated ths trade-off concludng that n many cases, even optmal 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 fundamental schedulablty analyss results for pre-emptve and non-preemptve, fxed prorty and EDF, schedulng of sporadc task sets. We have also brefly covered extensons to ths work that relax the rather unrealstc assumpton that schedulng and preempton costs are neglgble. In real systems, partcularly those runnng on processors wth cache, the costs of pre-empton can be relatvely hgh when compared wth the non-pre-emptve WCETs of the tasks. Thus technques for lmtng pre-empton are needed, as well as analyss that accounts for the cost of pre-empton overheads. 7.1 Lookng back: success stores Despte the advent of multcore processors and substantal research nto multprocessor schedulng, research nto realtme schedulng on sngle core systems stll thrves. One of the hardest challenges n ths area s to progress from smple