On Schedulability Bounds of Static Priority Schedulers

Transcription

1 On cedulablty Bounds of tatc Prorty cedulers Jana Wu, Jy-Carn Lu, and We Zao Department of Computer cence Texas A&M Unversty {anaw, lu, Wle utlzaton bound based scedulablty test s smple and effectve, t s often dffcult to derve te utlzaton bound For ts analytcal complexty, utlzaton bound results are usually obtaned on a case-by-case bass In ts paper, we develop a general framework tat allows one to effectvely derve scedulablty bounds of dfferent workload patterns and scedulers We ntroduce an analytcal model tat s capable of descrbng a wde range of tasks and scedulers beavors We propose a new defnton of utlzaton, called workload rate Wle smlar to utlzaton, workload rate enables flexble representaton of dfferent scedulng and workload scenaros and leads to unform proof of scedulablty bounds We derve a parameterzed scedulablty bound for statc prorty scedulers wt arbtrary prorty assgnment Exstng utlzaton bounds for dfferent prorty assgnments and task releasng patterns can be derved from te closed-form formula by smple assgnments of proper parameters Key words: Workload rate, utlzaton, scedulablty test, statc prorty sceduler, network calculus

2 Table of Contents Introducton 2 Related Work 3 3 General Framework 5 3 Task and ceduler Model 5 3 Task, Task et, Workload, and ervce Functon 5 32 Workload Constrant Functon 6 33 ervce Constrant Functon 6 32 cedulablty Bound 9 4 cedulablty Bound for Rate Monotonc ceduler 3 4 -aped Workload Constrant Functons 3 42 ervce Constrant Functon of RM wt -saped Workload Constrant Functons 5 43 cedulablty Bound for RM 5 5 Performance Evaluaton and Comparson 8 5 Analyss of te RM cedulablty Bounds 8 52 Parametrc Fttng of Exstng Utlzaton Bounds for RM systems 20 6 Extensons 22 6 Harmonc Can Tasks Arbtrary tatc Prorty cedulers Non-preemptve ystems 24 7 Fnal Remarks 24 References 26 APPENDIX A: Lst of Lemmas 29 APPENDIX B: Proof of Teorem 2 30 APPENDIX C: Proof of Teorem 3 32 APPENDIX D: Proof of Corollary 3 33 APPENDIX E: Proof of Teorem 4 36 APPENDIX F: Dervaton of Leoczky s Bound [26] 40 APPENDIX G: Proof of Corollary 4 43 APPENDIX H: Proof of Teorem 5 44 APPENDIX I: Proof of Corollary 6 50 APPENDIX J: Proofs of Lemmas Lsted n Appendx A 5

3 NOMENCLATURE ymbols Defntons Frst occurrng page C t Te ncrement of an s-saped workload constrant functon at te Page 6 perod, =, 2,, L C Te constant ncrement of an s-saped workload constrant functon Page 6 t after te L perod d Te worst case (e, largest delay of all obs n a task Page 4 A postve nteger defned by te relatonsp of k = λ Page 3 D Relatve deadlne of a task Page 4 / η = V k λ, Γ Page 3 f (t Workload functon Page 4 F( I Workload constrant functon Page 4 g (t ervce functon Page G( I ervce constrant functon Page V (, Γ Heterogenety functon of task set Γ Page 8 k Normalzed deadlne of a task, k = D / P Page 7 λ Degree of deadlne nverson Page 0 L Number of C s n an s-saped workload constrant functon Page 6 Perod of a perodc task or te perod parameter n an s-saped Page 6 workload constrant functon Γ Task set Page 4 t Job absolute deadlne Page 4 d t Job release tme Page 4 r t Job completon tme Page 4 f T A task Page 4 θ calng parameter used for workload rate measurement Page 9 V calng parameter used for eterogenety measurement Page 8 W (θ, Γ Workload rate of task set Γ Page 9 W ( θ cedulablty bound Page 2

4 Introducton Meetng strngent tmng requrements s crtcal to many computer applcatons (eg, dgtal control and samplng, voce over IP, vdeo conferencng, etc Tese real-tme systems are usually equpped wt a mecansm known as te scedulablty test, wc determnes weter eac of te admtted tasks can meet ts deadlne A new task wll not be admtted unless t passes te scedulablty test Te scedulablty test can be eter drect or ndrect Drect scedulablty test wll explctly calculate te worst-case delays of te tasks n order to determne te permssblty of a new task Ts type of test as a ger level of accuracy, but usually results n te g run-tme computng cost n calculatng te delays An ndrect scedulablty test does not compute te delay drectly, but tests selected system parameters to determne te scedulablty of a new task Te utlzaton based scedulablty test s te most common approac, n wc a task can be admtted only f te utlzaton s lower tan a pre-derved bound Te maor advantages of ts scedulablty test are as follows: It s very effcent Unlke te drect scedulablty test tat need to evaluate te scedulablty of eac task n te system (and te new one upon te arrval a new task, te utlzaton based scedulablty test only need to ceck weter te total system utlzaton (ncludng te new task s lower tan te prederved utlzaton bound 2 It provdes an operaton margn for system admnstrators and tus mproves te stablty of a real-tme system Durng desgn pase, te system desgner can set an upper bound of utlzaton tat s lower tan te pre-derved bound By provdng suc safety margn, te system can work smootly even some tasks accdentally volate ter load constrants, or wen some system parameter canges, eg clock skew Varous utlzaton bounds ave been derved n te lterature, and some ave been appled to mplementaton of msson crtcal applcatons ome of te most notable results nclude 69% (and extensons for te Rate/Deadlne Monotonc sceduler (RM/DM [22], [27], [32], [36], [37], [4], ; 00% for Earlest Deadlne Frst sceduler (EDF [32]; and 33% for te Tmed Token protocol (TTP [4], [35], [46], [47]; etc Utlzaton bounds of RM/DM n multprocessor systems ave also been derved n [7], [8], [9], [0], [2], [38] ome mportant utlzaton bounds for non-perodc systems are derved n [], [2], [3], [45] A more detaled summary of te lterature s gven n te next secton In general, t s dffcult to generalze te utlzaton bound metod to non-perodc system due to te followng two problems: a Ambguty n defnng utlzaton for non-perodc tasks Utlzaton s a measurement of te resource consumpton rate wtn a certan tme wndow (referred to as a measurng

5 wndow Typcally, for perodc systems, task perods ave been used as te measurng wndow It s dffcult to extend ts defnton to te doman of non-perodc tasks because one cannot ave a well-defned noton of perod One certanly could defne a long-term stable utlzaton wt te measurng wndow lengt beng nfntely large, but ts type of defnton cannot correctly reflect te resource demand wtn te deadlne Because of ts, n [] and [7], te autors proposed to defne te utlzaton by settng te lengt of te measurng wndow as te relatve deadlne of te task To derve utlzaton bounds, we must ave a flexble, robust noton of utlzaton tat can be appled to a broad range of workloads and scedulers Te defnton sould correctly reflect te resource demand and facltate dervaton of te bounds b Ad oc-ness n te dervaton of te utlzaton bound Most utlzaton bound results are obtaned case-by-case, and te metod developed for one system cannot be easly appled to anoter Te g complexty of te underlyng optmzaton problem s manly attrbuted to te fact tat one must fnd an optmal (lower bound of te utlzaton n an nfnte space of non-scedulable task sets In ts paper, we ntroduce an analytcal model tat s capable of descrbng a wde range of tasks and scedulers beavors In addton to address te problems mentoned above, we broaden te utlzaton bound dervaton tecnques usng te followng approac We propose a new defnton of utlzaton, called workload rate, wc measures te resource demand wtn a tme wndow of lengt proportonal to te deadlne of a task, so tat t can be used to caracterze bot perodc and non-perodc tasks n te same framework Takng te relatve deadlne of te task lengt of te measurng wndow to defne te utlzaton was frst proposed n [] and [7] everal oter key system parameters are caracterzed wt respect to te workload rate n formulaton of te utlzaton bound solutons On te bass of te network calculus framework [2], [3], [4], [5], [7], [8], and [9], we derve some key relatonsps between workload and servces, to arrve at a lower bound of workload rate for arbtrary servces and scedulers In prevous work, te searc for utlzaton bound was usually made along te boundary between te spaces of scedulable and non-scedulable task sets Knowng tat fndng te boundary of te two spaces s already a maor undertakng, we drectly derved te scedulablty bound by solvng a mnmzaton problem over te entre task set populaton As a result, te 2

6 utlzaton results (of scedulablty testng are applcable to a muc broader range of task models and scedulers To furter llustrate te effectveness of our new metodology, we explctly derve a parameterzed workload rate bound for statc prorty scedulers wt arbtrary prorty assgnment A closed-form formula s obtaned, and te bound s parameterzed for dfferent prorty assgnments and for varous task releasng patterns We sow tat wen te parameters are set properly, our general workload rate bound s reduced to te utlzaton bounds of te followng platforms: o o o o Te system as perodc tasks wose deadlnes are equal to perods Tasks are sceduled by a rate monotonc sceduler [32] Te system as perodc tasks wose deadlnes are less tan perods Tasks are sceduled by a rate monotonc sceduler [27] and [4] Te system as perodc tasks wose deadlnes are multples of perods Tasks are sceduled by a rate monotonc sceduler [22] Te system as multframe tasks wose deadlnes are equal to perods Tasks are sceduled by a rate monotonc sceduler [37] To our knowledge, no current lterature covers as a wde range of systems as our metodology We ten generalze te parameterzed bound to statc prorty sceduler wt arbtrary prorty assgnment sceme Ts scedulablty bound for arbtrary statc prorty sceduler s furter mproved based on te concept of armonc can tasks [6] and [23] Te rest of te paper s organzed as follows Te related work s dscussed n ecton 2 ecton 3 ntroduces te system model In ecton 4, scedulablty bounds are dscussed and a general bound for arbtrary sceduler and tasks are derved ecton 5 derves a closed-form bound for RM wt a specal type of task set In secton 6, te closed-form bound for RM s generalzed to arbtrary statc prorty scedulers and s furter mproved for te armonc can tasks Conclusons and dscussons are gven n ecton 7 2 Related Work In ter semnal work [32], te autors derved te well-known 69% utlzaton bound for te RM on sngle processor systems, were relatve deadlnes of perodc tasks are equal to ter perods A rc collecton of utlzaton bounds ave been derved snce ten for dfferent systems Ts result as been extended to arbtrary deadlne assgnment scemes n Error! Reference source not found, [27] and [4] 3

7 In [24] and [30], te autors mproved te bound by explotng te rato between te longest and sortest task perods Te work n [6] and [23] furter mproved te bound result wt te concept of te armonc can tat explots te dvsblty between perods Te autors n [22] ntroduced an algortm tat transforms te perodc task nto a armonc task set, wc as a workload bound of, and proved tat te algortm performs better tan te bound derved n [30] and [32], wt te cost of ger complexty Utlzaton bounds of statc prorty scedulers on te tme token protocol n FDDI networks were derved n [4], [35], [46], and [47] Te utlzaton bound for statc prorty scedulers n a network envronment ave been studed n [45] Utlzaton bounds for non-perodc tasks ave been addressed n [], [2], [3], and [34] Utlzaton bounds for RM/DM n mult-processor systems ave been studed n [7], [8], [9], [2], and [34] Generalzng te defnton of utlzaton from perodc tasks to aperodc tasks as been studed n [], [2], [3], [7], [36], and [45] In dervng te utlzaton bound for RM wt multframe and general realtme task models, te autors n [36] and [37] proposed a maxmum average utlzaton tat allows calculaton of utlzaton n an nfnte measurng wndow In te analyss of te utlzaton bound n a mult-node network envronment wt leaky bucket packet sources, te autors n [45] used a utlzaton defnton tat s based on te sustanable rate n te leaky bucket functon To derve te utlzaton bound for non-perodc tasks and multprocessor systems, te autors n [], [2], [3], and [7] proposed a utlzaton defnton tat s based on relatve deadlnes of tasks, nstead of perods Te lnear programmng metod tat as been proposed for fndng utlzaton bounds wen task parameters are known a pror as been studed n [6], [25], and [39] Te work n [] ntroduced a new scedulablty test wc s smlar to utlzaton based admsson control pecfcally tey proved tat for perodc tasks wt RM, a task set s scedulable f Π ( u + 2, were u s te utlzaton of te t task Generalzaton of a perodc task model was proposed n [35] and [36], n wc te autors derved a bound result for multframe tasks wc allows obs n te same task to ave dfferent sze, provded tat te relatve deadlne s same as perod lengt Usng workload constrant functon to model tasks can be traced back to [7], [8], e leaky bucket constrants of network traffc Ts concept was expanded n [45] to analyze te utlzaton bounds of statc prorty scedulers Te general model for real-tme tasks proposed n [36] sares a smlar concept, and t corresponds to a specal group of workload constrant functons n multframe forms Te dea of modelng scedulers wt servce constrant functons orgnated n [3], [4], [5], [9], and [40] Workload constrant and servce constrant functons ave been used for drect scedulablty test n [2], [9], [20], [42], and [43] among many oters, but none of tem s used for utlzaton based test 4

8 3 General Framework For utlzaton bound based scedulablty test, a task set s scedulable wen te utlzaton of te task set s lower tan a pre-derved bound Our goal n ts paper s to develop a general analyss framework for suc bound based algortms 3 Task and ceduler Model 3 Task, Task et, Workload, and ervce Functon We assume tat a sngle processor computng system s to serve a task set Γ = { T, T2,, T n }, were T s te t task Wen te context s clear, we may omt ndex n te subsequent dscussons Eac task s composed of a sequence of obs Te worst-case executon tme of a ob s called te ob sze, wc s measured n second A ob can start ts executon after ts release tme, t r, and must be fnsed by ts absolute deadlne t d = t r + D were D s called relatve deadlne For a ob, te tme elapsed from te release tme t r to te completon tme t f s called te delay of te ob, and te worst-case (e, largest delay of all obs n a task s denoted by d Wtn a task, te obs ave te same relatve deadlne, but may not necessarly ave te same sze Jobs wtn a task are executed n a frst come, frst served order To caracterze te resource demand of task T analytcally, we defne f ( t, te workload functon for T, as follows, f ( t = te summaton of te szes of all te obs from T released n [0, t] (3- mlarly, to caracterze te actual processor tme receved by task T, we defne g( t, te servce functon for T, as follows, g( t = te total executon tme rendered to obs of task T durng [0, t] (3-2 Based on te defntons of d, f ( t and g( t, t s easy to see tat ( ( τ τ d t 0 f t g t = sup nf ( ( + (3-3 In a real-tme system, a maor goal of te scedulablty test algortm s to ceck te trutfulness of One may want to use (3-3 to calculate d D (3-4 d and ten compare te result wt D to test te scedulablty However, ts metod may not be sutable for onlne operaton because te exact forms of f ( t and g( t may not be avalable wen scedulablty test s made Furtermore, even f f ( t and g( t are avalable, 5

9 tey are often too cumbersome to andle A practcal soluton s usng some alternatve forms of f ( t and g( t tat can be obtaned durng scedulablty test 32 Workload Constrant Functon Many works ave been done on te alternatves of f ( t For example, n perodc task model, a typcal alternatve of f ( t s F( t = t / P C, (3-5 were C s te maxmum ob sze and P s te mnmum nter-ob separaton tme Toug ts alternatve s accurate for perodc tasks, t may over-estmate te resource demand for non-perodc tasks [37], and would lead to pessmstc scedulablty decsons 0 I t A better alternatve of f ( t s te workload constrant functon F( I tat satsfes tat for any f ( t f ( t I F( I (3-6 Te F( I n form of (3-6 was frst ntroduced n [7] and [8] and as been wdely used [2], [4], [5], [29], and [45] By conventon, F (0 = 0 and F( I s non-decreasng F( I s an upper bound of total sze of obs can be released n any tme wndow [ t I, t] We use I n (3-6 because F defned on te doman of tme ntervals, wle f ( t s defned n te doman of tme Te workload constrant functon defned n (3-6 s tgter tan te alternatve functon defned n (3-5, snce for systems wt perodc task model, t s easy to see tat F( I = I / P C s a workload constrant functon satsfes (3-6 f ( t In ts paper we wll use te workload constrant functon F( I defned n (3-6 as alternatve of 33 ervce Constrant Functon Wt F( I defned, let us now consder alternatves of g( t Toug one can fnd many dfferent types of alternatves of g( t, we are only nterested n tose tat can facltate te delay bound analyss Opton : trct ervce Constrant Functon We say G( I s a strct servce constrant functon for task T f for any 0 I t g( t g( t I G( I (3-7 Clearly, wt ts defnton, we ave g( I G( I for all I 0 and togeter wt te fact f ( I F( I, we ave 6

10 ( ( τ τ t ( τ τ ( = + + = ˆ (3-8 d supt 0 nf f ( t g( t sup 0 nf F( t G( t d Tat s to say, we can derve a delay bound based on F( I and G( I usng (3-8 However, toug ts G( I s easy to obtan, ts coce of G( I s problematc snce ˆd can be very loose as llustrated n te followng example EXAMPLE : Consder te task workload and servce functons llustrated n Fgure Te task contnues to release obs untl tme 6, and after tat, tere s no more ob releasng Te sceduler begns to serve te obs from tme 6 and fnsed all of tem at tme 0 Te worst case delay appens on te ob arrved at tme and s 6 Now consder te substtuton functon G( I defned n (3-7 By (3-7 and te fact tat for any I 0, te total servce receved by te task n tme wndow [0, 0 + I] s 0, we know tat for any I 0 G( I = 0 (3-9 Ten by substtutng (3-9 nto (3-8, we ave d d =, wc s a very loose (meanngless bound ˆ Fgure : An example task workload and servce functon By a close examnaton of example, one can notce tat te problem of te loose bound can be trace back to te fact tat G( I as a strong requrement tat (3-7 must old for all t 0, 0 I t and tus leads to very low values An nterestng queston s tat weter we can relax te requrement on G( t Te answer s yes as llustrated n te followng opton Opton 2: ervce Constrant Functon We say G( I s a servce constrant functon provded by a sceduler to task T f for any t 0, tere exsts 0 I t suc tat g( t f ( t I + G( I (3-0 7

11 Typcally, we assume G( I s non-decreasng and G(0 0 (3-0 means tat for any tme nstant t, we can fnd a tme nstant t ave been served, 2 for te obs released n nterval [ t llustrated n Fgure 2 I, 0 I t, suc tat all te obs released n tme nterval [0, t I ] I, t], at least G(I amount ave been served, as Note tat n [3], te autors used a defnton n form of g( t nf ( f ( t I G( I +, wc s equvalent to (3-0 except for te case tat f and/or G are not contnuous Neverteless, for purpose of smplcty, we use (3-0 n ts paper Te followng teorem proves tat (3-8 s true wt te servce constrant functon gven n (3-0 Teorem [3]: Gven a task wt workload constrant functon F( I and servce constrant functon G( I, ( ( τ τ d I 0 F I G I Proof: By defnton, for any fxed tme t 0 tme nstant t 0 I t sup nf ( ( + (3-, and τ < d, we ave f ( t > g( t + τ Ten by (3-0, for + τ, tere exsts an 0 I t + τ, suc tat g( t + τ > f ( t + τ I + G( I and n turn f ( t > f ( t + τ I + G( I nce f ( t s non-decreasng, t must be true tat t > t + τ I and ence by (3-6, we get F( I τ > G( I Rewrtng t wt I ' = I τ, we know tat for any fxed tme t 0, and τ < d, F( I ' G( I ' τ Fgure 2: Explanaton of te ervce Constrant Functon > Ten follows nf ( F( I G( I d τ + τ and tus (3- In te followng dscusson, unless oterwse stated, we wll focus on te servce constrant functon defned n (3-0 unless stated oterwse Typcally, te servce constrant functon can be derved based on propertes of te sceduler and workload constrant functons For statc prorty scedulng systems, we ave te followng teorem Teorem 2: For te statc prorty scedulng system, G ( I, a servce constrant functon for task T, s ( G ( I = sup x F ( x (3-2 0 x I = 8

12 Note tat te sup operaton guarantees tat G ( I s non-negatve, and non-decreasng Teorem 2 can be proved based on te defnton of servce constrant functon pecfcally, gven any tme nstant t, one can defne I suc tat tme t I s te last tme nstant before t suc tat a task wt prorty lower tan T s sceduled Ten from te property of statc prorty sceduler, one can prove tat task T wll receve at least sup ( 0 x I x F ( x = For detaled proof, see Appendx B seconds of servces n nterval [t I, t], and tus te teorem Note tat te noton of workload and servce constrant functon defned n (3-6 and (3-0 are not entrely new mlar defntons ave been proposed n te lterature, eg, te burst-ness constrant functon n [7], te arrval curve and servce curves n [2] and [3], te rate controllng functon n [30], and te workload constrant functons n [29], [44], and [45], ust to name a few But, lttle, f any, of ts effect as been explored for scedulablty bound analyss, as we wll do n te next subsecton 32 cedulablty Bound From Teorem, we ave te followng suffcent scedulablty test condton Corollary : A task s scedulable f for any t 0 Proof: Clearly, by Teorem, f (3-3 olds for all t 0, we ave F( t G( t + D (3-3 d D Toug one can use (3-3 for eac task to decde ter sceduablty test, t maybe tme consumng snce te (3-3 need to be cecked for all t 0 In ts paper, we wll take a dfferent approac smlar to te utlzaton bound based test Recall tat for utlzaton bound based scedulablty test, a task set s scedulable wen te utlzaton of te task set s lower tan a pre-derved bound Our goal ere s to develop a smlar bound based test algortm Generally speakng, utlzaton s te resource consumpton rate n a measurng tme wndow For perodc systems, te most effectve measurng wndow s task perod For non-perodc tasks, n [] and [7], te autors proposed to use te relatve deadlne of te task as te lengt of te measurng wndow n order to defne te utlzaton for non-perodc tasks Wle ts coce s smple and convenent for some cases, we fnd tat t s too restrctve to meet our goal: a versatle utlzaton bound analyss system In stead, we propose to defne te lengt of te measurng wndow as a lnear scale of te relatve deadlne Tat s, te measurng wndow can be expressed as θ D, were θ > 0 s called te scalng parameter and D s te relatve deadlne of te task To avod confuson wt te lterature, we refer to ts generalzed utlzaton as te scaled workload rate, and t can be formally expressed as 9

13 n F ( θ D W ( θ, Γ = (3-4 = θ D Wen te context of dscusson s clear, te term scaled may be omtted nce F ( θ D s an upper bound of te sze of obs tat can be released n any tme wndow of lengt θ D, W ( θ, Γ can be treated as an upper bound of te ob releasng rate averaged n a wndow of lengt θ D Introducng θ nto te modelng process parameterzes te utlzaton measurement For example, wen θ =, (3-4 reduces to te defnton provded n [] and [7] Ts parameterzed measurement of utlzaton enables flexble representaton of dfferent scedulng and workload scenaros, and more mportantly, leads to unform analyss system of scedulablty bounds For a gven system, we say W ( θ s scedulablty bound f an arbtrary task set s scedulable wen te followng condton olds: Te callenge s ow to derve W ( θ, Γ < W ( θ (3-5 W ( θ for a broad range of workload patterns and scedulng dscplnes Let te space of all task sets be denoted as Ω, e, Ω = {Γ} Ω can be parttoned nto two subsets, Ω s and Ω ns, were and Ω s = {Γ Γ s scedulable} (3-6 Ω ns = {Γ Γ s not scedulable} (3-7 W ( θ s a lower bound of te workload rate of tese task sets tat belong to Ω ns Tat s ( W θ W ( θ = nf (, Γ (3-8 Γ Ω ns Many prevous studes ave mplctly followed (3-8 to derve scedulablty bounds For example, te mnmzaton process s often aceved by searcng along te boundary between Ω s and Ω ns Instead of tryng to fnd an analytcal representaton of te boundary drectly, wc may be qute callengng task, we wll transform all te tasks n Ω, wt a specal transformaton functon Y(Γ, nto a regon close to te boundary between Ω s and Ω ns and ten perform a mnmzaton n tat regon If te transformaton functon Y(Γ s properly selected, te bound results obtaned on te transformed regon may be very close to te bound obtaned from te exact boundary between Ω s and Ω ns, f not te same Fgure 3 llustrates te concept For example, n [27], [32], [36], [37], and [4], te boundary s defned by te task sets tat fully utlzes te resources 0

14 To guarantee te correctness of te bound result obtaned on te transformed tasks, t s clear tat Y need to satsfes tat for any Γ Ω ns, W ( θ, Γ W ( θ, Y( Γ (3-9 Tat s, te transformaton wll not ncrease te workload rate of any non-scedulable task set Wt (3-9 olds, we ave ( ( W ( θ = nf W ( θ, Γ nf W ( θ, Y( Γ (3-20 Γ Ω ns Note tat (3-9 s only requred for non-scedulable tasks For scedulable tasks, te transformaton functon may ncrease ts workload rate, but t s easy to understand tat ts wll not affect te correctness of (3-20 wc Γ Ω n Now we wll sow ow to construct Y Let nf ( G ( I D / F ( I = + and be te value of at α I > 0 F ( θ D ( α s mnmzed Tat s, for =, 2,, n, θ D Fgure 3: Illustraton of te Transformaton Functon F ( θ D F ( θ D ( α ( α θ D θ D, (3-2 Let { T ', T2 ',, T n '} denote te transformed task set Y ( Γ, F ' te workload constrant functon, and D ' te relatve deadlne for T ', =, 2,, n We construct te functon Y as follows: for =, 2,, n, for all and for and I 0 = and I 0 D ' = D, (3-22 F '( I = F ( I, (3-23

15 F '( I = α F ( I (3-24 Tat s, te functon Y canges te F to α F and keeps te oter F s and D s untouced Now, we wll fnd a lower bound of te scedulablty bound wt te elp of te ust constructed transformaton functon By (3-4, we ave F ( θ D F ( θ D W ( θ, Y ( Γ = + (, (3-25 n α = θ D θ D By Lemma, for any Γ Ω, tere must exst a task T suc tat α < By substtutng α < nto (3-25, we get, for any Γ Ω ns, By substtutng (3-26 nto (3-25, we ave, for any ns F ( θ D ( α 0 (3-26 θ D = Γ Ω ns, n F ( θ D W ( θ, Y( Γ = W ( θ, Γ (3-27 θ D Furtermore, by substtutng (3-27 nto (3-8, we get ( ( W ( θ nf W ( θ, Y( Γ nf W ( θ, Y( Γ (3-28 Γ Ω ns Ten, by substtutng (3-25 nto (3-28, we ave te followng teorem about te workload bound Γ Ω Teorem 3: A lower bound of scedulablty bound for a sceduler wt respect to Ω s gven by were nf ( G ( I D / F ( I α I > 0 F ( θ D F ( θ D W ( θ = nf + mn (, (3-29 = + Proof: ee Appendx C n Γ Ω =, 2,, n α = θ D θ D Note tat Teorem 3 s a general result It olds for any workload constrant functon and any work conservng sceduler By substtutng specfc forms of F ( I and G ( I nto (3-29 and solvng te optmzaton problem, one can obtan te scedulablty bounds for dfferent scedulers, as we wll demonstrate ts n te next secton 2

16 4 cedulablty Bound for Rate Monotonc ceduler In ts secton, scedulablty bounds for RM wll be derved usng te general framework dscussed n ecton 3 We wll frst ntroduce a specal type of workload constrant functon tat s capable of representng a wde range of task workloads Ten we wll derve te servce constrant functons for RM based on Teorem 2 Te detaled types of workload and servce constrant functons wll ten be plugged nto Teorem 3 to obtan a parameterzed, closed-form, scedulablty bound for RM Te closed-form bound result reduces to classcal bounds wen parameters are properly set 4 -aped Workload Constrant Functons Knowng tat workload constrant functons can exst n many dfferent forms and dervng sceduablty bound for arbtrary workload constrant functon may be very callengng, we start wt a specal workload constrant functon, namely te starcase-saped (s-saped workload constrant functon As ts name suggests, an s-saped workload constrant functon conssts of segmented peces, and resembles a starcase Te values of an s-saped workload constrant functon ncrease only at border ponts of segments We assume tat te segment lengt s fxed, and te ncrements may not be dentcal for te frst L segments were L s a parameter n te functon F( I L = 4 C C C 3 C 4 C C I Fgure 4: An example on s-saped workload constrant functon Formally, an s-saped workload constrant functon can be expressed as follows, a C a L = F( I = L C + ( u L C a > L = (4-3

17 were a = " I /! #, C s te ncrement at te begnnng of te t segment, and C s te constant ncrement after te t L segment Fgure 4 sows an example of te s-saped workload constrant functon Wen L =, an s-saped constrant functon reduces to te general real-tme task model, as te one defned n [36] Te s-saped workload constrant functons cover a broad range of tasks and ave analytcal propertes tat facltate te dervaton of workload rate bounds We say tat an s-saped functon F( I s smoot wen $ (4-2 2 L C C C C Tat s, f an s-saped constrant functon s smoot, ten ts ncrements over tme are monotoncally nonncreasng Te smootness property greatly smplfes te scedulablty analyss Any non-smoot s-saped workload constrant functon can be converted nto a smoot one by te followng procedure tep Locate te frst C n te functon suc tat C < C + ; tep 2 Replace bot C and C + wt (C + C + /2; tep 3 Repeat teps and 2 untl no suc C exsts; tep 4 Replace te C s tat are less tan C wt C It can be easly verfed tat te result of te above process wll produce a constrant functon tat s s-saped and stll meets te defnton gven n (3-6 For example, consder te non-smoot s-saped constrant functon n Fgure 4 wt segment lengt = 8, C = 5, C 2 = 5, C 3 =, C 4 = 5, and C = 2 Ts functon can be transformed to a smoot one wt = 8, C = 5, C 2 = 5, C 3 = 3, C 4 = 3, and C = 2 In te rest of ts dscusson, unless oterwse specfed, we only consder smoot s-saped workload constrant functons Example 2: -saped workload constrant functons for multframe tasks As defned n [3], a ( E,,,, E E P 0 multframe task s expressed n te form of ( N separaton tme and te executon tme of te t ob s ( mod N E were P s te mnmum ob For nstance, ((3,, 3 denotes a task wose mnmum ob separaton tme s 3, and ts executon tme alternates between 3 and A multframe task s sad to be Accumulatvely Monotonc (AM [37], f te total executon tme for te frst frames s te largest among all sze frame sequences for all, =, 2, 4

18 Recall tat an s-saped constrant functon s caracterzed by parameters, L, C, ( E,,,, E E P 0 and C Gven a multframe task ( N 2 C,, L C,, we can construct ts correspondng F( I by assgnng te followng parameter values: = P, L =, C = φ, and C = φ φ, = 2, 3,, were φ, =, 2,, s defned as φ = 0,, 2, ( f ( P f ( P = max % (& + & (4-3 It s easy to verfy tat for t 0, I 0, te followng nequalty olds for F( I ust constructed, f ( t + I f ( t F( I, (4-4 and t s a vald workload constrant functon By defnton, wen te multframe task s AM we know tat 0 φ = E and φ = E + E And tus, we ave C E 0 = and C = E Note tat te newly constructed s-saped functon s not necessarly smoot 42 ervce Constrant Functon of RM wt -saped Workload Constrant Functons We say a sceduler s RM for tasks wt s-saped workload constrant functons f task prortes are assgned n descendng order of task szes Tat s task T as a ger prorty tan T f and only f Based ts defnton and Teorem 2, te followng corollary gves te servce constrant functon for RM wt s-saped workload constrant functons Corollary 2: For te RM wt s-saped workload constrant functons, G ( I, a servce constrant functon for task T, s ( G ( I = sup x F 0 ( x x (4-5 I Proof: By Teorem 2 and te RM prorty assgnment rule we ave 43 cedulablty Bound for RM In ts subsecton, we wll derve a closed form scedulablty bound for RM wt s-saped workload constrant functons Frst, we wll derve a lower bound of te scedulablty bound for te specal case of s-saped workload constrant functons based on Teorem 3 Corollary 3: Gven a sceduler and collecton of task set wt s-saped workload constrant functons, a scedulablty bound wt respect to Ω s gven by 5

19 ' ' ' ( ( ( F ( θ D G ( m + D W ( θ = mn mn + mn + + +,,, (4-6 Outlne of proof: Γ Ω =, 2,, n m= 0,, =, 2,, n, θ D θ D + m mn θ D, By comparng (4-6 wt (3-29, we only need to prove mn I > 0 m= 0,, F ( I θ D θ D + m mn θ D, { } { } - - G ( I + D F ( θ D G mn ( m + D / 2 0 / 2 0 (4-7 nce F, =, 2,, n are s-saped functon and G ( I s non decreasng wt I, t can be verfed tat mn G ( I + D 6 F ( θ D 5 G mn ( m + D 6 F ( θ D (4-8 I > 0 m= 0,, F ( I θ D F ( m θ D Ten by te s-saped functon property 9, one can prove 2 L C C C C F ( θ D F ( m θ D θ D + m mn θ D, { } (4-9 By substtutng (4-9 nto (4-8, we ave (4-7 establsed and tus te corollary For complete proof, see Appendx D Now, wt Corollary 3, we wll derve a parameterzed scedulablty bound for RM wt s-saped constrant functons based on two key system parameters, normalzed deadlne and workload varance For task T wt a gven s-saped constrant functon, we defne ts normalzed deadlne, k, as follows: k = D /, (4-0 were D s te relatve deadlne of T and s te segment parameter n te constrant functon k can be vewed as te deadlne usng as te measurement unt, and t caracterzes tgtness of te deadlne requrements Te smaller te k, te more dffcult t s to scedule te task Furtermore, knowng tat s te unt tme nterval to defne te -sape functon, decreasng te value also reduces te dfferences between F and f, wc s one of te man reasons wy te utlzaton s nversely proportonal to te It wll be clear n ecton 5 and 6 tat parameter k plays a crtcal role n scedulablty test Followng te conventon n te lterature [4], [22], [27], [32], [35], [36], [37], [4], [46], and [47], we assume tat all te tasks ave te same value of k Tat s, for =, 2,, n k = k, (4- Recall tat te workload rate W ( θ, Γ s an upper bound of te ob releasng rate averaged n a wndow of fnte lengt θ D nce an s-saped functon allows bust ob releasng n sort wdows, as 6

20 long as te ob releasng rate slows down n longer wndow lengts, usng te workload rate measured n a wndow of lengt θ D for scedulablty bound analyss may over-estmate te actual resource demand of te task, and tus result n under-estmated bounds To overcome ts problem, n our scedulablty bound analyss, we want to take account nto consderaton te varance of te workload rate measured n dfferent wndows We proposed a task parameter, workload eterogenety V ( : for ts purpose, wc s defned as, ; ; ; F ( / V ( = ; ;, (4-2 F (( + F ( were : s a postve nteger Intutvely, V ( : s te rato between te workload rate measured n [0, : ], and te one measured n [ :, ( : + ] as sown n Fgure 5 Clearly, for perodc task, V ( : = F ( F (( + F( = > ( + B ( + 2 A ( + 3 I Fgure 5: Workload Heterogenety Takng one step furter, we defne te eterogenety of a task set as < < < < (, Γ = mn (, (,, ( ( V V V2 V n (4-3 Note tat te workload varance s frst ntroduced n [36] and [37] n te form of case were : = 2 C / C for a specal By substtutng (4-5 nto (4-6, and ten optmze te consequent nequalty for te dfferent parameters, we ave te followng closed-form scedulablty bound for RM wt tasks avng s-saped workload constrant functons Teorem 4: Gven a RM and collecton of tasks wt s-saped workload constrant functons, a scedulablty bound wt θ = s gven by 7

21 W ( C D D D E E E F G n η D E G η + H H mn G, G n G I k J H+ k H H F I k ; k I I η J J J = KF D E D E F G G kη + H H n Fnkη G I J H k = 2, 3, L I kη J (4-4 were η = V(M kn, Γ Proof: ee Appendx E (4-4 forms a tree dmensonal desgn space caracterzed by k, η and n It s compatble wt exstng known results for fxed-prorty systems, and represents a muc broader range of real-tme systems To make a sde-by-sde performance comparson between (4-4 and prevously known bounds, we set θ to, and derve te relatonsp between workload rate and te classcal utlzaton as follows OQ PR F ( kp k W (, Γ = = C (4-5 n n = = kp k P U V Te rgt and sde of (4-5 s obtaned by te dentty of F ( kp = kp / P T C, accordng to (4- For te eterogenety functon, we set te measurng wndow W = M kn, because ts settng s te same as te measurng wndow of workload rate and can smplfy te dervaton of te bound 5 Performance Evaluaton and Comparson 5 Analyss of te RM cedulablty Bounds Fgure 6: cedulablty Bounds of RM 8

22 Fgure 7: Comparson of New cedulablty Bounds wt Prevous Results On te bass of te results developed n ecton 4, we evaluate te system performance n ts secton, usng te scedulablty bound as te prmary performance measure Te followng factors affect te scedulablty bound: Te normalzed deadlne, k Wen normalzed deadlnes of tasks become tgter, te expected scedulablty bound decreases Te eterogenety of a task set, η Ts parameter gauges ftness of te workload constrant functon n capturng te spread of ob szes It gves one te ablty to mprove resource allocaton wtout requrng usng te worst case ob sze n te perodc model Te system performance s expected to mprove wt ncrease of X value Te senstvty of te scedulablty bound wt respect to two key factors, k and Y s analyzed, and te results are plotted n Fgure 6 Te 3-D graps of scedulablty bounds vs te two parameters are examned for a tousand tasks, or n =,000 For ease of comparson wt exstng results, we converted te workload rate to te classcal utlzaton based on (4-5, and we vared k from 0-4 to 0 4 and Y from to 0 2 Te followng observatons can be made: As expected, te tgter te deadlnes, te lower te scedulablty bounds Te senstvty s especally sgnfcant wen k s small (e, less tan 0 For example, n Fgure 6, wen k canges from 0 - to 0, te scedulablty bound ncreases from 0% to 953% Te senstvty becomes less sgnfcant for large k values In Fgure 6, wen k canges from 0 to,000, te scedulablty bound ncreases from 953% to 999%, only a 46% gan 9

23 2 Large eterogenety leads to mproved scedulablty bounds Te senstvty s ger wen X s small, (e, 0, and becomes lower for larger Y values For example, n Fgure 6, k =, wen X canges from to 0, te scedulablty bound ncreases from 693% to 953% Wen X canges from 0 to 00, te scedulablty bound ncreases from 953% to 995%, a merely 42% gan 52 Parametrc Fttng of Exstng Utlzaton Bounds for RM systems Teorem 4 gves te scedulablty bounds for a wde range of systems based on tree parameters normalzed deadlne, eterogenety of task set, and degree of deadlne nversons n a tree dmensonal space It s muc more general tan results reported n [22], [27], [32], [36], and [37] tat are merely one pont and tree curves n te 3-D grap on a surface constructed from our utlzaton bound n Fgure 7 In ts subsecton, we llustrate ow to reduce Teorem 4 nto tose prevous studes Except for te slgt dfference between (6-8 and (6-9, we matc all oter results mentoned above Frst, we consder te classcal rate monotonc sceduler wt perodc tasks, wose deadlnes are equal to ter perods In [32], Lu and Layland derved a scedulablty bound as follows: U Z [ \ ] n = n ^ 2 _ (5- Now, we re-derve (5- by usng Teorem 4 Note tat by (4-5, te workload rate reduces to te classcal defnton of utlzaton n ts case nce ts s a perodc system, by (4-2 and (4-3, we ave X = V(, Γ = Because te relatve deadlne s equal to te lengt of perod, we ave k = ubsttutng X = and k = nto (4-4, we ave ` a b c U = W ( = n d 2 n e (5-2 As n approaces nfnty, U and W wll approxmately become 693% Ts s represented by one sngle pont n Fgure 7 wc s supermposed on te Fgure 6 to glgt te matc econd, we analyze te perodc system n wc task s deadlnes are sorter tan perods, and tasks are sceduled by a rate monotonc sceduler In [27] and [4], Leoczky, a, Peng, and n derved an utlzaton bound 20

24 U f g k k 2 = g k m n n l on (2 k + k < k 2 (5-3 We now re-derve (5-3 usng Teorem 4 By (4-2 and (4-3, we ave p = Because te relatve deadlnes are less tan te perod lengts, we know tat k < ubsttutng p = nto (4-4, we get q r k 2 u W ( = r sv x xs v t n y w ty w z n (2 k + k 2 < k k (5-4 However, by (4-5, n ts case te workload rate and te classcal utlzaton ave te relatonsp U W ( = (5-5 k Tus, by substtutng (5-5 nto (5-4 and rearrangng t, we ave U { k k 2 = kw ( = } ~ n ƒ n (2 k + k < k 2 (5-6 Ts s exactly te same as (5-3 We llustrate ts result by a curve n te 3-D grap n Fgure 7 Trd, we analyze te perodc system n wc task deadlnes are multples of perods and tasks are sceduled by a rate monotonc sceduler In [22], Leoczky obtaned an utlzaton bound for k = 2, 3, ˆ + U = k( n ( Š (5-7 k n k Note tat by (4-5, te workload rate n ts case reduces to te classcal utlzaton We wll re-derve (5-7 troug Teorem 4 nce ts s a perodc system, we ave p = Because te relatve deadlne s a multple of perod, we ave k a postve nteger larger tan By substtutng p = nto (4-4, we get Œ Ž + U = W ( = kn ( (5-8 k n k 2

25 Note tat (5-8 s not as tgt as (5-7 (n vs n - Ts s due to te fact tat Teorem 4 s obtaned for general s-saped functons However, te exact bound of (5-7 can be easly obtaned wt our general metod ee Appendx F for ts dervaton In Fgure 7, a curve llustrates ts matc of results Fourt, we analyze te AM multframe system n wc task s deadlnes are equal to perods and tasks are sceduled by a rate monotonc sceduler In [36] and [37], Mok and Cen obtaned an utlzaton bound as follows + U = rn (, (5-9 r n r were r = E E, and 0 mn =,, n{ / } E = C, 2 E = C, are te sze of te frst and 2 second obs as descrbed n Example 2 By (4-5, te workload rate s te same as te utlzaton defned n [36] and [37] Now we re-derve (5-9 from Teorem 4 nce ts s a multframe system, by (4-3, (4-2, and Example 2, we ave p = V(, Γ = mn { C / C } = mn { E / E } = r Because relatve deadlnes are equal to 2 0 =,, n =,, n perods, we ave k = ubsttutng λ =, p = r, and k = nto (4-4, we get š + W ( = rn ( œ, (5-0 r n r wc s exactly te same as (5-9 Agan, ts matc s drawn on Fgure 7 6 Extensons 6 Harmonc Can Tasks In [6], te autors proved tat te utlzaton bound for perodc task mproves wen te perods of tasks are dvsble Ts observaton s applcable to Teorem 4 as stated next Corollary 4: Gven RM and a task set Γ wt s-saped workload constrant functons, Γ s scedulable f W ž ž ž Ÿ Ÿ Ÿ λ η + k k mn, η n' ( + k λ; k η λ λ = ž Ÿ λ k kη + λ n' ηn' ( k = λ, and s a postve nteger λ kη ž Ÿ n' were n s te number tasks wt non-dvdable segment lengts Proof: ee Appendx G (6-22

26 62 Arbtrary tatc Prorty cedulers In ts secton, we extend Teorem 4 for arbtrary statc prorty scedulers To model arbtrary statc prorty scedulers, we wll ntroduce a new parameter, λ, te degree of deadlne nverson for task T, as follows, to ( D max =, 2,, λ = (6-2 D Note tat λ Wen λ =, deadlnes of tasks wt prorty ger tan T are less tan or equal D Hence, no deadlne nverson occurs to T Wen λ >, deadlne nverson occurs to task T Takng one step furter, we let λ denote te degree of deadlne nverson for te wole task set ( λ = λ λ λ (6-3 max, 2,, n Wen λ =, te sceduler becomes deadlne monotonc Furtermore, wen λ = and D = P te sceduler becomes rate monotonc Corollary 5: For an arbtrary statc prorty sceduler wt s-saped workload constrant functons, G ( I, a servce constrant functon for task T, s ( G ( I = max x F ( x (6-4 0 x I λ Proof: Corollary 5 s stragtforward snce te prortes of tasks are assgned based on te lengts of task segments By substtutng Corollary 5 nto Teorem 3 and mnmzng te subsequent nequalty, we ave te followng results on te scedulablty bound of arbtrary statc prorty sceduler and tasks wt s-saped workload constrant functons Teorem 5: Gven a statc prorty sceduler and tasks wt s-saped workload constrant functons,, a scedulablty bound s gven by W «λ ««η + k k ª mn, η n ( + k λ; «k η λ λ = ª λ ª k «kη + λ n ηn ( k = λ, and s a postve nteger λ kη n Proof: ee Appendx H (6-5 23

27 63 Non-preemptve ystems In a non-preemptve system, a g prorty task T can be blocked by a lower prorty task T, >, for a lengt of J max, te maxmum ob sze of T Tus, ts non-preempton effect wll lead to a prorty nverson for te lengt of J max Durng ts nterval, te system acts as f T as a ger prorty tan T In te worst case, te system n ts nterval s operatng n a mode wt λ λ =, 2,, n D =, 2,, n D = = max { }/ mn { } By substtutng ts nto Teorem 5, we easly obtan te scedulablty bound for non-preempton case Corollary 6: Gven a non-preemptve statc prorty sceduler and a task set Γ wt s-saped workload constrant functons, Γ s scedulable f W ± ² ² ² ³ ³ ³ µ λ µ µ η + k k n' mn, η n' ( + k λ ; ² ³ µ k η λ λ = ¹ ² ³ λ k µ kη + λ n' º ηn' ( k = λ, and s a postve nteger λ kη (6-6 were = max { D }/ mn { D } and n s te number tasks wt non-dvdable segment λ =, 2,, n =, 2,, n lengts Proof: ee Appendx I 7 Fnal Remarks Dervng te utlzaton bound s an mportant, but callengng, task for sound desgn of real-tme computng systems Most bounds reported n earler work were obtaned by searcng along te boundary between te spaces of scedulable and non-scedulable task sets Ts approac s urdled by te dffculty of fndng an effectve analytcal representaton of te boundary To overcome ts callenge, n ts paper we follow te network calculus framework to derve a general scedulablty bound by solvng a mnmzaton problem over te entre task set populaton Our approac greatly enances te applcablty of utlzaton-based scedulablty testng to a wde range of task models and scedulers In partcular, we ave derved closed-form formulae for scedulablty bound of rate-monotonc, and non-rate monotonc scedulng systems Te bounds are parameterzed for dfferent prorty assgnments and for varous workload patterns We sow tat our general scedulablty bounds reduce to te utlzaton bounds derved n [26], [27], [32], [36], [37], and [4] by smple fttng of proper parameters Te senstvty of system performance, e, scedulablty 24

28 bounds, can be easly caracterzed wt respect to normalzed deadlne, eterogenety of ob szes, and te degree of deadlne nverson An nterestng tradeoff relatonsp between te tgtness of deadlne and te degree of deadlne nverson s explctly defned n te closed-form formulae Te work reported n ts paper can be extended n many dfferent ways Ts paper focuses on an nfnte number of task sets and derves te worst-case scedulablty bound In practcal systems, t s common tat te task collecton may be fnte and some of te parameters are know beforeand For example, te lengt of te segments can be only cosen from a fnte set {, 2, 3,, m } In tese cases, te scedulablty bound can be furter mproved wt lnear programmng metods [6], [25], and [39] In [22], te autors proposed an algortm to transform te task perods to armonc sets and ten measure te scedulablty by testng weter te transformed utlzaton s less tan Te same concept can also be appled to our system model to furter mprove utlzaton bound result Te scedulablty analyss and consequent dervaton of scedulablty bounds can also be extended to parallel and dstrbuted systems A task n suc systems may be processed by several nodes, were proper modelng of nter-node traffc s a must Te work can also be extended to applcatons n wc statstcal guarantees are preferred In tat case, we may ave to consder randomzed workload constrant functons and derve scedulablty bounds tat guarantee te deadlnes of tasks wt a predefned probablty 25

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