Éditeur Inria, Domaine de Voluceau, Rocquencourt, BP 105 LE CHESNAY Cedex (France) ISSN

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1 Uté de recherche INRIA Lorrae, techopôle de Nacy-Brabos, 615 rue du jard botaque, BP 101, VILLERS-LÈS-NANCY Uté de recherche INRIA Rees, IRISA, Campus uverstare de Beauleu, RENNES Cedex Uté de recherche INRIA Rhôe-Alpes, 46 aveue Félx Vallet, GRENOBLE Cedex 1 Uté de recherche INRIA Rocquecourt, domae de Voluceau, Rocquecourt, BP 105, LE CHESNAY Cedex Uté de recherche INRIA Sopha-Atpols, 2004 route des Lucoles, BP 93, SOPHIA-ANTIPOLIS Cedex Édteur Ira, Domae de Voluceau, Rocquecourt, BP 105 LE CHESNAY Cedex (Frace) ISSN

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3 51 {majorg the celg fucto} w w, q ( q + 1)C, q C. T j j j < => w, q q ( q + 1)C + C j j < C j T j < j q = C j qt C T j j j j C j T j < j As C j U 1 T j j => r max ( w, q qt ) r C C = =. j j T j j < j

4 50 Theorem 23 - A upper boud r of the worst-case respose tme of a task usg EDF s D r D j + T j = C. j D j T j D T j a + D Proof. From last theorem, we have: r max 1 D + j C a S T j a D j a + D j a S, elmatg the floor fucto, we have: a + D r ( a) 1 D j C, T j j a D j a + D. Thus that leads to: D D j + T j r ( a) max C C j. T j j a T D j a + D D j a + D j C j As U 1, we have a [ 0, L] : T D j a + D j D D j + T j r ( a) D D j + T j C T j j C T j j D j a + D D j T j D = r. B.2 Suffcet Codto for fxed prorty drve preemptve schedulers Usg fxed prorty drve schedulers, the followg feasblty SC establshes upper-bouds o the worst-case respose tmes. Ths result s establshed by frst a overvaluato o w, q ; ad secod by elmatg the term fucto of q. Theorem 24 - Usg statc schedulers, a upper boud of he worst-case respose tme r of a task τ of a geeral task set s foud by: r = C j 1 j C j T j < j (21) Furthermore, the geeral task set s feasble f:, r D. Proof. {from the state of the art, a NSC for statc prorty drve scheduler s (see Secto 3.2.2)} (, q) w, = ( q + 1)C, q C j q D w q j < T j +

5 49 Theorem 22 - Usg EDF, a upper boud of the worst-case respose tme of a task τ ca be foud settg the absolute deadle of τ o every absolute deadles of the tasks the sychroous busy perod ad s gve by: max a + D 1 D + j C. a S T j a D j a + D j D Where S = S p, S p D p + kt p D k ℵ D p L + D k D p =,, < ad L p = 1 T p T p s the legth of the sychroous processor busy perod (see Aex A). Proof. Let τ be a task released at tme t. The hgher prorty workload (see Secto 3.1.3) arrved up to tme t ca be reformulated as t s W ( a, t) m j a + D D =, + j C. T j T j D j a + D j Where s j = 0 f j ad s j = a a ---- else. The worst-case respose tme of task τ s the gve by r ( a) = max( C, L ( a) a) where L ( a) = W ( a, L ( a) ). t > 0, we have the: W ( a, t) 1 D j a + D + a + D D j C. j Hece, L ( a) a + D 1 D + j D C. As T j C 1 D + j D j a + D j D j D T j C, we have: j a + D r max 1 D + j C. a [0,L) T j a D j a + D j a + D Let f ( a) 1 D = + j C. s a step fucto whose value crease T j a f ( a) D j a + D j whe a + D D j = kt j.e such that the absolutes deadle of τ ( a + D ) cocde to the absolute deadle of a task released the sychroous scearo ( D j + kt j ) leadg to a S j, j = 1. Notce that f ( a) s mootoously decreasg betwee two successve values S. Thus we oly eed to fgure f ( a) for tmes S. Hece, a upper-boud of the worst-case respose tme of task τ s the gve by max a + D 1 D + j C. a S T j a D j a + D j T j From the last formula, we ca ow gve a polyomal tme computato upper boud o the worst-case respose tme of ay task τ. Ths boud s ot tght but ca be calculated o le.

6 48 Theorem 21 - A geeral task set sorted by creasg order of D = m( D, ) s feasble D + T j D j usg EDF f:, C j D D j D T j h Proof. Let h ( t) be the processor demad for task τ, f t D ( t) 1 we have t -- t 1 t D = h ( t) ad = 0 else. t D, we have: t h ( t) t -- t 1 t D C -- t + D = C -- t + D C T t t C I the same maer, we have: h( t) h j ( t) = t = t j = t 1 + D j t t D j T j 1 C j -- t D j t + T j D j C. T j j t As 1 -- t + D C s a o decreasg fucto, ts maxmum value s obtaed for t t = D..e. h ( t) D + D C. t D Hece, t [D, D + 1 ), h( t) t D What s more, t max( D ), D j D D + T j D j C. (19) T j j h( t) t t j t + T j D j C T j j max( D ) j max( D ) + T j D j C j T j. (20) As a NSC for the feasblty of a task set s to check that t 0, h( t) t (see Theorem 6). Applyg ths to eq. 19 ad eq. 20, a suffcet feasblty codto s to check that, D + T j D j C T j j D D j D

7 47 Aexe B. Polyomal suffcet codtos (SC) Let us ow troduce preemptve cotext some possble polyomal suffcet, but ot ecessary, feasblty codtos for geeral task sets. The ma objectve of ths aex s to propose several approaches that wll eable us to avod the cost of the pseudo-polyomal NSC whe possble. B.1 Suffcet Codtos for dyamc prorty drve preemptve schedulers Usg EDF, the frst SC s establshed, usg classc overvaluato of h( t) ad dervg a upper boud o the authorzed processor utlzato (see Theorem 20). The secod SC s establshed by a h( t) overvaluato of eablg us to check ths expresso o exactly worst-case pots (see Theorem t 21). A thrd ad fourth SC wll establsh upper-bouds o the worst-case respose tmes, based o the sychroous patter of arrval oly (ad ot those defed Secto 3.1.3). More precsely, the thrd SC uses overvaluato o W ( a, t) that eables us to avod the cost of the recursve aalyss (see Theorem 22). The, the fourth SC uses a extra overvaluato that eables us to obta a polyomal computato tme upper boud o the worst-case respose tme of a task (see Theorem 22). Theorem 20 - A geeral task set (wth EDF, f: D = m( D, ) ) s feasble, usg 1 U 1 m ( D ) 1 D = 1 C (18) Proof. from the state of the art, a NSC s: t 0 t D h( t) = C t D t As, D, elmatg the floor fucto ad by the defto of U. t 0 h( t) t + D C t + D D C t U D t D t D Therefore a suffcet feasblty codto s to check that t 0, t U C.e t = 1 1 U As t s suffcet to check the feasblty o the absolute deadles ad as t 1 D C = s a creasg fucto by t, a least upper boud o the processor utlzato t 1 D C = 1 s obtaed at t = m( D ). Hece a feasblty codto s to check that = 1 U m( D ) 1 D C = 1 C

8 46 Note that the algorthm s also vald whe preempto s ot allowed. I ths case, however, the equato for the computato of L ( a) s slghtly dfferet, sce we must take to accout the effect of o-preempto, amely, possble prorty versos. I partcular, the equato becomes: L L ( a) B m ( a) a + D , 1 D k = Ck D k a + D T k T a ---- k T C, k ad B = max D j > a + D { C j 1}. If our goal s oly to check the feasblty of the task set (ad ot to do a full worst-case respose tmes aalyss), the value of L computed by the prevous algorthm mght be uecessarly large. Ths s because whe checkg the feasblty, by Lemma 1 we ca focus our atteto oly o sychroous deadle busy perods ( s = 0, ). Ufortuately, ths way we loose the property show Lemma s 10, sce we caot apply the same swappg argumet. For ay task τ, the ew value L s the computed the followg way: a = ( L D ) ; whle L ( a) a a = ( L ( a) D ) ; ed whle s L = L ( a) ; Other detals of the hgher prorty busy perod legth computatos, ether for statc or for dyamc prorty models, are gve the specfc sectos, where the dffereces betwee the preemptve ad the o-preemptve models (actually very few) are also remarked.

9 45 Lemma 11 -, L ( a) s o-decreasg a. Proof. By defto, L ( a) s the smallest soluto of Equato (17), hece for ay t < L ( a) t < ( ) a ---- C. W a, t Gve a' a, we thus that t < L ( a) t < ( ) a ---- C W ( a', t) a' ---- C, W a, t sce W ( a, t) s o-decreasg a. It follows that L ( a' ) L ( a). By usg the results of Lemma 8, Lemma 10 ad Lemma 11, we ca compute the values of L the followg way. Let E = { mt + D, m 0} = { e, let be the legth of the = 1 1, e 2, } L sychroous busy perod, ad assume that D D + 1,. The algorthm for the computato of L starts by placg a stace of τ such a way to have ts deadle at the largest e k smaller tha or equal to L + 1 C + D (t s easy to see that these are the scearos whch gve the logest deadle busy perods for τ ). If the resultg deadle- e k busy perod cludes ths last stace of τ we have foud L, otherwse the computato s doe aga by choosg a ew sutable deadle e k', ths tme accordg to the value of L ( e k D ). Note that the algorthm always stops. The pseudo-code follows: L + 1 = L ; for = dowto 1 let k be such that e k L + 1 C + D < e k + 1 ; a = e k D ; ed for whle L ( a) a let k be such that e k L ( a) C + D < e k + 1 ; a = e k D ; ed whle L = L ( a) ;

10 44 t = W ( a, t) a ---- C. (17) Wth L we deote the legth of the logest such deadle busy perod for task τ. We have the L = max a < L ( a) { L ( a) }. I order to speed up the computato of lemma. L, we ca utlze two ce propertes show the followg Lemma 10 - D D j L L j Proof. By defto of L, d = s + m + D for whch L = L ( d D ) : L = = L m d D, k Ck + ( m + 1)C D k d k T k T k L m d D, k L Ck ( m + 1)C m d D j + +, C j. D k d k k j T k T k T j T j Sce D, the last s stace wth deadle before or at has release tme smaller tha, 1 D j τ j d L the L = L m d D, k Ck + ( m + 1)C + d D j C, j D k d k k j T k T k If we cosder ow the scearo whch τ s released sychroously (.e., s becomes 0 ) ad τ j has start tme s j = ( d D j ) ( d D j ) T j T j, so that there s a stace wth deadle d, we have L L m d D, k Ck + d D j C. j D k d k j T k T k The rght quatty of the equalty s smaller tha or equal to the ew deadle- d busy perod, whch tur s by defto smaller tha or equal to L j. The thess follows. T j T j 1. We assume D j d. If ths s ot true, the thess trvally holds.

11 43 have a absolute deadle smaller tha or equal to d. Cosequetly, the umber of τ s staces to be take to accout s whch gves a sort of hgher prorty workload. Also, ths case we are ot merely terested ay deadle- d busy perod, but oly o those whch clude the last stace. More precsely, gve a τ s stace released at tme a, hece wth deadle d = a + D (see fgure 8), we are terested deadle- d busy perods that starts at t = 0 ad cludes ths stace (.e., whe ts legth s bgger tha a 1 ). It s ot dffcult to see that the logest busy perods are obtaed scearos whch all tasks but τ are released sychroously (see Lemma 3). Let L ( a) deotes the legth of such deadle- d busy perod that eds o the completo tme of the τ s stace released at tme a (wth d = a + D ), whle all other tasks are cosdered to be released sychroously ( s j = 0, j ). The deadle busy perods whch we are terested are oly those for whch m t d D ----, , L ( a) > a. 0 a τ d=a+d L (a) deadle-d busy perod Fgure 8 L ( a) s the computed by meas of the teratve computato ( 0) ( m + 1) ( m) L ( a) = 0, L ( a) = W( a, L ( a) ) + ( 1 + a T )C, where W ( a, t) m t a + D D, j = C j. D j a + D j T j T j The computato s halted whe two cosecutve values are foud equal. Note that ths way we fd the smallest soluto of the equato 1. Note that f the deadle- d busy perod s shorter tha a, the t s ot terestg ether for the feasblty checkg (accordg to Lemma 1 the deadle d caot be mssed), or for the computato of the worst-case respose tme of τ (see Secto 3.1.3).

12 42 A.2 Hgher Prorty Busy Perods So far, the cocept of busy perod has bee troduced as such, that s, as a perod whch the processor s ot dle. More geerally, t s possble to specalze the cocept by makg t relatve to a gve level of prorty. The dea s to cosder processor busy perods whch oly task staces wth prorty, ether statc or dyamc, at least at a gve level are executed. Ths form of busy perod s schedulg algorthm depedet, ad partcular t s useful for the computato of tasks worst-case respose tmes. I the case of fxed prorty systems, we talk of level- busy perods [LEH90]: DEFINITION 3 - A level- busy perod s a processor busy perod whch oly staces of tasks wth prorty greater tha or equal to that of execute. τ For the computato of worst-case respose tmes t s ecessary to compute the legth of sutable level- busy perods. The approach s very smlar to that see for the computato of the sychroous busy perod legth. I ths case, though, oly the workload of the tasks wth prorty greater tha or equal to that of τ s take to accout. The legth of the logest level- busy perod s deoted by the smallest soluto of the equato L, ad ca be easly computed by fdg L = j hp( ) { } L C j, T j for the preemptve case, ad L = max j lp( ) { C j 1} + j hp( ) { } L C j, T j for the o-preemptve case, respectvely. Note that the latter case, the frst term o the rght-had sze takes to accout the effect of o-preempto, that s, the o-preemptable executo of a lower prorty task stace ca delay the executos of hgher prorty task staces as much as max j lp( ) { C j 1}. As usual the equatos ca be solved by meas of teratve computatos. Smlarly, for dyamc prorty systems, ad partcular for deadle scheduled oes, we talk of deadle busy perods: DEFINITION 4 - A deadle- d busy perod s a processor busy perod whch oly task staces wth absolute deadle smaller tha or equal to d execute. Aga, the approach for the computato of deadle- d busy perod legths s as usual. We oly eed to be careful takg to accout the rght umber of staces for each task, whch requres slghtly more chages. I fact, ths case gve a terval of tme [ 0, t ), the umber of staces of τ released wth t s t, however oly 1 + ( d D ), provded that D d, 0 otherwse,

13 41 the recursvely appled to W( t), W( W( t) ),, utl we fd a value equal to the prevous oe. Formally, L s the fxed pot of the followg teratve computato: L ( 0) = C L m + 1 ( ) = W( L ( m) ). (15) The computato s halted whe two cosecutve values L ( m) ad L ( m + 1) are foud equal. L s the assged the value L ( m). Note that ths way we fd the smallest soluto of the equato L = W ( L). (16) The covergece of Equato (15) s proved the followg lemma. Lemma 9 - If C 1, Equato (15) coverges a fte umber of steps. Proof. C W( P) = P ---- C = C P ---- P. It follows that L P. Furthermore, the workload fucto W( t) s a o-decreasg step-fucto, hece L ( m) s o-decreasg m. Fally, at each step L ( m) s ether creased by at least C m or remas uchaged. Thus the fal value s reached a fte umber of steps. If we have the further hypothess that C c, where c s a costat smaller tha 1, the the complexty of L s computato becomes pseudo-polyomal [SPU96]. To show ths, we just eed few algebrac mapulatos: L L L = ---- C C = C C + L ---- C + cl. Hece C L c Each step of the teratve formula Equato (15) takes O( ) tme, thus the whole computato takes O( ( C tme. 1 ) C m ) 1. Note that fdg a full polyomal tme algorthm for the computato of L would mply a full polyomal tme procedure for the feasblty assessmet of a geeral task set (see secto 3.1.2), whch s stll a ope questo [BHR90].

14 40 Aexe A. Busy Perods, Deftos ad Propertes The oto of processor busy perod s very smple, but at the same tme also very powerful, sce the propertes (feasblty ad worst-case respose tmes) of fxed ad dyamc prorty o-dlg schedulers ca be exactly characterzed by usg ths cocept. More precsely, we frst focus o the oto of processor busy perod that does ot deped o the schedulg algorthm, ad the o the oto of hgher prorty busy perod, that s schedulg algorthm depedet. A.1 Processor Busy Perods DEFINITION 1 - A processor busy perod s a terval of tme whch the processor s kept cotually busy by the executo of pedg staces. Note that the defto othg s sad about what delmts the terval at ts sdes. However, uless otherwse stated we wll usually ted a tme terval delmted by two dstct processor dle perods,.e., ay perods such that o outstadg computato exsts. 1 Amog all processor busy perods, we fd partcularly terestg the frst oe obtaed by releasg all tasks sychroously from tme t = 0. DEFINITION 2 - Gve a geeral task set, we call sychroous busy perod the processor busy perod begg at tme t = 0 ad delmted by the frst processor dle perod, whe all tasks are cocretely released from tme t = 0 o at ther maxmum rate. The mportace of the sychroous busy perod s that, ot surprsgly eough, t s the most demadg oe for most of the o-dlg schedulg algorthms. Also, we ca easly prove that s the largest possble busy perod. Lemma 8 - Gve a geeral task set, let L be the legth of ts sychroous busy perod. If L' s the legth of a processor busy perod the schedule of ay derved cocrete task set, the L' L. Proof. The gve processor busy perod must be preceded by a dle tme, ad ts begg must cocde wth the release tme t of a task stace. If all staces of ay task released after t are shfted left as much as possble, possbly up to t, we obta a sychroous busy perod startg at t. Sce the workload betwee t ad t + L' caot decrease wth the shft-left argumet, the legth of the ew busy perod caot dmsh. Hece, L' L. Note that the property establshed by the prevous lemma does ot deped o the schedulg algorthm assumed, ether statc or dyamc, preemptve or o-preemptve. The oly assumpto s that the algorthm s o-dlg. The legth L of the sychroous busy perod ca be computed by meas of a smple procedure. Gve ay terval [0, t[, the dea s to compare the geerated workload W( t) wth the legth t of the terval: f W( t) s greater tha t the the durato of the busy perod s at least W( t). The argumet s 1. Note that a processor dle perod ca have zero durato, whe a ew task stace s released at the ed of a busy perod.

15 39 [LL73] [LM80] [LS95] [LW82] [MA84] C.L Lu, James W. Laylad, Schedulg Algorthms for multprogrammg a Hard Real Tme Evromet, Joural of the Assocato for Computg Machery, Vol. 20, No 1, Jav J. Y.T.Leug et M.L.Merrl, A ote o preemptve schedulg of perodc, Real Tme Tasks, Iformato processg Letters, Vol. 11, um. 3, Nov L. Leboucher, Jea-Berard Stefa, Admsso cotrol for ed-to-ed dstrbuted bdgs, COST231, Lectures Notes Computer Scece, Vol 1052, pp192,208, Nov J. Leug, J. Whtehead. O the complexty of fxed-prorty schedulg of perodc, real-tme tasks. Performace Evaluato, vol 2, p , P. R. Ma, "A model to solve Tmg-Crtcal Applcato Problems Dstrbuted Computg Systems, IEEE Computer, Vol. 17, pp , Ja [MOK83] A.K. Mok, Fudametal Desg Problems for the Hard Real-Tme Evromets, May 1983, MIT Ph.D. Dssertato. [RCM96] I. Rpoll, A. Crespo, A.K. Mok, Improvemet Feasblty Testg for Real-Tme Tasks, Real-Tme Systems, 11, 1996, pp [SRL90] [SPU95] [SPU96] L. Sha, R. Rajkumar, ad J.P. Lehoczky, Prorty Ihertace Protocols: A Approach to Real-Tme Sychrozato, IEEE Trasactos o Computers, 39(9), M. Spur, Earlest Deadle schedulg real-tme systems, Doctorate dssertato, Scuola Superore S. Aa, Psa, M. Spur, Aalyss of deadle scheduled real-tme systems Research Report 2772, INRIA, Frace, Ja [SPU96-2]M. Spur, Holstc aalyss for deadle scheduled real-tme dstrbuted systems Research Report 2873, INRIA, Frace, Apr [STA95] J.A. Stakovc, M. Spur, M. D Natale, ad G. Buttazzo, Implcatos of Classcal Schedulg Results for Real-Tme Systems, IEEE Computer, Jue [TBW94] K. Tdell, A. Burs, A. J. Wellgs, A extedble approach for aalyzg fxed prorty hard real tme tasks, Real-Tme Systems, 6, 1994, p [TBW95] K. Tdell, A. Burs, A. J. Wellgs, Aalyss of hard real-tme commucatos, Real- Tme Systems, 9, 1995, p [THW94] K. Tdell, H. Hasso, A. J. Wellgs, aalyzg real-tme commucatos: cotroller Area Networks (CAN), Real-Tme Systems, 1994, p [TIN95] K. Tdell, Holstc Schedulablty Aalyss for Dstrbuted Hard Real-Tme Systems, Euromcro Joural, Specal Issue o Embedded Real-Tme Systems, Feb [YUA91] Xaopg Yua, A decomposto approach to No-Preemptve Schedulg o a sgle ressource, Ph.D. thess, Uversty of Marylad, College Park, MD [YUA94] Xaopg Yua, Maas C. Saksea, Ashok K. Agrawala, A decomposto approach to No- Preemptve Real-Tme Schedulg, Real-Tme Systems, 6, 7-35 (1994). [ZRS87] [ZS94] W. Zhao, K. Ramamrtham, J. A. Stakovc. Schedulg Task wth Resource requremets a Hard Real-Tme System, IEEE Tras. o Soft. Eg., Vol. SE-13, No. 5, pp , May Q. Zheg, K.G. Sh, O the Ablty of Establshg Real-Tme Chaels Pot-to-Pot Packet-Swtched Networks, IEEE Trasactos o Commucatos, 42(2/3/4), 1994.

16 38 Refereces [AUD91] N. C. Audsley, Optmal prorty assgmet ad Feasblty of statc prorty tasks wth arbtrary start tmes, Dept. Comp. Scece Report YCS 164, Uversty of York, [BAK91] T.P. Baker, Stack-Based Schedulg of Real-Tme Processes, Real-Tme Systems, 3, pp , [BHR90] S. K. Baruah, R. R. Howell, L. E. Roser, Algorthms ad Complexty Cocerg the Preemptve Schedulg of Perodc Real-Tme tasks o oe processor, Real-Tme Systems, 2, pp , [BHR93] S. Baruah, R. Howell, ad L. Roser. Feasblty Problems for Recurrg Tasks o Oe Processor. Theoret. Comput. Sc. 118 (1993), pp (84 K). [BMR90] S. K. Baruah, A. K. Mok, L. E. Roser, Preemptvely Schedulg Hard-Real-Tme Sporadc Tasks o Oe Processor, Proceedgs of the 11th Real-Tme Systems Symposum, p , [CL90] [COF76] M. Che ad K. L, Dyamc Prorty Celgs: A Cocurrecy Cotrol Protocol for Real- Tme Systems, Real-Tme Systems, 2, pp , E.G. Coffma, Jr., Itroducto to Determstc Schedulg Theory, : E.G. Coffma, Jr., Ed., Computer ad Job-Shop Schedulg Theory, Wley, New York, [DER74] M. Dertouzos, Cotrol Robotcs: the procedural cotrol of physcal processors, Proceedgs of the IFIP cogress, p , [GJ79] M. R. Garey, D. S. Johso, Computer ad Itractablty, a Gude to the Theory of NP-Completeess, W. H. Freema Compay, Sa Fracsco, [GMR95] L.George, P.Muhlethaler, N. Rverre, Optmalty ad No-Preemptve Real-Tme Schedulg Revsted, Rapport de Recherche RR-2516, INRIA, Le Chesay Cedex, Frace, [HLR96] J.F. Hermat, L. Lebouche, N. Rverre, O comparg fxed/dyamc prorty drve schedulg algorthms Rapport de Recherche, INRIA, Le Chesay Cedex, Frace, to appear [HV95] [JP86] [JSM91] [KLS93] [KN80] [LEH90] R. R. Howell, M. K. Vekatrao, O o-preemptve schedulg of recurrg tasks usg serted dle tme, Iformato ad computato Joural, Vol. 117, Number 1, Feb. 15, M. Joseph, P. Padya, Fdg respose tmes a real-tme system, BCS Comp. Jour., 29(5), pp , K. Jeffay, D. F. Staat, C. U. Martel, O No-Preemptve Schedulg of Perodc ad Sporadc Tasks, IEEE Real-Tme Systems Symposum, Sa-Atoo, December 4-6, 1991, pp D. I. Katcher, J. P. Lehoczky, J. K. Strosder, Schedulg models of dyamc prorty schedulers, Research Report CMUCDS-93-4, Carege Mello Uversty, Pttsburgh, Aprl Km, Naghbdadeh, Preveto of task overrus real-tme o-preemptve multprogrammg systems, Proc. of Perf., Assoc. Comp. Mach., 1980, pp , J.P. Lehoczky, Fxed prorty schedulg of perodc task sets wth arbtrary deadles, Proceedgs 11th IEEE Real-Tme Systems Symposum, Lake Buea Vsta, FL, USA, pp , 5-7 Dec

17 37 7. Cocluso I ths paper, we have focused o the schedulg of geeral task sets (.e., o-cocrete perodc or sporadc task set such as, ad D are ot related) as a cetral fgure for the descrpto of possble processor loads, several cotexts. Although a lot of results were already kow, some ew results have bee establshed ether preemptve or o preemptve cotext. Our hope s that, gve a real-tme problem, ths work mght be helpful to pck a soluto from the plethora of results avalable. I partcular, the optmalty of preemptve/o-preemptve, fxed/dyamc prorty drve schedulg algorthms, the respectve feasblty codtos ad worst-case respose tmes have bee examed (see Secto 6 for a syoptc). Some classc extesos such as jtter ad resource sharg have also bee cosdered. Although ths work s ot oreted toward a comparso of these results, t appears that preemptve ad o-preemptve schedulg are closely related. Ideed, the optmal schedulg algorthms are smlar both cases. The ma dffereces, owg to the absece of preempto, are frst that a task stace wth lower prorty ca possbly cause a prorty verso before a hgher prorty busy perod; ad secod that whe estmatg the respose tme of a task stace, the atteto must be o the hgher prorty busy perod precedg the executo start tme of the stace, ad ot o the hgher prorty busy perod precedg ts completo tme. These dffereces slghtly chage the feasblty codtos ad the worst-case respose tme expressos but the whole aalyss s very smlar. Moreover, fxed ad dyamc schedulg dffer, as already kow, may aspects such as mplemetablty, effcecy, complexty, etc.. Ths s ot surprsg gve that the optmalty property of EDF s more geeral tha wth ay fxed prorty scheduler (the terested reader s referred to [HLR96] for a formal comparso of the effcecy ad the complexty of fxed/dyamc prorty drve schedulg preemptve cotext). However, the aalyss ca be ufed by usg the cocepts of processor busy perod, that are scheduler depedet, as well as hgher prorty busy perod, that are scheduler depedet. Ideed, t appears that these cocepts are geeral ad very useful for the detfcato of the worst possble desty of arrval ad worst-case respose tmes. I partcular, we have troduced the cocept of deadle-d busy perod for dyamc prorty drve schedulg that we cojecture as a terestg parallel of the level- busy perod already used fxed prorty drve schedulg. The ma dffereces we detect are: frst that the mpact of the prorty versos caused by the absece of preempto dsappear swftly usg EDF (after max{ D } ay deadle-d busy perod) whe t perssts throughout the level- busy perod the fxed prorty case. Ths remark s favour of EDF whe preempto s ot allowed, ad s ot surprsg sce ay prorty verso, usg EDF, refers to a absolute deadle (ad ot a fxed prorty level) that s dyamcally maaged. secod that ay geeral task set whch s feasble by EDF o-preemptve cotext s ecessarly feasble wth EDF preemptve cotext. Ths property does t hold whe fxed prorty assgmet are cosdered (e.g., Secto a task set that s feasble o-preemptve cotext but ot feasble preemptve cotext s gve). Hece coversely to EDF prorty assgmet, there s o obvous relatoshp betwee the feasblty of a fxed prorty assgmet preemptve ad o-preemptve cotexts, The ma questo whch stll remas ope s whether there exsts a fully polyomal soluto to the feasblty problem for geeral task sets. Eve f otos lke processor demad, deadle busy perod ad level- busy perod have bee helpful for the mprovemets of the kow solutos, both the dyamc ad the fxed prortes models the state of the art s represeted by pseudo-polyomal algorthms. Whether the problem s NP-hard s also ot kow at preset [BHR90].

18 36 where w, ( q + 1)C B, q + J = + j C j w q j hp( ) T j (12) Furthermore, the geeral task set s feasble f:, r D No-preemptve case As for the o-preemptve dyamc case, resource sharg s qute smple to maage o-preemptve fxed prorty cotext sce, owg to the absece of preempto, there s stll o eed of ay partcular protocol lke the prorty celg. The oly possble prorty versos are the caused by the absece of preempto (see Theorem 15). O the other had, the worst patters of arrval, cosderg release jtter o-preemptve cotext, are also smlar to those detfy preemptve cotext,.e., whe tasks experece ther shortest ter-release tmes at the begg of the schedule. Therefore, to deal wth the cotext of ths secto, Equato (11) ad Equato (12) oly have to be replaced by: r = max q ( w, q + C + J q ) (13) where w, qc 1, q + J = j C j + max k lp( ) { C k 1}. w q j hp( ) T j (14) Note that a smlar approach has bee descrbed [TBW95] for the aalyss of real-tme etworks whch packetzed messages have access to the physcal medum accordg to ther fxed prortes. 6. Sythess As stated Secto 2.3, the goal of ths paper was to fll Table 1 for geeral task sets (.e., to put together optmalty propertes, feasblty codto & worst-case respose tmes o-dlg, preemptve/o-preemptve, fxed/dyamc prorty drve cotexts). To that ed, we ca ow summarze the foregog Table 2 where whte cells deote exstg results whe grey cells deote a exteso of exstg results or ew results. TABLE 2. Results for geeral task sets Dyamc prortes Secto 3.1 Preemptve schedulg No-preemptve schedulg Fxed prortes Secto 3.2 Dyamc prortes Secto 4.2 Fxed prortes Secto 4.3 Optmalty Theorem 1 Theorem 7 Theorem 11 Secto Feasblty codtos Worst-case Respose tmes Theorem 6 Theorem 10 Theorem 14 Theorem 15 Secto Theorem 10 Secto Theorem 15

19 35 As doe Secto 3.1 the effcecy of the feasblty aalyss ad the worst-case respose tme computato ca be mproved by usg the cocept of deadle busy perod No-preemptve case The feasblty aalyss of a geeral task set o-preemptve cotext (see Theorem 14) or presece of blockg factor (see Theorem 18) are qute smlar. Ths s ot surprsg sce both refer to prorty versos w.r.t. absolute deadles. It s also terestg to otce that resource sharg s qute smple to maage o-preemptve cotext sce, owg to the absece of preempto, there s o eed of ay partcular protocol. The oly possble prorty versos are caused by the absece of preempto. O the other had, the worst patters of arrval, whe release jtter s cosdered, are smlar to those detfed preemptve cotext,.e., whe tasks experece ther shortest ter-release tmes at the begg of the schedule. Therefore, to deal wth the cotext of ths secto, Equato (10) Theorem 18 oly has to be replaced by: t L, t + J 1 D C T + max D J > t{ C 1} t D t + J wth the coveto that max D > t{ C 1} = 0 f : D J > t. J 5.2 Fxed prorty drve schedulers Preemptve case Takg to accout shared resources ad release jtter presece of fxed prorty drve schedulg leads to the same reasog tha presece of dyamc prorty drve schedulg. Frst, whe = D, Sha, Rajkumar ad Lehoczky [SRL90] have exteded the suffcet codto of [LL73] (see Theorem 8) for the Prorty Celg Protocol: where B deotes the logest blockg tme of τ a lower prorty task (see Secto 5.1.1). The same ehacemet ca also be appled to the Lehoczky s busy perod aalyss. Moreover [TBW94], presece of geeral task sets, exteds ths aalyss further by cosderg the oto of release jtter ( J for τ ). Ther aalyss s a exteso of Theorem 10 that results the followg fal codto: C j T j = 1 j B ( 2 1 1), 1 Theorem 19 - ([TBW94]) The worst-case respose tme r of a task τ of a geeral task set presece of shared resources ad release jtters s foud a scearo whch all tasks are at ther maxmum rate ad released sychroously at a crtcal stat t=0. r s computed by the followg recursve equato (where hp() deotes the set of tasks of hgher prorty tha task τ ): r = max q ( w, q + J q ) (11)

20 34 5. Shared resources ad release jtter The above aalyses assume that all tasks do ot share resources. I real operatg systems, however, ths eeds ot be the case. We must therefore exted the aalyss to deal wth prorty verso problems whe dealg wth depedeces. Furthermore, ad for reasos such as tck schedulg or dstrbuted cotext (e.g. the holstc approach troduced by [TIN95] for fxed prorty schedulg, exteded for dyamc prorty schedulg [SPU96-2]), tasks may be allowed to have a release jtter. I ths secto, we gve some hts o how the aalyss descrbed prevously must be modfed order to exted the model accordgly. The terested reader may refer to the gve refereces. 5.1 Dyamc prorty drve schedulers Preemptve case If the tasks are allowed to share resources, the aalyss must take to accout addtoal terms, amely blockg factors, owg to evtable prorty versos. Note that: the maxmum durato of such versos ca be bouded f shared resources are accessed by lockg ad ulockg semaphores accordg to a protocol lke the prorty celg [CL90], [SRL90] or the stack resource polcy [BAK91]. I partcular, for each task τ t s possble to compute the worst-case blockg tme B, the maxmum tme a task τ may be blocked by lower prorty tasks whe accessg a shared resource. the legth L of the processor busy perods s uaffected by the presece of blockg stead. Prorty versos may oly cause the schedule to devate from ts ordary EDF characterstc. The requred modfcatos o the aalyss are oly few. The stace beg checked, or aother oe whch precedes t the schedule, may experece a blockg that has to be clude as a addtoal term. O the other had, f a task τ s delayed for a maxmum tme J (ts release jtter) before beg actually released, the two cosecutve staces of τ may be separated by the terval J. [SPU96] exames the feasblty of a geeral task sets presece of shared resources ad release jtter. Theorem 18 - ([SPU96]) a geeral task set presece of shared resources ad release jtters s feasble (assumg that tasks are ordered by creasg value of D -J ), usg EDF, f: t L, t + J 1 D C. (10) T + B k( t) t D t + J where L s the sze of the sychroous processor busy perod (see aex A) ad k( t) = max{ k ( D k J k t) }. The proof geeralzes Theorem 6 showg that the worst processor busy perod s stll the sychroous processor busy perod ad that the worst patter of arrval, cosderg release jtter, s whe tasks experece ther shortest ter-release tmes at the begg of the schedule. Cosderg shared resources, t s show that the worst patter of arrval arses wth the blockg factors of the task wth the largest D k J k value amog those cluded the sum. Smlarly, [SPU96] develops the same argumets for the computato of the worst-case respose tme.

21 33 Suppose ow that τ s feasble at prorty but has a prorty Φ τ ( τ ) <. By Lemma 7, chagg the prorty Φ τ ( τ ) of τ to caot crease the respose tmes of the other tasks, ad by hypothess τ s feasble at prorty. Hece τ s stll feasble whe tasks sb are assged prortes, + 1,,. From the above theorem, we ca derve a optmal prorty assgmet based o the approach descrbed [AUD91], whch s thus also vald o-preemptve cotext. Let τ = { τ 1,, τ } be a geeral task set. I order to determe a optmal prorty assgmet for those tasks, we proceed the followg way. We check f at least oe task s feasble (accordg to Theorem 15) f assged prorty. Two cases are possble: o task s feasble, the clearly o feasble prorty assgmet exsts for τ. at least oe task s feasble. If several tasks are feasble, we ca choose oe at radom. Ideed, by Theorem 17, f a feasble prorty assgmet exsts the oe wll exst wth prorty for the selected task. We the repeat the steps to prorty 1,, 1, uless the task set s foud uschedulable the meawhle. The pseudo-code of the algorthm follows. τ = { τ 1,, τ } beg for j = dowto 1 uassged = TRUE; for all tasks τ k τ f s feasble at prorty j the τ k assg to prorty j ; τ k τ = τ { τ k } ; uassged = FALSE; ed f f uassged= TRUE the ext; /* o prorty assgmet exsts for ed f ed for ed for ed τ Clearly, as for the preemptve case [AUD91], the tme complexty of ths procedure s O( 2 ).

22 32 Geeral task sets (, ad D are ot related) A optmal prorty assgmet for geeral task sets scheduled preemptve systems has bee descrbed [AUD91]. We wat to prove that the strategy s optmal also o-preemptve systems. I order to do ths, we frst eed to show that decreasg the prorty of a task s harmless for the other tasks. Lemma 7 - Decreasg the prorty of a task ca oly decrease or leave uchaged the respose tmes of the other tasks. Proof. Let τ = { τ 1,, τ } be a set of geeral tasks wth creasg prortes. Let τ k be the task whose prorty s to be decreased, ad let τ' = ( τ, 1, τ, k 1 τ, k + 1, τ, m τ, k τ, m + 1, τ ) be the ew prorty orderg. I τ' we ca dstgush three subsets: A = { τ 1,, τ k 1 }, B = { τ k + 1,, τ m }, ad C = { τ m + 1,, τ }. The worst-case respose tmes of the tasks A ad C are ot affected by the ew prorty orderg. Ideed, ther prortes do ot chage, whle the ew prorty of task τ k produces the same worst-case scearo (see Theorem 15). Thus the respose tmes of those tasks are uchaged. The tasks of subset B see ther prortes to decrease by 1 (havg more prorty) thus possbly decreasg ther worst-case respose tmes (obvous from the formula of Theorem 15). The followg theorem s spred from [AUD91], where t s prove a preemptve cotext. Let τ be a task set of geeral tasks, τ = { τ 1,, τ }, ad let Φ τ ( ) be a partcular prorty assgmet fucto, such that for ay task τ, = 1, Φ τ ( τ ) s the prorty of the task τ τ. Theorem 17 - Let sb = { τ,, τ } be a subset of τ. Suppose that tasks sb are feasble whe k =, task τ k s assged prorty k. If there exsts a prorty assgmet fucto Φ τ ( ) that eables τ to be feasble the there exsts a prorty assgmet fucto that assgs the tasks sb prorty, + 1,, such that τ s stll feasble. Proof. We prove the theorem by ducto. Suppose τ be feasble at prorty. Suppose that a feasble prorty orderg Φ τ ( ) exsts that assgs τ prorty m <. By Lemma 7, f the prorty of τ s chaged from m to the respose tmes of the other tasks s ot worseed. As τ s feasble at prorty level, the resultg prorty orderg s stll feasble. Suppose the property true for tasks τ + 1,, τ : those tasks reassged prorty + 1,, stll eable τ to be feasble. Let Φ τ ( ) to be updated accordg to the ew prorty assgmet.

23 31 Theorem 16 - DM s a optmal prorty assgemet for perodc or sporadc task sets wth D,, f D < D j C C j. Proof. We wll show that wheever a vald schedule exsts, a vald DM schedule exsts, too. I that purpose, let τ = { τ 1,, τ } be a set of tasks wth a gve feasble prorty assgmet. Let τ ad τ j be two tasks of adjacet prortes, wth τ j the hghest prorty oe (.e., hp( ) = hp( j) { j} ). Assume that D < D j. By hypothess we also have C C j. As the task set s schedulable, the worst case respose tme of each task s less tha or equal to ts relatve deadle. Furthermore, as, D, the worst-case respose tme of ay task s foud ts frst stace ( Q = 0 Theorem 15). Thus we have: r j = w j + C j, wth w j = ( 1 + w j T k )C k + max, ad k hp( j) k lp( j) { C k 1} r = w + C, wth w = ( 1 + w T k )C k + max. k hp( ) k lp( ) { C k 1} Note that w < w + C D < D j T j, so 1 + w T j = 1. Let us swap the prortes of τ ad τ j, so as to have a deadle mootoc orderg betwee the two tasks. Sce the prorty of τ has rased, ts ew worst-case respose tme caot crease, that s, the schedule remas vald for τ. Vce versa, the prorty of τ j has lowered, so ts worst-case respose tme may crease. The ew value s r j ' = w j ' + C j, wth w j ' equal to the smallest soluto of the equato w = ( 1 + w T k )C k + max k lp' ( j) { C k 1}, wth h p' ( j) = hp( j) { }. If k h p' ( j) we evaluate the rght term of the equato w = w C j + C, we have w 1 C j + C Ck + max k lp' ( j) { C k 1} k hp' ( j) k hp( j) { } k hp( j) k hp( j) w T k w T k T k w T k w Ck T k max k lp( ) { C k 1} + C C j = w + C C j. k hp( ) Ck + max k lp( j) { } { C k 1} = Ck + C + max k lp( ) { C k 1} = Ck + C j + max k lp( ) { C k 1} + C C j = It follows that w j ' w + C C j, hece r j ' = w j ' + C j w + C D < D j, that s, also τ j remas feasble wth the ew prorty assgmet. Sce the worst-case respose tmes of all other tasks are ot affected by the prorty swap, the ew prorty assgmet s globally feasble. A deadle mootoc feasble prorty assgmet ca be fally acheved wth a fte umber of smlar steps.

24 Optmalty Case D, Cotrary to the preemptve case, whe preempto s ot allowed DM s o loger a optmal prorty assgmet ths case. However, we frst show that the optmalty remas f slghtly stroger codtos are mposed. To prove that DM s o loger optmal, we oly eed to gve a couter example. Let τ = { τ 1, τ 2, τ 3 } be a perodc task set wth τ 1 ( C 1 =3, T 1 =5, D 1 =5), τ 2 ( C 2 =2, T 2 =10, D 2 =6), ad τ 3 ( C 3 =1, T 3 =10, D 3 =7). Assume the prortes are assged accordg to DM (.e., τ 1, τ 2 ad τ 3 have decreasg prortes). I a sychroous scearo, the frst occurrece of task τ 3 s ot executed by tme D 3 = 7 (see Fgure 7). Its respose tme (see Theorem 15) s r 3 = 2C 1 + C 2 + C 3 > D 3, hece the deadle s mssed. The schedule s ot vald. Yet, t s easy to see that by assgg prortes decreasg order to τ 1, τ 3, ad τ 2, respectvely, the schedule becomes vald, ad the task set feasble. Ideed, we have: r 1 = C 1 + ( C 2 1) D 1, r 3 = C 1 + C 3 + ( C 2 1) D 3, r 2 = C 1 + C 2 + C 3 D 2. τ τ τ 3 0 Task τ 3 msses ts deadle 7 Fgure 7: τ s ot feasble wth DM s prorty assgmet Remark: the schedule produced by the preemptve DM wth the same sychroous patter would have led smlarly to a deadle mss for task τ 3. Sce DM s optmal preemptve cotext whe = 1, D, we may coclude that τ s ot feasble whe preempto s allowed. Thus we have foud a task set that s feasble o-preemptve cotext wth a fxed prorty assgmet, but ot feasble preemptve cotext. Let us show ow that the optmalty of DM s kept whe deadle ad executo tme orders are smlar.

25 29 The result of the prevous lemma lets us exted the approach developed by [AUD91] ad [LEH90], to fgure out the worst-case respose tmes of a task τ the o-preemptve cotext. Uless stated otherwse, we wll oly exame scearos as descrbed Lemma 6. Theorem 15 - Gve a geeral task set τ = { τ 1,, τ } wth arbtrary fxed prortes, the worst-case respose tme of ay task τ s gve by r = max q = 0,, Q { w, q + C q }, where, = qc , C j + max k lp( ) { C k 1}, w q j hp( ) w q T j (9) ad Q = L, where L s the legth of the logest level- busy perod o-preemptve cotext (see Aex A.2). Proof. Gve a task τ, cosder ts stace released at tme q. w, q s the smallest tme such that the workload the terval [ 0, w, q ] due to all task staces whch precede the executo of the τ s stace cosdered s maxmum ad equal to w, q,.e o other task stace ca delay the ( q + 1) -th stace of τ at tme w, q. I Equato (9), qc stads for the durato of the q staces of τ released before q. ( 1 + w, q T j )C stads for the maxmum workload of tasks wth j hp( ) j hgher prorty tha τ the terval [ 0, w, q ] ad max k lp( ) { C k 1} s the maxmum delay resultg from tasks wth lower prorty tha τ (worst-case accordg to Lemma 6). Oce t has gaed the processor at tme w, q, the ( q + 1) -th stace of τ completes ts executo by tme w, q + C. Its respose tme s therefore w, q + C q. L s the maxmum legth of ay level- busy perod (see Secto A.2). Thus, accordg to Lemma 6, we do ot eed to exame staces released after L, that s, the last oe to be cosdered s that released at tme L. The worst-case respose tme of τ s fally r = max q = 0,, Q { w, q + C q }. Note that, smlarly to preemptve case, the computato of worst-case respose tmes has a pseudopolyomal tme complexty, sce L s upper bouded by L, whose legth s pseudo polyomal wheever U c, wth c a postve costat smaller tha 1 (see Aex A).

26 28 tmes. I order to acheve ths goal, we frst show that the oto of level- busy perod, troduced by [LEH90], s also useful the o-preemptve cotext. As for the dyamc case, the ma dffereces wth the preemptve cotext are: Owg to the absece of preempto, a task stace wth later absolute deadle ca possbly cause a prorty verso, whch must be accouted for. Always owg to the o-preemptablty of ay task stace executo, our atteto wll be o the busy perod precedg the executo start tme of the stace, ad ot o the busy perod precedg ts completo tme, as s the case the preemptve model. Let τ = { τ 1,, τ } be a geeral task set wth arbtrary fxed prortes. Lemma 6 - The worst-case respose tme of τ s foud a level- busy perod obtaed by releasg all tasks τ j wth j hp( ) { } sychroously from tme t = 0, ad by releasg the logest task τ k wth k lp( ), f ay, at tme t = 1. Proof. Cosder the schedule produced by the o-preemptve hghest prorty frst algorthm for a gve scearo (see fgure 6). Let t 2 be the completo tme of oe of the τ s staces. Let t 1 be the last tme before t 2 such that there are o pedg staces before t 1 wth prorty hgher or equal to that of τ. By defto, there s o dle tme [ t 1, t 2 ], ad the oly tasks that have staces executed [ t 1, t 2 ] are those wth dexes hp( ) { }. I addto, the stace of a lower prorty task, f ay, may execute at t 1 owg to the o-preemptablty of executos (ote that ths case the lower prorty stace must have bee released before t 1 ). The terval betwee the completo tme of ths stace ( t 1, f there s o such stace) ad t 2 s a level- busy perod. The respose tme of the τ s stace cosdered ca be possbly worseed the followg ways. If all staces of τ [ t 1, t 2 ] are actually released from t 1 at ther maxmum rate, each executo fshes at the same tme, but has possbly a larger respose tme. Smlarly, f all tasks wth hgher prorty tha τ are released sychroously from t 1, the umber of hgher prorty staces caot decrease, thus gvg a possbly loger respose tme for τ s staces. Fally, f τ k s the task wth the maxmum executo tme amog all tasks wth lower prorty tha τ ( k lp( ) ), by releasg a stace of τ k at tme t 1 1, the effect of o-preempto s maxmzed, thus possbly worseg the respose tme of τ s staces. By substtutg t 1 wth 0 ad t 2 wth t 2 t 1 we have the thess. τ j, j hp( ) { } t 1 t 2 Level- busy perod Task τ k, C k = max j lp( ) { C j } Fgure 6

27 Fxed prorty drve schedulers Ths chapter frst cocetrates o feasblty codtos ad worst-case respose tmes computato for geeral task sets. Some exstg results cotuous schedulg are adapted to the cotext of dscrete schedulg. Optmalty fxed prorty drve schedulers s the studed. We wll otably show that the optmal prorty assgmet algorthm proposed by [AUD91] preemptve schedulg s stll vald whe exteded to the o-preemptve cotext Feasblty Codto ad Worst-Case Respose tmes Cotrary to the preemptve cotext, less results are kow about fxed prortes o-preemptve schedulg. I the cotext of cotuous schedulg, where tasks parameters ad tme are allowed to be o teger, oe ca derve from [THW94] ad [TBW95] the followg codto for the feasblty of a task set wth arbtrary prortes: Let τ = { τ 1,, τ } be a geeral task set wth arbtrary prortes assged by some algorthm. As the preemptve cotext, the feasblty codto s based o the computato of the worst-case respose tme of each task τ, ad by comparg ts value wth the relatve deadle D. The o-preemptablty of the schedule s take to accout by cosderg a blockg factor B = max j lp( ) { C j }, where lp( ) s the subset of dexes that detfy the tasks wth lower prorty tha τ. The worst-case respose tme r of τ ca thus be fgured out by meas of the followg recursve equato: r w = max { w, q + C q } where w, q qc, q + ϒ = res C j + B q = 0 Q j hp( ) T j Q s the smallest value such that w, Q + C ( Q + 1), ϒ res s the resoluto wth whch tme s measured, ad hp( ) s the subset of dexes that detfy the tasks wth hgher prorty tha τ. Although cotuous schedulg s more geeral tha dscrete schedulg, [BHR90] argue favor of dscrete schedulg, showg frst that t s reasoable to restrct task parameters to be teger, as ay scheduler s lmted to schedulg multples of some dscrete tme ut, ad as task parameters are expressed that tme ut. Secod, they show that oce the put has bee restrcted to be teger, a vald cotuous schedule exsts f ad oly f a vald dscrete schedule exsts,.e we ca cosder wthout loss of geeralty that tasks are scheduled at teger tmes. If we lmt our atteto to dscrete cotexts, the prevous feasblty codto s stll suffcet but o loger ecessary. For example, let the task set τ = { τ 1 ( C 1 = 2, T 1 = 5, D 1 = 3), τ 2 ( C 2 = 2, T 2 = 10, D 2 = 10) } be scheduled accordg to DM. The worst-case respose tmes computed by usg the prevous formula wth ϒ res = 1 our cotext are r 1 = 4 ad r 2 = 4. The task set s declared ufeasble, although t s easy to verfy that t s deed feasble. Namely, accordg to Theorem 15, whch s later show, r 1 = 3 ad r 2 = 4. Establshg a ecessary ad suffcet feasblty codto for ay geeral task set wth arbtrary fxed prortes, essetally meas to develop a procedure for exactly computg the task worst-case respose

28 26 possbly released at tme t. For ay task τ j, the maxmum umber of staces released [ 0, t] s 1 + t T j. However, at most 1 + ( a + D D j ) T j amog them ca have a absolute deadle before or at a + D. It follows that W ( a, t) m 1 t a + D D +, j = C j. j D j a + D Beg W ( a, t) a mootoc o-decreasg step fucto, the smallest soluto of Equato (8) ca be foud by usg the usual fxed pot computato: ( 0) L ( a) = 0 ( m + 1) L a T j ( m) ( ) = maxd j > a + D { C j 1} + W ( a, L ( a) ) + a ---- C. T j Accordg to the argumet of Lemma 5, we have defed the respose tme relatve to a, r ( a), as a fucto of the busy perod legth L ( a) whch s upper bouded by L, the legth of the sychroous busy perod (see Aex A). Hece we coclude that the computato of r ( a) ca be coheretly lmted to values of a smaller tha L. That s, the worst-case respose tme of τ s fally r = max{ r ( a): 0 a < L}. The umber of evaluatos of r ( a) ecessary to compute r ca be further reduced by observg that the rght sde of Equato (8) s a step fucto whose dscotutes a are for values equal to kt j + D j D, for some task τ j ad some teger k. The sgfcat values of a the terval [ 0, L ) ca be reduced accordgly. Moreover, as for the preemptve case, t s possble to restrct the terval where the worst-case respose tme of τ has to be looked for, to [ 0, L ] wth L beg the maxmum legth of a deadle busy perod, for τ o-preemptve cotext (see Aex A for a exact computato of L presece of Lemma 5 s patters of arrval). Oce aga, ote that, cotrary to what happes the feasblty secto, the computato of L mght mprove sgfcatly the worst-case respose tmes aalyss sce t already makes use of recursve expresso ad sce the followg property holds (f the tasks are sorted by creasg relatve deadle): 1], L L + 1 (see Aex A). Note that, smlarly to the feasblty codto, the computato of worst-case respose tmes has pseudo-polyomal tme complexty sce L s upper bouded by L, whose legth s pseudo polyomal wheever U c, wth c a postve costat smaller tha 1 (see Aex A).