Precise timing analysis for direct-mapped caches

Transcription

1 Preise timing nlysis for diret-mpped hes Sidhrt Andlm, Roopk Sinh, Prth Roop, Alin Girult, Jn Reineke To ite this version: Sidhrt Andlm, Roopk Sinh, Prth Roop, Alin Girult, Jn Reineke. Preise timing nlysis for diret-mpped hes. Design Automton Conferene, DAC, Jun 23, Austin, TX, United Sttes. ACM, 23, <.45/ >. <hl > HAL Id: hl Sumitted on Jul 23 HAL is multi-disiplinry open ess rhive for the deposit nd dissemintion of sientifi reserh douments, whether they re pulished or not. The douments my ome from tehing nd reserh institutions in Frne or rod, or from puli or privte reserh enters. L rhive ouverte pluridisiplinire HAL, est destinée u dépôt et à l diffusion de douments sientifiques de niveu reherhe, puliés ou non, émnnt des étlissements d enseignement et de reherhe frnçis ou étrngers, des lortoires pulis ou privés.

2 Preise Timing Anlysis for Diret-Mpped Ches ABSTRACT Sidhrt Andlm TUM CREATE, Singpore Alin Girult INRIA - Grenole, Frne lin.girult@inri.fr Sfety-ritil systems require gurntees on their worst-se exeution times. This requires modelling of speultive hrdwre fetures suh s hes tht re tilored to improve the verge-se performne, while ignoring the worst se, whih omplites the Worst Cse Exeution Time (WCET) nlysis prolem. Existing pprohes tht preisely ompute WCET suffer from stte-spe explosion. In this pper, we present novel he nlysis tehnique for diret-mpped instrution hes with the sme preision s the most preise tehniques, while improving nlysis time y up to 24 times. This improvement is hieved y nlysing individul ontrol points seprtely, nd rrying out optimistions tht re not possile with existing tehniques. Ctegories nd Sujet Desriptors: B.3.3 [Performne Anlysis nd Design Aids]: Worst-se nlysis Generl Terms: Verifition, Algorithms Keywords: Instrution, Diret-Mpped, Che Anlysis.. INTRODUCTION Hrd rel-time systems require urte gurntees on the funtionlity s well s the timing hrteristis of progrms. Trditionl speultive rhiteturl fetures suh s multilevel hes nd deep pipelines render the worst-se exeution. Two types of memory rhitetures re used in rel-time systems: speilized ompiler ssisted hes, lled srthpds [2], nd (widely ville) onventionl hes [3, 9, ]. This rtile fouses on the stti nlysis [2] for preditle diret-mpped instrution hes [9, ], where lotions in min memory re mpped to unique he lines. Che nlysis involves omputing the numer of he misses tht n hppen in the instrution he t speifi ontrol points in progrm. The progrm is usully trnslted to ontrol flow grph (CFG) [9], whih ontins ontrol points s its nodes. Anlytilly, the he nlysis prolem oils down to sttilly determining ll possile he sttes (nd therefore the numer of misses in the worst se) t eh node using suitle fixed point omputtion. While numer of he nlysis pprohes exist [, 9], some re not slle (ut more preise) while others overestimte he misses (ut re more slle). In onrete tehniques like [9], ll possile he sttes re enumerted expliitly t eh ontrol point, nd very preise results n Roopk Sinh, Prth Roop University of Auklnd, New Zelnd {r.sinh,p.roop}@uklnd..nz Jn Reineke Srlnd University- Srrüken, Germny reineke@s.uni-srlnd.de e otined. However, these tehniques suffer from stteexplosion, nd do not sle for lrge progrms. In [8], proilisti pproh for modelling he ehviour is presented, whih is used for design-spe explortion to redue overll nlysis time y exploiting the struturl similrities mong relted he onfigurtions. In [7], the ide of he onflit grphs is introdued where he lines re nlysed one t time. This pproh is more slle thn onrete pprohes, ut loses preision s ny reltions etween he lines re strted out. On the other hnd, strt tehniques like [, ] ollpse multiple possile he sttes t every ontrol point into single strt he stte. This strtion llows the lgorithm to reh fixed point muh fster, nd hene lrger progrms n e nlysed, ut t the ost of srifiing preision. In this rtile, we present novel he nlysis tehnique for diret-mpped hes tht mintins the sme preision s the onrete tehniques while signifintly improving slility (for lrge enhmrks, nlysis time is less thn 2 minutes, insted of 2 hours s in [9]). This improvement is hieved y nlysing individul ontrol points t one, pturing nd ggregting instrutions in reltive mnner, nd rrying out ertin optimistions tht re not possile in other tehniques. The key ontriution of this pper is new lgorithm for stti nlysis tht offers the sme preision s the most preise lgorithms while eing extremely slle in omprison. It lso ompres very fvourly with strtion-sed nlysis tehniques with respet to nlysis time. This pper is orgnised s follows. The he nlysis prolem is formlized in Se. 2. In Se. 3 we present the proposed pproh. Qulittive nd quntittive omprison with the onrete nd strt pprohes is presented in Setions 4 nd 5 respetively. Finlly, onlusions re presented in Se. 6. The Appendix further illustrtes key spets of the proposed lgorithm with dditionl experimentl results to demonstrte its dvntges. Also, in Appendix E, we disuss how the proposed pproh for diret-mpped hes n e extended to set-ssoitive hes, whih re lso widely used in preditle systems [3]. 2. THE CACHE ANALYSIS PROBLEM Our nlytil he model is sed on the model of [9], nd is defined elow. To improve redility, we represent the terms si loks nd memory loks s loks nd instrutions, respetively. Permission to mke digitl or hrd opies of ll or prt of this work for personl or lssroom use is grnted without fee provided tht opies re not mde or distriuted for profit or ommeril dvntge nd tht opies er this notie nd the full ittion on the first pge. To opy otherwise, to repulish, to post on servers or to redistriute to lists, requires prior speifi permission nd/or fee. DAC 3, My 29 - June 7 23, Austin, TX, USA. Copyright 23 ACM /3/5...$5. Definition (Che model). The he model for given progrm is defined s tuple CM = I, C, CI, G, BI, where I is finite set of instrutions in the progrm, C = {,,..., N } is n ordered set of he lines with N = C s the totl numer of he lines, nd CI : C 2 I is the diret-mpping funtion.

3 Also, G is direted grph G = B, init, E where B is finite set of si loks with init B s the initil lok, nd E B B is the set of edges. Finlly, BI : B (I { }) N is the lok to instrution mpping funtion, where represents no instrution. B4: m3 m4 B3: m B5: m2 B: m m2 m3 m4 B2: m5 B9: m8 B6: m2 m3 m4 : m3 m4 m6 : m6 m7 m8 () A ontrol flow grph (CFG) Figure : Che Inst. line m,m5 m2,m6 2 m3,m7 3 m4,m8 () Mpping of instrutions to he lines An exmple he model Figure shows the he model of smple progrm. It presents CFG in Figure (), whih ontins 9 loks (B = {B,..., B9}) with B s the initil lok. Eh lok exeutes one or more instrutions. The CFG hs 8 instrutions (I = {m,..., m8}) tht re mpped to 4 he lines (C = {,, 2, 3}), s presented in Figure (). Eh he line hs two unique instrutions mpped to it (e.g., CI( ) = {m, m5}). An instrution n e mpped only to unique he line, representing diret-mpped he. A he line, t ny time, n ontin only one of the instrutions tht re mpped to it (or is empty). The edges desrie progrm flow (sequentil nd rnhing) etween the ontrol points of the given progrm. We use the short-hnd B B2 to desrie CFG edges suh s (B, B2) E.. Eh lok lso ontins instrutions (from I) tht it exeutes using the mpping funtion BI. For exmple, B ontins four instrutions m, m2, m3 nd m4. We restrit the model suh tht loks n ontin t most one instrution mpped to ny he line (s per CI), whih llows us to represent the lok to instrution mpping using vetor indexed y C. E.g., BI(B) = [m, m2, m3, m4], with BI(B)[] = m desriing the instrution mpped to he line. If lok does not ontin n instrution mpped to he line i, then BI()[i] = (e.g., BI(B2)[] = ). The ontents of the he t ny time during the progrm exeution is lled he stte, nd is represented s vetor s = [inst,..., inst N ] where eh inst i represents the instrution ontined in he line i C. When exeution egins, we ssume tht there is no instrution (represented y ) in eh he line. The he is ssumed to e empty. For the exmple presented of Fig., the empty he stte is represented y s = [,,, ]. During exeution, or trversl of the CFG, instrutions re loded into the he s the si loks re exeuted (strting from the initil lok). E.g., fter exeuting the instrutions in B the he stte is s = [m, m2, m3, m4]. In this exmple, s is rehing he stte of B, while s is leving he stte of B. Sine ll 4 instrutions needed y B were not present in s, we sy tht there were 4 he misses. Given ny rehing he stte s of lok, we n ompute the numer of he misses y ompring the instrutions in s nd BI(). E.g., given rehing he stte s = [m, m2, m3, m4] of lok B2 (with BI(B2) = [m5,,, ]), there is single miss on he line euse the instrution m5 needed y the lok (s per BI()[] = m5) is not present in the he stte (s[] = m). It is generlly possile for lok to hve multiple rehing he stte. Here, we ompute the worst-se miss ount wm s the mximum numer of he misses possile, s defined elow. Definition 2 (The Che Anlysis Prolem). Given he model CM, the he nlysis prolem is the omputtion of the numer of worst se he misses (wm ), for ll si loks B, long ll possile exeutions. 3. PROPOSED APPROACH Our pproh for the stti nlysis of diret-mpped hes is sed on the intuition tht nlysing single si lok ref of the CFG t time llows us to () redue the numer of loks needed to ompute the worst nd est se miss ounts for ref, nd (more importntly), () we n strt he sttes omputed during the fixed point lgorithm w.r.t. the instrutions exeuted y ref. This my signifintly redue the numer of possile he sttes nd onsequently redue nlysis time. We use Fig. to illustrte these enefits. First priniple: During the nlysis of lok ref =, sine does not exeute ny instrution on he line (BI()[] = ), we n ignore lok B2 s the exeution of B2 n only ffet the he line. We ll B2 vuous lok euse it does not interfere with ny of the he lines used y, nd it n e removed when nlysing. Seond priniple: During the nlysis of lok ref =, the instrutions ontined in nother lok, sy B, n e strted suh tht they only refer to their effet on the nlysis of. Given tht BI() = [, m6, m3, m4] nd BI(B) = [m, m2, m3, m4], the instrutions in B n e strted s the vetor [,,, ]. Here, the first element mens tht the instrution is not of interest s the referene lok does not use this he line (BI()[] = ). The seond element mens tht for he line, the instrution in B is different from the instrution in (BI(B)[] = m2 m6 = BI()[]). Finlly, for the third nd the fourth elements mens tht for he line 2, the instrution in B is the sme s the instrution in (BI()[2] = m3 = BI(B)[2]). Also, when there is no instrution on he line i in lok the strt representtion is. The ility of reduing the numer of instrutions ( I ) in the CFG to just four reltive instrutions (,,, ) signifintly redues the memory foot print nd nlysis time, without srifiing preision. This is key optimiztion tht enles us to propose slle nlysis tehnique without srifiing preision. Alg. presents n overview of our pproh. Eh lok in the CFG is nlysed individully (desried using the for-loop on lines 5). For eh referene lok ref in B, on line 2, we first redue the CFG y removing the vuous loks nd then ompute the reltive instrutions w.r.t. ref. Next, on line 3, fixed point lgorithm is used to ompute ll possile he sttes of the referene lok. Finlly, on line 4, the numer of he misses in the worst se re omputed. We now present the detils of eh of these steps.

4 Algorithm Overview of the proposed pproh Input: A he model CM = I, C, CI, G, BI. Output: Compute the worst miss ount (wm) for ll si loks. : for eh ref B do 2: {Redued CFG, ompute reltive instrutions (Se. 3.)} (G r, BI r ) = Redue(CM, ref ) 3: {Compute rehing reltive he sttes (Setion 3.2.3)} RCS r ref = F P (CM, G r, BI r ) 4: {Compute he misses in the worst-se (Setion 3.3)} wm ref = MAXm(RCS r ref ) 5: end for 6: return wm for ll loks in B {Solution for the he nlysis prolem (Definition 2)} 3. CFG Redution Alg. 2 presents the pseudoode of the Redue lgorithm tht returns redued grph G r from given CFG G. Given G nd referene lok ref, we first represent the instrutions in eh lok w.r.t. the referene lok, nd then remove ny vuous loks. Line initilizes G r s opy of G. Then, for eh lok in G r, nd eh he line i, reltive lok to instrutions mpping BI r is reted (lines 2 7). Depending on whether the instrution originlly ontined in (in G) is of no-interest to the referene lok (line 3), different (line 4) or identil (line 5) to the instrution ontined in the referene loks, or is equl to (line 6). Algorithm 2 Redue: Redue the CFG nd strt inst. Input: Che model CM = I, C, CI, G, BI nd referene lok ref B. Output: G r = B r, init, E r nd BI r : B r ({,,, }) N : Initilize G r s opy of G with BI r () = for ll B r. 2: for eh B r nd for eh i C do 3: BI r ()[i] =, if (BI( ref )[i] = ){not of interest} 4: BI r ()[i] =, if (BI( ref )[i] BI()[i]) {different} 5: BI r ()[i] =, if (BI( ref )[i] = BI()[i]) {identil} 6: BI r ()[i] =, if (BI r ()[i] BI()[i] = ){no inst.} 7: end for 8: for eh B r do 9: if (BI() ({, }) N ) ( init) {hek for ll vuous loks, exluding initil lok} then : Remove from B r, nd djust E r : end if 2: end for Next, on lines 8, loks for whih the mpping funtion BI r returns or for every element of the vetor BI r () re delred s vuous nd re removed from the grph. The removl of vuous lok r involves djusting the edges E r of the grph G r suh tht eh predeessor of r now hs diret edge to eh of the suessors of r. More detils nd the full Redue Algorithm pper in Appendix A. Fig. 2 shows the redued CFG returned y the lgorithm Redue when ref =. Note tht the vuous lok B2 is removed from G r. Also, every lok now ontins the reltive instrutions. 3.2 Fixed point Computtion 3.2. Reltive he sttes Sine lok to instrutions mpping is desried using reltive instrutions, we lso ompute he sttes in reltive fshion. A reltive he stte is defined s follows. Definition 3 (Reltive he stte). A reltive he stte s r is vetor [inst r,..., inst r N ], where eh element inst r i {,,,, }. The set of ll possile reltive he sttes is denoted s CS r. Eh reltive instrution inst r i of reltive he stte s r = [inst r,..., inst r N ] is desried w.r.t. the instrution (BI( ref ) B4: B5: m2 3 B: m m2 B6: m2 B9 3 m8 : m m2 : m6 2 m7 3 m8 Figure 2: The redued grph G r otined from the Redue lgorithm, with the referene lok ref =. [i], i [, C ]) in the referene lok ref for the he line i C. inst r i = or inst r i = mens the instrution in the he is different or identil respetively to the instrution exeuted in the referene lok ref. E.g., given BI( ref )[i] = m, inst r i = or inst r i = mens the instrution on he i is not m or m respetively. inst r i = mens tht he line i is empty, wheres inst r i = mens tht the he hs n unknown instrution. Finlly, inst r i = mens tht the instrution on this he line is not of interest during the nlysis of ref. Also, reltive he stte efore exeuting lok is know s rehing reltive he stte of. Similrly, reltive he stte fter exeuting lok is know s leving reltive he stte of. More detils out reltive he sttes re presented in Appendix B The Trnsfer Funtion An importnt opertion in the fixed point omputtion is the trnsformtion of rehing reltive he stte into leving reltive he stte. This trnsformtion, lled the trnsfer funtion T : CS r B CS r, is illustrted s follows. s r BI r () T s r 2 s r =[,,, ] BI r () =[,,, ] T (s r, ) = sr 2 =[,,, ] For ny he line i, the instrution s r 2[i] in the leving reltive he stte is equl to BI r ()[i] only if there is n instrution in lok (BI r ()[i] = or BI r ()[i] = ). Otherwise, the instrution s r 2[i] is the sme s the instrution s r [i] in the rehing he stte. For the ove exmple, given BI r () = [,,, ] nd s r = [,,, ], fter exeution of lok, s r 2 = [,,, ]. For he lines nd 3, lok exeutes instrutions tht relte to the referene lok, nd for he lines nd 2, its does not exeute ny relevnt instrutions. Therefore, the ontents of the leving he sttes re updted to e the sme s the instrutions exeuted y the lok on he lines nd 3, nd remin the sme s the rehing he stte for he lines nd Fixed point omputtion Alg. 3 shows the fixed point lgorithm F P used to ompute ll possile rehing reltive he sttes for the referened lok in the redued grph G r. We illustrte the fixed point omputtion using T., whih shows the possile rehing nd leving reltive he sttes of eh lok in the redued grph (Fig. 2) in every itertion of F P. During initiliztion, the rehing he stte of the initil lok is set to s r (line 2) euse the he is onsidered

5 empty initilly. For every other lok, on line 2, the initil rehing he stte is unknown (s r ). As shown in T., The initil rehing reltive he stte of the initil lok B is set to s r = {[,,, ]}, while for every other lok, the initil rehing reltive he stte is s r = {[,,, ]} (itertion, olumn 3, T. ). Algorithm 3 F P : Fixed point omputtion Input: A he model CM = I, C, CI, G, BI, redued grph G r = B r, init, E r nd BI r : B r ({,,, }) N. Output: Rehing reltive he sttes of lok ref (RCS r ). ref : Crete initil he stte s r where for every i C, sr [i] = if BI r ( ref ) =, or s r [i] = otherwise. 2: RCS r = {s r init }, nd RCSr = {s r } for ll other Br. 3: i = 4: repet 5: for eh B r do 6: LCS ri = T (RCS ri, ) 7: end for 8: i = i + ; {Next itertion} 9: for eh B r do : if = init then : RCS ri = {s r init } ( LCS ri (, ) E r ) 2: else 3: RCS ri 4: end if 5: end for 6: until B r, RCS ri 7: return RCS r ref = LCS ri (, ) E r = RCS ri {Termintion ondition} Tle : Computing ll possile rehing reltive he sttes of the referene lok ref (). Itr. Blok Rehing Reltive Leving Reltive Che (i) () Che Sttes (RCS i ) Sttes (LCS i ) B {[,,, ]} {[,,, ]} B4 {[,,, ]} {[,,, ]} B5 {[,,, ]} {[,,, ]} B6 {[,,, ]} {[,,, ]} {[,,, ]} {[,,, ]} {[,,, ]} {[,,, ]} B9 {[,,, ]} {[,,, ]} 2 B {[,,, ]} {[,,, ]} B4 {[,,, ], [,,, ]} {[,,, ], [,,, ]} B5 {[,,, ], [,,, ]} {[,,, ], [,,, ]} B6 {[,,, ], [,,, ]} {[,,, ]} {[,,, ], [,,, ]} {[,,, ]} {[,,, ], [,,, ]} {[,,, ]} B9 {[,,, ], [,,, ]} {[,,, ], [,,, ]} B {[,,, ]} B4 {[,,, ], [,,, ], [,,, ]} B5 {[,,, ], [,,, ], [,,, ], [,,, ]} B6 {[,,, ], [,,, ], [,,, ]} {[,,, ], [,,, ], [,,, ]} {[,,, ],[,,, ]} B9 {[,,, ], [,,, ], [,,, ]} The repet-until loop (lines 4 6) is the fixed point itertion. In eh itertion i, eh rehing reltive he stte ontined in the set RCS ri for every lok is trnsformed into leving reltive he stte (in set LCS ri ) y pplying the trnsfer funtion (lines 5 7). E.g., for lok B in itertion, given the only rehing reltive he stte [,,, ], the orresponding leving reltive he stte is T ([,,, ], B) = {[,,, ]} (see T. ). Then we ompute, the rehing reltive he sttes of eh lok for the next itertion. For eh lok, the set of rehing reltive he sttes is the union of the sets of the leving reltive he sttes of ll of its predeessors (line 3). For init, the dditionl rehing he stte s r is lso dded to this set (line ). E.g., the predeessors of B4 re B nd B9 (see Fig. 2). Hene, their sets of leving he sttes (resp. {[,,, ]} nd {[,,, ]}) for itertion re ggregted together to form the rehing he stte of B4 in itertion 2. The itertions ontinue until the fixed point is rehed, i.e., when two onseutive itertions yield the sme sets of rehing reltive he sttes for ll loks (line 35). For the redued CFG shown in Fig. 2, the fixed point is rehed in the 5 th itertion. Also during the fixed point, s n optimistion, reltive he stte s r k is dropped if there exists s r j suh tht, if for ll he lines ( i), when the reltive instrution in s r j is the sme s the instrution in s r k (s r j[i] = s r k[i]) or, the reltive instrution in s r j ptures he miss (s r j[i] = ). E.g., given four possile rehing he sttes {[,,, ], [,,, ], [,,, ], [,,, ]}, we n sfely ignore the first two he sttes, euse the third stte ptures the worst se ehviour. However, we must still rry the lst stte, resulting in the redued set {[,,, ], [,,, ]}. 3.3 Computing the numer of he misses The finl step in the he nlysis lgorithm is the omputtion of the numer of he misses in the worst se. This is done y nlysing the reltive he sttes of the referene lok ref s omputed y the fixed point lgorithm. For eh rehing he stte s r = [inst r,..., inst r N ], nd for every he line i C, inst i = represents he miss on i. The totl numer of misses when the rehing he stte s r is the numer of s ontined in s r. The rehing he sttes with the highest numer of s orrespond to the worst-se miss ounts of ref respetively. E.g., s shown in T., the two rehing he sttes of the referene lok s omputed y the fixed point lgorithm F B re {[,,, ] nd [,,, ]}. The first rehing he stte hs one ourrene of while the seond one hs two ourrenes of. Thus, the mximum miss ount for is 2, i.e., wm = QUALITATIVE COMPARISON T. 2 provides qulittive omprison etween the onrete [9], strt [4], nd the proposed pprohes. Figure 3 illustrtes the si lok of the CFG shown in Fig. 2 with its two predeessors B6 nd. We use this exmple to show the differenes in the wys the three pprohes represent nd ggregte he sttes. Tle 2: Qulittive omprison of the three pprohes. App.Fixed Pre-Time Optimistion Mx no. of s point ision t eh pro- -grm point Con.ll loks high slow none (I/N) N As. ll loks low fst merge he sttes onstnt Pro. one lok high med. () redue grph (3) N t time (2) redue he lines (3) reltive instrutions In the onrete pproh [9], he stte (s) is desried s vetor [inst,..., inst N ] where eh inst i represents single instrution ontined in he line i C or is empty ( ), i.e., inst i CI( i) { }. E.g., the sets of leving he sttes for B6 nd re {[m5, m2, m3, m4]} nd {[m5, m6, m7, m8]} respetively (see Fig. 3()). The set of rehing he sttes of is the union of these sets, s shown in Fig. 3(). This set represents the ft there re only two sttes in whih the he n e efore is exeuted. In the strt pproh [4], he stte is desried s vetor [set,..., set N ] where eh set i represents set of instrutions tht must e ontined in he line i C. Tht is, set i CI( i) { }.. E.g., the strt leving he sttes for B6 nd re [{m5}, {m2}, {m3}, {m4}] nd [{m5}, {m6}, {m7}, {m8}] respetively (see Fig. 3()). For

6 B6 {[m5,m2,m3,m4]} {[m5,m6,m7,m8]} {[m5,m2,m3,m4], [m5,m6,m7,m8]} () Conrete pproh [9] Figure 3: B6 [{m5},{m2},{m3},{m4}] eh of these strt he sttes, eh he line ontins preisely one instrution. Hene, these two strt he sttes re equivlent to the onrete he sttes [m5, m2, m3, m4] nd [m5, m6, m7, m8] shown in Fig. 3(). Next, the strt rehing he stte of is the pir-wise intersetion over the vetor elements of the ove strt he sttes, giving [{m5}, { }, { }, { }] (see Fig. 3()). A he line i, with n empty set represents ny instrution (tht is mpped to i, CI( i)) or no instrution. E.g., the empty set on he line, represents three possile sttes {m2, m6, }. Similrly, [{m5}, { }, { }, { }] represents [{m5}, {m2, m6, }, {m3, m7, }, {m4, m8, }]. This strt rehing he stte represents 27 possile onrete he sttes, whih re omputed using ross produt s: {m5} {m2, m6, } {m3, m7, } {m4, m8, } ={[m5, m2, m3, m4], [m5, m2, m3, m8], [m5, m2, m7, m4], [m5, m2, m7, m8],...} While two of these possile onrete sttes [m5, m2, m3, m4] nd [m5, m6, m7, m8] re vlid (s disussed ove), mny others, suh s [m5, m6, m3, m4], re not rehle euse instrutions m6 nd m3 nnot e loded into the he y ny one of the predeessor loks of. As the ove exmple shows, the strt pproh my lose preision due to joins, whih my introdue non-rehle he sttes. Yet, the fixed point lgorithm onverges fster for the sme reson. Finlly, in the proposed pproh, he sttes re reltive to the instrutions of the referene lok. As shown in Fig. 3(), the rehing reltive he sttes of re omputed y tking union of the predeessor s leving reltive he sttes, {[,,, ]}, {[,,, ]}}, s shown in Fig. 3(). The reltive he stte representtion llows us to optimize performne without srifiing preision. E.g., s shown in Fig. 3(), one of the rehing reltive he stte of is {[,,, ]}. It represents 6 possile onrete he sttes, whih re omputed using ross produt s: {[,,, ]} ={, m, m5} {, m2} {m3} {m4} ={[,, m3, m4], [, m2, m3, m4], [m,, m3, m4], [m, m2, m3, m4], [m5,, m3, m4], [m5, m2, m3, m4]} Thus, the trnsltion results in 6 onrete he sttes. However, the extr sttes, unlike in the strt pproh, do not effet the preision of the nlysis. Tle 3: Compring the he sttes nd WCET estimtes etween the three pprohes s we nlyse lok. App. he stte wm (Worst) Conrete {[m5, m2, m3, m4], [m5, m6, m3, m4]} 2 Astrt [{m5}, {m2, m6}, {m3, m7}, {m4, m8}] 3 Proposed {[ ], [ ]} 2 Comprison of the preision: By nlysing the rehing he sttes of, we n ompute wm nd ompre the preision mong the three pprohes. We present this omprison using Tle 3. For lok (BI() = [, m6, m3, m4]), given the set of possile rehing he sttes for eh pproh (in olumn 2), the WCET estimte (wm) is presented in olumn 3. We note tht the onrete pproh hs higher preision (smller WCET), ompred the [{m5},{m6},{m7},{m8}] [{m5},{ },{ },{ }] () Astrt pproh [4] Illustrtion of the omining he sttes. B6 {[,,, ]} {[,,, ]} {[,,, ], [,,, ]} () Proposed pproh strt pproh. Also, the proposed pproh mintins this high preision. The optimised use of reltive he sttes is only possile in the proposed pproh euse it nlyses one lok t one. Both the onrete nd strt pprohes nlyse ll loks in the CFG together. Comprison of the omplexity: The time omplexity of eh of the three pprohes depends on the numer of loks in the CFG, the omplexity of the merging nd equivlene opertions, nd the mximum numer of he sttes tht n e reted (or the height of the lttie). The numer of loks of the CFG is fixed t B, while we ssume tht the merging nd equivlene opertions etween he sttes n e performed in O(N) times (N is the numer of he lines). Therefore, the omplexities re of the form O( B N HeightOfLttie). To ompute the height of the lttie, we ssume tht eh he line hs n equl shre I /N (I is the set of instrutions) of instrutions mpped to it, nd tht I nd N re suffiiently lrge so tht we n ignore the presene of nd in the omputed he sttes. For the onrete pproh, there re ( I /N ) N possile rehing he sttes. Hene the omplexity of the pproh is O( B N ( I /N ) N ). For the strt pproh, euse of the point-wise intersetion opertion used to merge he sttes for every he line, the height of the lttie is onstnt t for single he line, or N for ll he lines. The omplexity of this pproh is therefore O( B N 2 ). Finlly, for the proposed pproh, eh he line hs preisely 5 reltive instrutions {,,,, } mpped to it. However, rell tht if he line hs ssigned to it, it mens the he line is not used (nd hene nnot ontin ny other instrution during the nlysis). Similrly, is used only for the initil node. The numer of he sttes possile (over N he lines) in our pproh is therefore 3 N. The omplexity of our pproh is therefore O( B N 3 N ) for one ll to the fixed point lgorithm. Given tht we repet this proess B times (one for every lok), nd tht the CFG redution opertion is liner to the size of the CFG (gin, B ), the omplexity of our pproh is therefore O( B 3 N 3 N ). As n e seen ove, the omplexity of the strt pproh is signifintly lower thn the onrete pproh s well s our pproh. Also, the omplexity of the proposed pproh does not depend on the numer of instrutions in the progrm, whih helps us hieve muh etter performne thn the onrete pproh, s disussed in the next setion. 5. BENCHMARKING AND RESULTS We ompre the preision nd nlysis time of the onrete, strt, nd proposed pprohes over set of enhmrks onsisting of five ontrol pplitions from [6]. In ddition, we reted two exmples: BuleSort progrm nd Syntheti exmple. The enhmrking results re presented in T. 4. The first olumn presents the exmples followed y their desription (olumn 2). The numer of C lines of the progrm nd its memory footprint is presented in olumns 3 nd 4 respetively. The lrgest exmples re CruiseController

7 nd RilRodCrossing, with more thn 4 lines of ode. Eh progrm is ompiled to exeute on the MiroBlze (MB) proessor []. We hoose MB due to the vilility of timing nlysis tools [5]. The CFG for eh exmple is extrted utomtilly from its ompiled inry, nd eh loop of the CFG is unrolled one for more preise he nlysis [4]. For MiroBlze with 64 MB min memory, the size of the he n e onfigured from 28 ytes to 64 KB []. Using these proportions, in our experiments we explore he sizes etween.% nd % of the progrm s size. Tle 4: Benhmrk progrms nd their hrteristis. Exmple Desription LOC Size BuleSort (BS) Bule sort lgorithm 28 2KB Syntheti (SY) Brnhing nd loops 8 4KB Flsher (FL) Distriuted lights 384 9KB DrillSttion (DS) Drilling sttion 8 62KB ConvBelt (CB) Airport onveyor elt 28 44KB RilRodCros. (RR) Ril rod nt KB CruiseCntroler (CC) Cruise ontrol model KB For the first two exmples (BS, SY), the WCET estimtes from the proposed pproh ws identil to tht of the onrete pproh (see Tle 5 in Appendix D). However, for the rest of the exmples, the onrete pproh filed to terminte (nlysis time is more thn 2 hours, represented using T.O in Tle 5). Thus, we only fous on ompring etween the proposed nd the strt pprohes. For he size of % of the progrm size, normlised WCET estimtes w.r.t. the results from the strt pproh re presented in Figure 4(). Aross ll the enhmrks, we oserve tht the WCET omputed y the proposed pproh is lwys less thn or equl to the estimtes from the strt pproh. On verge, the WCET estimte from the proposed pproh is 5% smller thn the strt pproh. For.% reltive he size, the WCET nlysis results re presented in Figure 4(). Here, the proposed pproh does not gin extr preision (w.r.t. the strt pproh), euse the he size is too smll. In terms of nlysis time, the proposed pproh lwys tkes less thn 3 minutes for eh exmple, whih is signifintly fster thn the onrete pproh. However, the strt pproh is even muh fster (lwys tkes less thn 4 seonds) thn the proposed pproh. For the lrgest exmple (RR), the proposed pproh tkes 42 seonds ompred to the 4 seonds tken y the strt pproh, ut the WCET estimte is tighter y 9% ( 25725/3855). Finlly, Fig. 5 shows the WCET vs nlysis time for the ontrol pplitions (lst 5 enhmrks). Sine the onrete pproh filed to terminte, its WCET is represented y the WCET of the proposed pproh (sine oth yield identil results). On verge, ompred to the strt pproh, the WCET from the proposed pproh is 6% tighter. For the nlysis time, the proposed pproh lwys ompletes in less thn 3 minutes, ompred to the timeout fter 2 hours for the onrete pproh. In Appendix D, we explore eight other he sizes etween.2% nd.9%. For the RilRodCrossing exmple, on verge ross the eight he sizes, the WCET of proposed pproh gives 6 % muh tighter results on verge, nd up to 9 % tighter result thn the strt pproh. 6. CONCLUSIONS We proposed new he nlysis pproh for preise nlysis of instrutions in diret-mpped hes. The proposed pproh presents new strtion, nd ompred to the onrete pproh it signifintly redues the nlysis time without srifiing the preision. This is unlike the onrete strt proposed () % reltive he size ().% reltive he size Figure 4: Compring the WCET estimtes (the smller the etter) of the strt nd proposed pprohes Timeout Anlysis time in seonds strt proposed onrete WCET w.r.t. the strt pproh strt proposed Avg. onrete = + 2 hours Avg. proposed = 6 se Avg. strt = 2 se Figure 5: Compring WCET nd nlysis time for the lst five exmples (% reltive he size) or the strt pprohes, where slility or preision must e srified. Overll, the proposed pproh enles preise nd effiient WCET nlysis even for lrger progrms. In the future, we will e extending our pproh for nlysing set ssoitive hes nd design spe explortion of hes. 7. REFERENCES [] MiroBlze Proessor Referene Guide. (Lst essed: //22). [2] D. Bui, E. Lee, I. Liu, H. Ptel, nd J. Reineke. Temporl isoltion on multiproessing rhitetures. In Pro. of DAC, pges , Sn Diego, Cliforni, June 2. [3] H. Ding, Y. Ling, nd T. Mitr. WCET-entri prtil instrution he loking. In Pro. of DAC 2, pges 42 42, Sn Frniso, Cliforni, June 22. [4] C. Ferdinnd, F. Mrtin, R. Wilhelm, nd M. Alt. Che Behvior Predition y Astrt Interprettion. Siene of Computer Progrmming, 35:63 89, Novemer 999. [5] M. Kuo, R. Sinh, nd P. S. Roop. Effiient WCRT Anlysis of Synhronous Progrms using Rehility. In Pro. of DAC, Sn Diego, USA, June 2. [6] M. Kuo, L. H. Yoong, S. Andlm, nd P. Roop. Determining the worst-se retion time of IEC 6499 funtion loks. In Pro. INDIN, pges 4 9, July 2. [7] Y.-T. S. Li, S. Mlik, nd A. Wolfe. Performne estimtion of emedded softwre with instrution he modeling. ACM Trns. Des. Autom. Eletron. Syst., 4(3): , July 999. [8] Y. Ling nd T. Mitr. Stti nlysis for fst nd urte design spe explortion of hes. In Pro. of CODES+ISSS 8, pges 3 8, Atlnt, GA, USA, 28. [9] H. S. Negi, T. Mitr, nd A. Royhoudhury. Aurte Estimtion of Che-Relted Preemption Dely. In Pro. of CODES+ISSS 3, pges 2 26, CA, USA, 23. [] J. Reineke, D. Grund, C. Berg, nd R. Wilhelm. Timing preditility of he replement poliies. Rel-Time Systems, 37(2):99 22, Novemer 27. [] H. Theiling, C. Ferdinnd, nd R. Wilhelm. Fst nd Preise WCET Predition y Seprted Che nd Pth Anlyses. Rel-Time Systems, 8:57 79, 999. [2] R. Wilhelm et l. The Worst-Cse Exeution-Time Prolem Overview of Methods nd Survey of Tools. Trnstions on Emedded Computing Systems, 7(3): 53, 28.

8 APPENDIX A. REDUCING THE CFG AND COMPUTING RELATIVE INSTRUCTIONS Given he model CM = I, C, CI, G, BI nd the referene lok ref B, the ojetive of the lgorithm is: () to ompute new redued grph G r = B r, init, E r whih ontins only these loks tht re relevnt for the nlysis of lok ref (desried erlier in Exmple ) nd, (2) to ompute the funtion BI r whih desries the reltive instrutions exeuted y the loks in B r w.r.t. ref, referred s the reltive instrution mpping funtion (desried erlier in Exmple 2). The lgorithm ontins the following three steps. Step : Initilise (lines 2 to 4) On line 2, we initilise G r to hve the sme ontent s G, i.e., sme set of loks (B r = B), initil lok ( init), edges (E r = E). On lines 3 to 4, we initilise the funtion BI r suh tht it does not ontin ny reltive instrutions for ny lok. For the exmple CFG (presented in Figure ()), the grph G r is presented in Figure 6(). Algorithm 4 Redue: Redue the CFG nd ompute reltive instrutions Input: Che model CM = I, C, CI, G, BI nd referene lok ref B. Output: G r = B r, init, E r nd BI r : B r ({,,, }) N : {Step : Initilise} 2: B r = B, E r = E, G r = B r, init, E r {Copy ll loks nd edges} 3: for eh B r do 4: BI r ( ) = {Initilise the vetor BI r ( )} 5: end for 6: {Step 2: Reltive instrution mpping} 7: for eh B r do 8: for eh i C do 9: if (BI( ref )[i] = ){not of interest} then : BI r ( )[i] = : end if 2: if (BI( ref )[i] BI( )[i]) {different instrution} then 3: BI r ( )[i] = 4: end if 5: if (BI( ref )[i] = BI( )[i]){sme instrution} then 6: BI r ( )[i] = 7: end if 8: if (BI( )[i] = ){no instrution} then 9: BI r ( )[i] = 2: end if 2: end for 22: end for 23: {Step 3: Remove vuous loks nd updte edges} 24: for eh B r do 25: if (BI( ) ({, }) N ) ( init) {hek for ll vuous loks, exluding initil lok} then 26: {Compute predeessors nd suessors of } 27: P reds = { 2 2 E r } 28: Su = { 2 2 E r } 29: {Remove inoming nd outgoing edges of } 3: for eh 2 P reds do 3: E r = E r \ { 2 32: } end for 33: for eh 2 Su do 34: E r = E r \ { 2 35: } end for 36: {Add new edge from eh predeessor to eh suessor } 37: for eh p P reds do 38: for eh s Su do 39: E r = E r { p s } 4: end for 4: end for 42: B r = B r \ { } {Remove lok } 43: end if 44: end for Step 2: Reltive instrution mpping (lines 7 to 22) Given referene lok ref B nd he line i, the instrution of ny loks B r n e expressed s different () when BI( )[i] BI( ref )[i] (heked on line 2), or sme () when BI( )[i] = BI( ref )[i] (heked on line 5), or no instrution ( ) when there is no instrution in on he line i, BI( )[i] = (heked on line 8), or not of interest ( ) when there is no instrution in ref on he line i, BI( ref )[i] = (heked on line 9). This reltion is ptured using the reltive instrution mpping BI r ( ). For illustrtion, refer to Exmple 2 in Setion 3. For the grph G r in Figure 6(), the reltive mpping (w.r.t. ) for every lok is shown in Figure 6(). Note tht for ll loks in B r, if the referene lok ( ref ) does not hve n instrution on he line i (BI( ref )[i] = ), then for he line i, the reltive instrution for ll loks is (BI r ()[i] = ). This shows tht during the nlysis of ref, the he line i is not onsidered. For exmple, in Figure 6(), the referene lok does not hve n instrution on he line (BI()[] = ). Thus, for he line, the reltive instrution for ll loks is (BI r ()[i] = ). Step 3: Remove vuous loks nd updte edges ( lines 24 to 2). As desried in Exmple, we n remove vuous loks whih do not ffet the preision of the referene lok ref. We define lok B r to e vuous, if BI r ( ) ({, }) N. This hek is done on line 25. It is possile for the initil lok ( init) to e vuous. However, if the initil lok hs more thn one suessors, removing the initil lok my result in multiple initil loks. Thus, to simplify the nlysis, we do not remove the initil lok even when it is vuous (line 25). If lok is vuous, we first ompute the predeessors (line 27) nd the suessors (line 28) of. Seondly, we remove the inoming edges to from eh predeessor lok (on lines 3 to 32) nd, we remove the outgoing edges from to eh suessor lok (on lines 33 to 35). Thirdly, we rete trnsition from eh predeessor to eh suessor of. This is hieved using the nested loop on lines 37 to 4. Finlly, on line 42, the vuous lok is removed. For the grph shown in Figure 6(), loks B2 nd B3 do not ontin ny reltive instrutions (BI r (B2) = [,,, ] nd BI r (B3) == [,,, ], thus, they re removed for the grph nd the updted grph G r is presented in Figure 6(). This is the redued grph G r (w.r.t. to ref = ). B. RELATIVE CACHE STATES We desrie the ontents of he using the notion of reltive he sttes. It is desried s vetor [inst r, inst r,..., inst r N ], where eh element inst r i is desried w.r.t. the instrution (BI( ref )[i]) in the lok ref. Definition 4 (Reltive he stte). Given referene lok ref, reltive he stte s r ({,,,, }) N, is vetor [inst r, inst r,..., inst r N ], where eh element inst r i {,,,, }. Also, the set of ll possile reltive he sttes (w.r.t ref ) is denoted s CS r. Before we illustrte reltive he sttes, we introdue two key terms essentil to he nlysis. For ny si lok, the rehing reltive he sttes represent the set of reltive he sttes prior to the exeution of si lok nd the reltive leving he sttes represent the set of he sttes

9 B4 2 m3 3 m4 B3 m 2 3 B5 m2 2 3 B m m2 2 m3 3 m4 B2 m5 2 3 B9 2 3 m8 B6 m2 2 m3 3 m4 m6 2 m7 3 m8 m6 2 m3 3 m4 () Grph G r fter step B4 B3 m 3 B5 m2 3 B m m2 B2 m5 3 B9 3 m8 B6 m2 m6 m6 2 m7 3 m8 () Grph G ref fter step 2, given ref = B4: B5: m2 3 B: m m2 B6: m2 B9 3 m8 : m6 2 m7 3 m8 : m m2 () Grph G ref fter step 3, given ref = (redued grph). fter the exeution of the si lok. We denote the rehing reltive he sttes of si lok s RCS r nd the reltive leving he sttes of si lok s LCS r. B. Illustrtion 2 3 B r 2 3 Figure 7: 2 3 r 2 3 B6 r Figure 6: Illustrtion of Algorithm 4 with ref =. r B9 r Illustrtion of the reltive he sttes. A frgment of the redued grph (Figure 6()) is presented in Figure 7. Using this grph we illustrte reltive rehing nd leving he sttes of lok. Given the referene lok ref = with BI( ref ) = [, m6, m3, m4] nd the reltive instrution mpping of eh lok (desried y the funtion BI r ), we illustrte the reltive he stte s we strt exeuting the initil lok B. Initilly, the he is empty nd the referene lok does not hve n instrution on he line (BI r ( ref )[] = ). Thus the instrution on he line is not of interest ( ) resulting in the initil he stte of [,,, ], denoted s s r. Note this initil he stte is unlike the onrete pproh, where the initil stte is represented using the vetor [,,, ]. In our pproh, given the referene lok ref (in this se ref = ), we re only interested in the he line, 2, 3. Thus, we ignore the he stte w.r.t he line, s it does not ffet the preision of the referene lok. Also, y ignoring, llows for memory effiient implementtion. In this se, the reltive he stte n e represented using vetor with only three elements, insted of four elements, sving 25% of memory. After exeution of lok B, where the reltive instrutions re desried y BI r (B) = [,,, ], he lines, 2, 3 will ontin reltive instrutions,,. Hene fter B exeutes, the reltive he stte s r = [,,, ]. Here, s r [] = represents the instrution on he line is not of interest. s r [] = represents tht the instrution in the he is not m6, euse BI( ref )[] = m6. s r [2] = represents tht the instrution in the he is m3, euse BI( ref )[2] = m3. Similrly, s r [3] = represents the instrution in the he is m4 (BI( ref )[3] = m4). The reltive he stte s r is the stte of the he prior to the exeution of the si lok B. Thus, RCSB r = {[,,, ]} is the set of rehing reltive he sttes of lok B. Similrly, LCSB r = {[,,, ]} is the set of reltive leving he sttes of lok B. Now, the ontrol rehes rnh due to whih, it is possile to exeute either lok B6 or lok. In this se, RCSB6 r = LCSB r = RCS. r After exeuting loks B6 (or ), the stte of the he is [,,, ] (or [,,, ]). Blok hs two inoming edges: from loks B6 nd. To ompute RCS, r we need to join LCSB6 r nd LCS. r In this se, the join funtion is union over the set of reltive he sttes. Thus, RCS r = LCSB6 r LCS, r resulting in RCS r = {[,,, ], [,,, ]}. C. COMPUTING ALL POSSIBLE REACHING RELATIVE CACHE STATES OF THE REF- ERENCE BLOCK The first step for he nlysis involves the omputtion of ll possile rehing reltive he sttes of ref (RCS r ref ), using the fixed point omputtion lgorithm presented in Algorithm 5. As illustrted in Figure 7, the initil stte of the he is empty(s r ), nd is sed on the instrutions of the referene lok. Similrly, like the onrete nd the strt p-

10 prohes, we must lso introdue the unknown he stte (s r ), whih is explined lter during this lgorithm. In generl, the empty/unknown he stte is different for eh ref B. Thus, for given referene lok ref, we first ompute the empty he stte s r, nd unknown he stte s r on lines 2 to 8. For eh he line i, if the referene lok ref does not hve n instrution (in this se, BI r ( ref ) = ), then the he stte on i is not of interest during the nlysis of the referene lok ref. Thus, the reltive instrution on he line i of s r nd s r is set to (line 4). Otherwise (BI r ( ref ) ), on line 6, for he line i, the empty he stte is set to e (s r [i] = ), nd the unknown he stte is set to e (s r [i] = ). Using reltive he sttes s r nd s r, we initilise the rehing reltive he sttes for ll loks (lines to 7). Sine we ssume tht initilly the stte of the he is empty, on line 3 for the initil lok init we set its rehing s RCS r init = {s r }. Here, the nottion RCS ri represents the rehing reltive he sttes of lok in itertion i. E.g., RCS r init represents the rehing reltive he sttes of lok init for itertion. For rest of the loks, the initil stte of the he is unknown. Thus, on line 5, we set their rehing he sttes s {s r }. After initilistion, we ompute the reltive leving he sttes of eh lok, on lines 9 to 22. We pply the trnsfer funtion(t ) to every lok nd its orresponding rehing reltive he sttes. The itertion index (i) is inremented (line 23) to signl the strt of the next itertion. Next, on lines 25 to 34, the rehing reltive he sttes of eh lok re omputed. For the initil lok init, we know tht the rehing reltive he stte is lwys empty. Thus, on line 27, we lwys set its rehing s RCS ri init = {s }. For rest of the loks, we first initilise the rehing reltive he stte s empty set (line 29), nd on lines 3 to 32, the rehing he sttes re omputed y looking t the reltive leving he sttes of the predeessors ( ) of the lok nd using the union opertion. The itertive proess, repet-until loop on lines 8 to 35, is repeted until fixed point is rehed, i.e., if two onseutive itertions hve the sme sets of rehing reltive he sttes for ll loks (line 35). D. RESULTS strt proposed onrete Gin Exmple WCET ATWCET ATWCET AT(ol5/ (lks)(se) (lks)(se) (lks)(se) ol2) BuleSort Syntheti Flsher T.O T.O 2 DrillSttion T.O T.O 6 ConvBeltModel T.O T.O 5 RilRodCrossing T.O T.O CruiseController T.O T.O Algorithm 5 F P : Fixed point omputtion for the proposed pproh Input: A he model CM = I, C, CI, G, BI, redued grph G r = B r, init, E r nd BI r : B r ({,,, }) N. Output: Rehing reltive he sttes of lok ref (RCS r ref ). : {Initilise s r nd sr } 2: for eh i C do 3: if BI r ( ref )[i] = then 4: s r [i] =, sr [i] = {When ref hs no instrution (in this se, BI r ( ref ) = ) on he line i, initilise empty/unknown he sttes s not of interest.} 5: else 6: s r [i] =, sr [i] = {When ref hs n instrution on he line i, initilise empty/unknown he sttes s /.} 7: end if 8: end for 9: i = {itertion ounter} : {Initilise RCS r for ll loks } : for eh B do 2: if = init then 3: RCS r = {s r init } {for the initil lok, the initil stte of 4: the he is lwys empty} else 5: RCS r the he is unknown} 6: end if 7: end for = {s r } {for rest of the loks, the initil stte of 8: repet 9: {Compute LCS r for ll loks} 2: for eh B r do 2: LCS ri 22: end for = T (RCS ri, ) 23: i = i + ; {Next itertion} 24: {Compute RCS r for the next itertion i + } 25: for eh B r do 26: if = init then 27: RCS ri = {s r init } 28: else 29: RCS ri = 3: for eh LCS ri, where (, ) E r do 3: RCS ri 32: end for 33: end if 34: end for = RCS ri 35: until B r, RCS ri 36: return RCS r ref LCSri = RCS ri {Termintion ondition} we oserve tht the WCET estimtes from the proposed pproh is lwys less thn or equl to the estimtes from the strt pproh. With % reltive he size, on verge, the WCET of proposed pproh gives 3 % muh tighter results nd upto 9% tighter result thn the strt pproh. Tle 5: Quntittive omprison etween the strt, proposed nd onrete pprohes The WCET nd the nlysis time for strt is presented in olumns 2 nd 3 respetively of Tle 5. Similrly, the following olumns present results for the proposed nd onrete pprohes. Finl olumn presents the WCET estimte of the proposed pproh w.r.t. the strt pproh. The proposed pproh is tighter y 3% on verge, nd up to 9% tighter thn the strt pproh. Using the he size etween.2% nd.9% of the progrm size, for eh exmple, the WCET nlysis results re presented in Figures 8() 8(h). Aross ll the enhmrks,

11 Astrt Astrt Astrt Astrt ().2% reltive he size ().3% reltive he size ().4% reltive he size (d).5% reltive he size Astrt Astrt Astrt Astrt (e).6% reltive he size (f).7% reltive he size (g) % reltive he size (h).9% reltive he size Figure 8: Compring the WCET of the strt nd proposed pprohes etween.2% nd.9% reltive he sizes B B2 B3 B B2 B3 B B2 B3 new old d e f new old {} {} {} {d} {} {} {} {e} {} {} {} {f} new old new old f e f new { } {} {, } old { } new old B4 () onrete pproh B4 () strt pproh B4 () extending the proposed pproh Figure 9: Extending nd ompring the proposed pproh to set ssoitive hes E. EXTENDING OUR APPROACH TO SET ASSOCIATIVE CACHES The proposed pproh n esily e extended for timing nlysis of set ssoitive hes. For 4 wy set ssoitive he with the lest reently used replement (LRU) poliy [3], omprison of the join opertion etween the onrete nd strt pprohes is presented in Figure 2 of [3]. Using this ext exmple, we reprodue Figures 9() nd 9(). Here, lok B4 hs three predeessor loks (B, B2 nd B3). The stk(s) next to the trnsitions represent the stte of the he, for eh he line. The order represents the history, the reent instrution is on top of the stk, nd the lest reent instrution is t the ottom of the stk. For the onrete pproh (Fig. 9()), the rehing he sttes of B4 re omputed s the union of ll the leving he sttes of the predeessor loks, resulting in three he sttes. This is very preise, ut will not sle for lrge progrms. In the se of the strt pproh (Fig. 9()), the instrutions re omined sed on the upper ound of its ge. E.g., for the leving he sttes of B, B2 nd B3, instrution resides on top (first), top (first) nd seond positions, respetively. Sine, the upper ge ound (oldest) is the seond position, the join funtion strts the ge of instrution s seond position. Similrly, instrution nd re omputed s third position. In ontrt, instrution d, only exists in the leving he stte of B, nd not in B2 nd B3. In this se, we nnot gurntee its presene, so it is removed from the stk. Similrly, instrutions e nd f re removed during the join opertion. This strtion, improves slility, ut lks preision. For the proposed pproh (Fig. 9()), the instrutions in the he re presented w.r.t. the instrutions of the referene lok, if identil, or otherwise. E.g., let us ssume tht instrutions,, d re strted s, nd, e, f re strted s. This strtion is similr to the ide of reltive he sttes presented erlier in Se One gin, the join opertion performs the union over the rehing reltive he sttes, mintining oth preision nd slility. Sine the join opertions of eh pproh is similr to their ounter prts in diret-mpped nlysis, we elieve the omplexity of the three pprohes my e similr to T. 2. Also, the WCET nd the nlysis time my reflet the trends seen in Figures 4 nd 5.