Fixed-Priority Schedulability of Sporadic Tasks on Uniprocessors is NP-hard

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1 Fixd-Priority Schdulability of Soradic Tasks on Unirocssors is NP-hard Pontus Ekbrg and Wang Yi Usala Univrsity, Swdn {ontus.kbrg Abstract W study th comutational comlxity of th schdulability roblm for soradic or synchronous riodic tasks on a rmtiv unirocssor. W show that this roblm is (wakly) NP-hard, vn whn rstrictd to ithr (i) task sts with imlicit dadlins and rat-monotonic riority ordring, or (ii) task sts with constraind dadlins, dadlin-monotonic riority ordring and utilization boundd by any constant c, such that 0 < c < 1. I. INTRODUCTION W considr tasks that ar soradic or synchronous riodic, xcutd by th Fixd-Priority () schduling algorithm on a singl rocssor that allows rmtions at no associatd cost or nalty. Th schduling algorithm is widly usd in raltim systms, but th comutational comlxity of dciding whthr a st of tasks is -schdulabl in this basic stting has bn a longstanding on roblm. In addrssing this w sttl th first roblm mntiond by Baruah and Pruhs [1] in thir list of on algorithmic roblms in ral-tim schduling. In this introduction w first dscrib modls and initions, followd by som tchnical rliminaris and, last, rlatd work and our contributions. A. Modl and Dfinitions Lt a soradic task τ b rrsntd as a tril (, d, ) of ositiv intgrs, rrsnts th task s worst-cas xcution tim, d its rlativ dadlin and its riod (or minimum saration tim), rsctivly. A task τ (, d, ) gnrats an unboundd squnc of jobs, ach job has an xcution tim of u to tim units and an absolut dadlin xactly d tim units aftr its rlas tim. Th rlas tims of two conscutiv jobs from th task τ ar saratd by at last tim units. W say that a job is rady whn it has bn rlasd but has not yt xcutd to comltion. A task st T is a (multi-)st of tasks and gnrats an intrlaving of th job squncs gnratd by ach of th tasks in T. W say that T has imlicit dadlins if d for all (, d, ) T, constraind dadlins if d for all (, d, ) T, and arbitrary dadlins if no rstrictions ar lacd on th rlation btwn d and. W considr schduling of soradic tasks on a rmtiv unirocssor. On such machins, th Fixd-Priority () schduling algorithm taks a fixd (total) riority ordring of th tasks in th task st and thn xcuts jobs strictly according to this riority ordring. In othr words, th schdulr attachs to ach job th riority of th task that gnratd it, and at any tim oint chooss among th rady jobs to xcut th job with th highst riority, rmting any lowr-riority job if ndd. If svral jobs from th sam task ar rady, ths ar rioritizd in FIFO ordr. Th rsults of this ar ar stablishd by rlating schdulability with th algorithm to schdulability with th Earlist Dadlin First (EDF) algorithm, for which hardnss rsults ar known. Th EDF schduling algorithm chooss among th rady jobs to xcut th job with th arlist absolut dadlin (tis brokn arbitrarily), rmting a job with a latr dadlin if ndd. Both th and EDF schduling algorithms hav bn xtnsivly studid in th litratur. Th EDF schdulr is known to b otimal on rmtiv unirocssors [2], but it is not imlmntd in most ral-tim orating systms. In contrast, th schdulr is th ault schdulr in many ral-tim orating systms, artly du to its asy and fficint imlmntations. (Buttazzo [3] discusss som ractical ascts and trad-offs btwn and EDF schduling.) W say that a task st T is -schdulabl with a givn riority ordring if and only if all jobs gnratd by T will always comlt xcution by thir dadlins whn using th schdulr with that riority ordring on a rmtiv unirocssor. Th -schdulability roblm is to dtrmin whthr a givn task st is -schdulabl with a givn riority ordring. Lt EDF-schdulabl and EDF-schdulability b ind similarly. Th synchronous riodic task is a common altrnativ workload modl to th soradic task. A synchronous riodic task is also ind by a tril (, d, ) of ositiv intgrs, with th only diffrncs that instad dnots th xact saration btwn conscutiv job rlass, and th first job of all such tasks ar rlasd synchronously at th sam tim oint. It is known that - and EDF-schdulability is unaffctd by th choic btwn ths two task modls. Thorm I.1 (Lhoczky [4], Baruah t al. [5]). Lt T b a st of soradic tasks, and lt T b a st of synchronous riodic tasks with th sam aramtrs as th tasks in T. Thn, on a rmtiv unirocssor (i) T is -schdulabl with a givn riority ordring if and only if T is -schdulabl with th sam riority ordring, and (ii) T is EDF-schdulabl if and only if T is EDF-schdulabl. Th quivalnc of - and EDF-schdulability for th

2 diffrnt tys of tasks in this stting mans that our rsults aly to both of thm. In th following, whn w simly writ task w intrchangably man a task that is ithr soradic or synchronous riodic. B. Prliminaris Hr w brifly rviw som rsults from ral-tim schduling thory that ar usd in this ar. In schduling, th riority ordring that assigns highr riority to tasks with shortr rlativ dadlins (tis brokn arbitrarily) is known as th Dadlin-Monotonic (DM) riority ordring. DM is an otimal riority ordring for task sts with constraind dadlins. Thorm I.2 (Lung and Whithad [6]). If a task st with constraind dadlins is -schdulabl on a rmtiv unirocssor with any riority ordring, thn it is also schdulabl with th DM riority ordring. Th Rat-Monotonic (RM) riority ordring instad assigns highr riority to tasks with shortr riods. For task sts with imlicit dadlins, RM is th sam as DM, and is thrfor an otimal riority ordring. Th sminal work by Liu and Layland [7] stablishd a sufficint condition for schdulability in this stting in th form of a utilization bound. Thorm I.3 (Liu and Layland [7]). A task st T of n imlicitdadlin tasks is -schdulabl on a rmtiv unirocssor with th RM riority ordring if is th utilization of T. U(T) n(2 1 n 1), (1) U(T) (,d,) T Not that lim n n(2 1 n 1) ln , which lads to th simlr sufficint condition (2) U(T) ln 2, (3) which w will mak us of instad of th on in Eq. 1. Liu and Layland s utilization-basd condition is only sufficint and only valid for task sts with imlicit dadlins. Josh and Pandya [8] gav an xact condition for -schdulability of task sts with constraind dadlins. This condition is basd on calculating a task s rsons tim, which is th maximum amount of tim that can ass btwn th rlas and finishing tim of any job gnratd by that task. Thorm I.4 (Josh and Pandya [8]). Lt τ low ( low, d low, low ) b a constraind-dadlin task schduld by th schdulr on a rmtiv unirocssor, and lt T h b th st of tasks with highr riority than τ low in th givn riority ordring. Th rsons tim r low of τ low is thn th smallst ositiv fixd oint to r low low + rbf(t h, r low ), (4) rbf(t, r) (,d,) T r (5) is th rqust bound function of a st of tasks T in a tim intrval of siz r. All jobs gnratd by τ low ar guarantd to mt thir dadlins with th givn riority ordring if and only if r low d low. Baruah t al. [5] gav an xact condition for EDFschdulability of task sts with arbitrary dadlins. Thorm I.5 (Baruah t al. [5]). A task st T is EDFschdulabl on a rmtiv unirocssor if and only if U(T) 1 and dbf(t, l) l 0, dbf(t, l) l, (6) (,d,) T { max 0, l d } + 1 (7) is th dmand bound function of T in tim intrval lngths l. Not that for a task st T with constraind dadlins, w hav ) l d dbf(t, l) + 1 (8) (,d,) T for l 0. As w considr constraind dadlins in this ar, w will us th simlr form in Eq. 8 for brvity. W also hav th following corollary. 1 Corollary I.6 (Baruah t al. [5]). A task st T with constraind dadlins is EDF-schdulabl on a rmtiv unirocssor if and only if U(T) 1 and l {0, 1,..., 1}, dbf(t, l) l. (9) is th hyr-riod of T. lcm{ (, d, ) T} (10) Th EDF-schdulability roblm (or fasibility roblm) is known to b strongly conp-comlt in th gnral cas [9]. W will bas our nw rsults on th following thorm about th hardnss in a mor rstrictd stting with boundd utilization. Thorm I.7 (Ekbrg and Yi [10]). Dciding whthr a task st is EDF-schdulabl on a rmtiv unirocssor is conpcomlt in th wak sns if rstrictd to task sts with constraind dadlins and utilization boundd from abov by any constant c, such that 0 < c < 1. 1 This is slightly diffrnt than th rsult rortd by Baruah t al. [5], th quivalnt is statd for arbitrary dadlins. In this corollary w rstrict attntion to constraind dadlins and thrfor gt a slightly smallr rang of valus for l to considr, which will b a usful rorty latr on. This variant follows dirctly by th rasoning of Baruah t al. [5]

3 Imlicit dadlins (d ) Constraind dadlins (d ) Arbitrary dadlins (d, unrlatd) Arbitrary utilization Wakly NP-comlt Wakly NP-comlt Wakly NP-hard (On ) Utilization boundd by a constant c Polynomial tim for c ln 2 and RM rioritis [7] (On ) Wakly NP-comlt for 0 < c < 1 Wakly NP-hard for 0 < c < 1 (On ) Arbitrary utilization Polynomial tim [7] Strongly conp-comlt [9] Strongly conp-comlt [9] EDF (Fasibility) Utilization boundd by a constant c Polynomial tim [7] Wakly conp-comlt for 0 < c < 1 [10] Wakly conp-comlt for 0 < c < 1 [10] Fig. 1. Comutational comlxity of th schdulability roblm for soradic or synchronous riodic tasks on a rmtiv unirocssor. Exct for th ntris markd On, ur and lowr bounds match, with all wakly NP- or conp-comlt roblms also having known sudo-olynomial tim solutions. S th nxt ag for a dscrition of th on roblms. C. Contributions and Rlatd Work Th rsults summarizd in th rvious sction rovid som ur bounds on th comutational comlxity of th -schdulability roblm. Thorm I.4 immdiatly yilds a sudo-olynomial tim algorithm for task sts with constraind dadlins. It also shows that th sam roblm is in NP (a st of small fixd oints is a witnss of schdulability). Thorm I.3 similarly givs a trivial olynomial-tim algorithm for th scial cas of task sts with imlicit-dadlins, RM rioritis and utilization boundd by ln 2. (S,.g., Baruah and Goossns [11] for mor in-dth discussions.) For th cas with arbitrary dadlins, thr ar xonntial-tim algorithms, as shown by Lhoczky [4], but no sudo-olynomial tim algorithms ar known. Dsit th walth of rsarch rlatd to ths roblms, lowr bounds on thir comutational comlxity hav bn lacking for dcads. To th bst of our knowldg, th only rvious rsult rlatd to lowr bounds on th -schdulability roblm was givn by Eisnbrand and Rothvoß [12]. Thy showd that with schduling it is NP-hard to vn aroximat th rsons tim of th lowst-riority task within a constant factor (i.., to aroximat th smallst fixd oint in Eq. 4 for a givn task). Whil this intuitivly sms to suggst that th -schdulability roblm is hard, Eisnbrand and Rothvoß [12] as wll as Baruah and Pruhs [1] oint out that this dos not follow. Th rason is that th rductions mloyd by Eisnbrand and Rothvoß can crat som highrriority tasks that ar clarly unschdulabl in ordr to mak it hard to calculat th rsons tim of a singl lowr-riority task. Rothvoß [13] latr conjcturd that th -schdulability roblm is NP-hard, vn with imlicit dadlins and RM rioritis. Our contributions ar to rovid lowr bounds for th schdulability roblm that in svral cass match th known ur bounds. Our main rsults ar as follows. (1) Th -schdulability roblm is wakly NP-hard, vn whn rstrictd to task sts with imlicit dadlins and RM riority ordring. (2) Th -schdulability roblm is wakly NP-hard, vn whn rstrictd to task sts with constraind dadlins, DM riority ordring and utilization boundd by any constant c, such that 0 < c < 1. Itm (1) sttls a strongr vrsion of On Problm 1 as listd by Baruah and Pruhs [1] and rovs th conjctur of Rothvoß [13]. Itm (2) shows that with constraind dadlins, th comutational hardnss of -schdulability rmains vn if w rstrict attntion to task sts with vry low utilization, similar to th cas for EDF-schdulability. As th RM and DM riority ordrings ar known to b otimal in thir rsctiv sttings, w hav as a corollary that th hardnss rsults of both itms (1) and (2) rmain th sam if w wr to instad ask if a givn task st is -schdulabl with any riority ordring, instad of with a givn on.

4 Figur 1 combins our nw lowr bounds with th known ur bounds, and contrasts ths with th known bounds on EDF-schdulability. Exct for thos ntris markd as On, th lowr bounds match th ur bounds, with wakly NP- and conp-comlt roblms having known sudo-olynomial tim algorithms. On : Th lowr bounds hr carry ovr from th corrsonding roblms with constraind dadlins. Howvr, whil thr ar xonntial-tim algorithms for -schdulability with arbitrary dadlins (s Lhoczky [4]), thr ar no known sudo-olynomial tim algorithms. In contrast to th cas with constraind dadlins, which is sn to b in NP bcaus of Thorm I.4, it is not clar that th arbitrary dadlins cas admits asily vrifiabl witnsss, and so to th bst of our knowldg, its mmbrshi in NP is also on. On : If th -schdulability roblm is rstrictd to task sts with utilization boundd by a constant c and RM riority ordring, thn Thorm I.3 yilds a trivial olynomialtim algorithm for c ln 2. It is on whthr thr xists a olynomial-tim algorithm for th cas whn w hav RM riority ordring but ln 2 < c < 1, or for th cas an arbitrary riority ordring can b givn and 0 < c < 1. II. THE HARDNESS OF -SCHEDULABILITY In this sction w show that th -schdulability roblm is wakly NP-hard, vn whn rstrictd to ithr of th two scial cass mntiond in th rvious sction. W can rov ithr cas with only a minor variation to th roofs, so th following will targt both cass. W slit this roof into a chain of two rductions. (1) Th first is a olynomial-tim many-on rduction from th EDF-schdulability roblm for task sts with constraind dadlins and utilization boundd by any constant c, such that 0 < c < 1, to a scial cas of th sam roblm that is furthr rstrictd to task sts all tasks hav airwis corim riods. (2) Th scond is a olynomial-tim many-on rduction from that scial cas of th EDF-schdulability roblm (though hr w must hav 0 < c ln 2) to th comlmnt of th -schdulability roblm. This works by finding a duality btwn dmand bound functions and rqust bound functions that xists whnvr th tasks hav airwis corim riods. As th sourc roblm of th first rduction is conp-hard by Thorm I.7 for any c such that 0 < c < 1, th NP-hardnss of th -schdulability roblm will follow. For th first rduction w will us a rsult from numbr thory about th Jacobsthal function g(n). Jacobsthal [14] ins g(n) to b th smallst numbr such that all intrvals of g(n) conscutiv intgrs a, a + 1, a + 2,..., a + g(n) 1 contain at last on numbr that is corim to n. Whil g(n) is vry irrgular, it grows slowly. Th bst known ur bound on th Jacobsthal function is du to Iwanic [15], from which w hav g(n) K log 2 (n) (11) for n 2 and som unknown ositiv constant K. W will us this bound to show that suitabl corim numbrs can b found for th rduction. As a convninc w assum, without loss of gnrality, that K N +. A. Rducing EDF-schdulability with Boundd Utilization to a Scial Cas with Pairwis Corim Priods Lt c b any constant such that 0 < c < 1. Thn lt T c dnot an instanc of th EDF-schdulability roblm rstrictd to task sts with constraind dadlins and utilization boundd by c. Givn any T c, our rduction roducs a task st Tc that is furthr rstrictd so that all tasks in Tc hav airwis corim riods, whil Tc is EDF-schdulabl if and only if T c is. First, in n T c, (12) κ (10KnP(T c )) 3, (13) K is th constant from Iwanic s bound on th Jacobsthal function (s Eq. 11), and is th hyr-riod of a task st T (s Eq. 10). Thn, lt th tasks in T c b dnotd as {( 1, d 1, 1 ),..., ( n, d n, n )}, and lt thos tasks b indxd by non-dcrasing riods, so that j i if j < i. Now, th task st Tc roducd by th rduction is ind as i d i i T c {( 1, d 1, 1),..., ( n, d n, n)}, (14) κ i, (15) κd i, (16) min{m κ i m is corim to j, j < i}. (17) Not that by this construction, Tc is a coy of T c task aramtrs hav bn scald by κ, xct that th riods might b vn largr to nsur that thy ar airwis corim. As a consqunc, Tc also has constraind dadlins and w must hav U(Tc ) U(T c ) c. W bgin by stablishing a lmma about how incrasing th rlativ magnituds of th riods by a modrat amount dos not affct EDF-schdulability. Thn w show in anothr lmma that th conditions of th first lmma aly to th task st T c, rlativ to T c, and that thy thrfor hav th sam EDF-schdulability. Last w show how to comut T c in olynomial tim. Lmma II.1. Lt T b a task st with constraind dadlins and lt T b anothr task st satisfying T {(k i, kd i, k i + δ i ) ( i, d i, i ) T}, k N + and 0 δ i < k i. Thn, T is EDF-schdulabl if and only if T is EDFschdulabl.

5 Proof: W show th two dirctions saratly. T is EDF-schdulabl T is EDF-schdulabl: Assum that T is EDF-schdulabl. Thn, for all l 0 w hav l dbf( T, l) d ) i + 1 ẽ i i (ẽ i, d i, i) T ) l kdi + 1 k i k i + δ i ( i,d i, i) T l k d ) i + 1 k i ( i,d i, i) T k dbf(t, l/k) l, follows from Thorm I.5 and th EDFschdulability of T. By th sam thorm, T must also b EDF-schdulabl. T is EDF-schdulabl T is EDF-schdulabl: Assum that T is EDF-schdulabl. Thn, for all l 0 w hav ) l di dbf(t, l) + 1 i ( i,d i, i) T ( i,d i, i) T ( i,d i, i) T ( i,d i, i) T i i ( )( ) ) k + k l di ( ) + 1 k + k i kl + kl kd i kdi k i + ki kl + kl kd ) i + 1 k i + δ i kl + kl d i i (ẽ i, d i, i) T ) dbf ( T, kl + kl k l + l, + 1 ) i ẽ i k i + 1 follows from Thorm I.5 and th EDFschdulability of T. Now, bcaus dbf(t, l) is intgrvalud, w must hav dbf(t, l) l + l, but for all l {0, 1,..., 1} w thn hav dbf(t, l) l + l l, and T is EDF-schdulabl by Corollary I.6. ) i Now w show that task st Tc mts th conditions of Lmma II.1, rlativ to T c, and thrfor rsrvs th EDFschdulability of T c. Lmma II.2. Tc EDF-schdulabl. if is EDF-schdulabl if and only if T c is Proof: Th lmma holds, by construction and Lmma II.1, i κ i < κ i P(T c ), (18) for all 1 i n. W show that Eq. 18 holds by strong induction ovr i. Th bas cas (i 1) trivially holds as 1 κ 1. For 1 < i n, not that i is corim to j for all j < i if and only if i is corim to N, N i 1 j1 j. W know that all intrvals of g(n) conscutiv intgrs contain a numbr corim to N, g is th Jacobsthal function. Also not that w hav N < i 1 j j1 i 1 j1 ( κ j + i 1 2κ j j1 (2κ i ) n, κ ) j P(T c ) follows from th induction hyothsis and from th ordring of task indics by non-dcrasing riods. Using Iwanic s bound on g from Eq. 11, w thn hav g(n) K log 2 (N) < K log 2 ((2κ i ) n ) Kn 2 log 2 (2κ i ) (19) 36Kn 2 log 2 ( 6 2κ i ) < 36Kn 2 3 2κ i < 36Kn 2 (2 3 κ i ) 72Kn 2 (10KnP(T c )) i 720K 2 n 3 P(T c ) i < 1000K 3 n 3 P(T c ) 2 i (10KnP(T c)) 3 i P(T c ) κ i P(T c ). It follows from th inition of g that thr xists a numbr corim to N in th intrval [ κ i, κ i + κi P(T c)). By th inition of i w thrfor hav i < κ i + κ i P(T c ), and Eq. 18 holds. Thr is nothing to show for i > n, so this concluds th induction st and th roof.

6 Finally, th rduction can b shown to b valid. Lmma II.3. Dciding whthr a task st of soradic or synchronous riodic tasks is EDF-schdulabl on a rmtiv unirocssor is conp-comlt in th wak sns if rstrictd to task sts with constraind dadlins, utilization boundd from abov by any constant c, such that 0 < c < 1, and tasks with airwis corim riods. Proof: W know from Thorm I.7 that this dcision roblm without th rstriction to airwis corim riods is conp-comlt for any constant c such that 0 < c < 1. W hav shown that thr xists a rduction 2 to th scial cas with airwis corim riods that by Lmma II.2 is a valid many-on rduction. What rmains to b shown is that th rduction can b comutd in olynomial tim. For this, th only challng is to comut th valus of th riods i. To s that this can b don in olynomial tim, w again us Iwanic s bound on th Jacobsthal function g. By inition, i is th smallst intgr not lss than κ i and corim to N i 1 j1 j. This numbr must b in th intrval [κ i, κ i + g(n)), but from Eq. 19 w hav g(n) < Kn 2 log 2 (2κ i ). As Kn 2 log 2 (2κ i ) is boundd by som olynomial in th siz of th rrsntation of T c, w know that i is containd in an intrval of olynomial lngth. W can sarch this intrval for i by looking at all intgrs κ i, κ i + 1,... until w find a numbr that is corim to N. Using standard algorithms for comuting gratst common divisors (.g., th Euclidan algorithm), this can b don in olynomial tim. B. Rducing EDF-schdulability with Pairwis Corim Priods to th Comlmnt of -schdulability Now w dscrib a olynomial-tim many-on rduction from th scial cas of th EDF-schdulability roblm that was shown to b conp-comlt in Lmma II.3 to th comlmnt of th -schdulability roblm. Lt c b any constant such that 0 < c ln 2. Givn an instanc Tc of th EDF-schdulability roblm with constraind dadlins, utilization boundd by c, and airwis corim riods, lt ˆl b th smallst numbr such that and ˆl d (mod ), (, d, ) T c (20) ˆl > max{ (, d, ) T c }. (21) Bcaus th tasks in Tc hav airwis corim riods, w know by th Chins rmaindr thorm that ˆl xists and that ˆl 2P(Tc ). Not that if dbf(tc, ˆl) > ˆl, thn Tc is infasibl by Thorm I.5 and th rduction is trivial. In th 2 This roof is actually somwhat non-constructiv. As w do not know th valu of K th constant in Iwanic s bound on th Jacobthal function w hav not dscribd a concrt rduction, but only showd that on xists. Whil this is a bit unusual, it is nough for th uross of dmonstrating comutational hardnss. following w assum, without loss of gnrality, that dbf(t c, ˆl) ˆl. (22) Th rduction thn roducs a task st T as T h τ low low d low low T T h {τ low }, (23) {(,, ) (, d, ) Tc } (24) ( low, d low, low ) (25) ˆl dbf(tc, ˆl) + 1 (26) ˆl (27) ϕˆl (28) for som ϕ N +. W will st th valu of ϕ latr in Thorm II.8, to targt ithr of th two scial cass of th -schdulability roblm that was mntiond in Sction I-C. Not that T h is a coy of Tc with dadlins st to qual riods. First w stablish two lmmas, starting with a lmma about th sizs of countrxamls to th EDF-schdulability of Tc. Lmma II.4. T c is EDF-schdulabl if and only if l {0, 1,..., ˆl}, dbf(t c, l) l. (29) Proof: Th ncssity of th condition in Eq. 29 follows dirctly from Thorm I.5. W show th sufficincy by contradiction. Assum for this uros that th condition in Eq. 29 holds but Tc is not EDF-schdulabl. Thn, by assumtion and Thorm I.5 thr must xist a λ N + such that dbf(tc, ˆl + λ) > ˆl + λ, but ( dbf(tc, ˆl ˆl + λ d + λ) (,d,) Tc ( ) ˆl d λ (,d,) Tc ( ) ˆl d (,d,) T c dbf(t c, ˆl) + λu(t c ) ˆl + λu(t c ) ˆl + λc < ˆl + λ, ) + 1 (,d,) T c λ holds bcaus (ˆl d)/ is an intgr for all (, d, ) Tc by th inition of ˆl (s Eq. 20), and follows by assumtion. Th sufficincy of th condition in Eq. 29 follows from this contradiction. Nxt w find a connction btwn dmand bound functions and rqust bound functions. W know from Thorms I.4 and I.5 that ths can rovid conditions for EDF- and schdulability, rsctivly, which maks this a ky st for th rduction.

7 Lmma II.5. For all l {0, 1,..., ˆl}, w hav dbf(t c, ˆl) dbf(t c, l) rbf(t h, ˆl l). Proof: Tak any l {0, 1,..., ˆl} and w hav dbf(tc, ˆl) dbf(tc, l) ) ) ˆl d l d (,d,) Tc ) ˆl d l d (,d,) Tc ( ) ˆl d l d (,d,) Tc ( ) ˆl d (l d) (,d,) Tc ˆl l (,d,) T c (,d,) T h ˆl l rbf(t h, ˆl l), qualitis markd follow bcaus (ˆl d)/ is an intgr and follows from th inition of T h. Rcall that w rstrictd th sourc roblm of our rduction to instancs Tc with U(Tc ) c ln 2. As is dmonstratd by th following lmma, this mans that th -schdulability of th constructd task st T hings only on th schdulability of its lowst riority task τ low for aroriat riority ordrings. Lmma II.6. Task st T is -schdulabl with RM and DM riority ordring if and only if undr this schduling all jobs gnratd by τ low ar guarantd to mt thir dadlins. Proof: Assum ithr RM or DM riority ordring. Thn τ low is th lowst-riority task in T as by Eqs. 21, 27 and 28 w hav low > max{ (, d, ) T h } and w also hav d low > max{d (, d, ) T h }. As th tasks in T h all hav imlicit dadlins, th RM and DM riority ordrings ar in fact th sam for T. Not also that by construction w hav U(T h ) U(T c ) c ln 2. As τ low can not affct th schduling of th highr-riority tasks in T h, w know by Thorm I.3 (s scially th simlifid condition in Eq. 3) that all jobs gnratd by tasks in T h will mt thir dadlins. Task st T is thrfor -schdulabl if and only if all jobs gnratd by τ low will mt thir dadlins. With a last lmma w show that th schdulability of th jobs gnratd by τ low, and thus th whol task st T, is invrsly rlatd to th EDF-schdulability of T c. Lmma II.7. Task st T is -schdulabl with RM and DM riority ordring if and only if Tc is not EDF-schdulabl. Proof: First not that by Lmma II.6 w know that T is -schdulabl undr RM or DM riority ordring if and only if all jobs gnratd by task τ low ar guarantd to mt thir dadlins. By Thorm I.4, this is th cas if and only if thr xists a fixd oint r low to th quation r low low + rbf(t h, r low ), such that 0 < r low d low. Not that r low, if it xists, is an intgr. W saratly rov th two dirctions of th lmma. T is -schdulabl Tc is not EDF-schdulabl: Assum that T is -schdulabl, and hnc that thr xists som r low such that 0 < r low d low and r low low + rbf(t h, r low ). Thn, T c dbf(t c, ˆl r low ) dbf(t c, ˆl r low ) + dbf(t c, ˆl) dbf(t c, ˆl) dbf(tc, ˆl) rbf(t h, ˆl (ˆl r low )) dbf(tc, ˆl) rbf(t h, r low )) dbf(tc, ˆl) + low r low ˆl r low + 1, follows from Lmma II.5 and from th inition of low. By Thorm I.5 it follows that Tc is not EDF-schdulabl. is not EDF-schdulabl T is -schdulabl: Assum that Tc is not EDF-schdulabl. Thn, by Lmma II.4 thr xists som l {0, 1,..., ˆl} such that dbf(tc, l) > l. Lt r d low l and not that r d low l ˆl l > ˆl dbf(t c, l) ˆl dbf(t c, l) + dbf(t c, ˆl) dbf(t c, ˆl) low + dbf(t c, ˆl) dbf(t c, l) 1 low + rbf(t h, ˆl l) 1 low + rbf(t h, r ) 1, follows by assumtion and follows from Lmma II.5. Bcaus both r and low + rbf(t h, r ) 1 ar intgrs, w must thn hav r low + rbf(t h, r ), (30) but by th inition of rqust bound functions (s Eq. 5) w also hav 0 < low + rbf(t h, 0). (31) Combining Eqs. 30 and 31 with th obsrvation that rbf(t h, r) is a non-dcrasing function in r, w can s that thr must xist som r low such that 0 < r low r d low and r low low + rbf(t h, r low ). By Thorm I.4, it follows that all jobs from τ low ar guarantd to mt thir dadlins, and thus by Lmma II.6 that T is schdulabl.

8 Finally w can show th NP-hardnss of th schdulability roblm. Thorm II.8. Dciding whthr a st of soradic or synchronous riodic tasks is -schdulabl with a givn riority ordring on a rmtiv unirocssor is NP-hard in th wak sns. This holds vn if rstrictd in ithr of th following ways. (i) All tasks hav imlicit dadlins and th riority ordring is RM. (ii) All tasks hav constraind dadlins, th riority ordring is DM and th utilization of th task st is boundd from abov by any constant c, such that 0 < c < 1. Proof: W hav dscribd a rduction from th EDFschdulability roblm rstrictd to task sts with constraind dadlins, airwis corim riods and utilization boundd by any constant c, such that 0 < c ln 2, to th comlmnt of th -schdulability roblm. By Lmma II.7, this is a valid many-on rduction with both th RM and DM riority ordrings. Not that th valu of ˆl can b comutd in olynomial tim using standard algorithms for th Chins rmaindr thorm, and th rst of th rduction can trivially b comutd in olynomial tim as wll. Bcaus th abov EDF-schdulabilty roblm is conp-comlt by Lmma II.3, th -schdulability roblm is (wakly) NP-hard. W can gt ithr of th two rstrictions (i) and (ii) by adating th valu of th constant ϕ in Eq. 28. Not that all lmmas shown so far ar indndnt of th valu of ϕ. If w st ϕ 1, thn T has imlicit dadlins and th NP-hardnss of th -schdulability roblm with rstriction (i) follows. Considr instad rstriction (ii) with any constant c, such that 0 < c < 1. W choos th constant c for bounding th utilization for th EDF-schdulability roblm that is th sourc of th rduction as c c/2. Not that w still hav c ln 2 and that our EDF-schdulability roblm is conp-comlt also with this c by Lmma II.3. Thn w st ϕ 2/ c. Now, T has constraind dadlins and w also hav U(T ) U(T h ) + U({τ low }) U(T c ) + low low c + 1 ϕ c 2 + c 2 c. Th NP-hardnss of th -schdulability roblm with rstriction (ii) follows. III. CONCLUSIONS Th comutational comlxity of dciding whthr a st of soradic or synchronous riodic tasks is -schdulabl on a rmtiv unirocssor has bn a longstanding on roblm, arguably going back to th sminal work of Liu and Layland [7]. W hav rovidd lowr bounds on th comlxity of this roblm, which in svral imortant sttings match ur bounds rovidd by classic rsults in ral-tim schduling thory. A rmaining on roblm is to dtrmin th xact comlxity of -schdulability with arbitrary dadlins. It has a wll-known xonntial-tim algorithm, but thr rmains a ga btwn this and our lowr bound, which shows that -schdulability is wakly NP-hard. Mmbrshi in NP is also on for th cas with arbitrary dadlins. Anothr outstanding on roblm is to dtrmin whthr -schdulability of task sts with imlicit dadlins and utilization boundd by som constant c < 1 admits a olynomialtim solution. W know from Liu and Layland [7] that this is (trivially) th cas whn c ln 2 and th riority ordring is RM, but it is unknown if this also holds for ln 2 < c < 1 or with arbitrary riority ordrings. REFERENCES [1] S. Baruah and K. Pruhs, On roblms in ral-tim schduling, Journal of Schduling, vol. 13, no. 6, , [2] M. L. Drtouzos, Control robotics: Th rocdural control of hysical rocsss, in Procdings of th IFIP congrss, vol. 74, 1974, [3] G. C. Buttazzo, Rat monotonic vs. EDF: Judgmnt day, Ral-Tim Systms, vol. 29, no. 1,. 5 26, [4] J. P. Lhoczky, Fixd riority schduling of riodic task sts with arbitrary dadlins, in Procdings of th 11th Ral-Tim Systms Symosium (RTSS), Dc 1990, [5] S. Baruah, A. K. Mok, and L. E. Rosir, Prmtivly schduling hardral-tim soradic tasks on on rocssor, in Procdings of th 11th Ral-Tim Systms Symosium (RTSS), 1990, [6] J. Y.-T. Lung and J. Whithad, On th comlxity of fixd-riority schduling of riodic, ral-tim tasks, Prformanc Evaluation, vol. 2, no. 4, , [7] C.-L. Liu and J. W. Layland, Schduling algorithms for multirogramming in a hard-ral-tim nvironmnt, Journal of th ACM, vol. 20, no. 1, , [8] M. Josh and P. Pandya, Finding rsons tims in a ral-tim systm, Th Comutr Journal, vol. 29, no. 5, , [9] P. Ekbrg and W. Yi, Unirocssor fasibility of soradic tasks with constraind dadlins is strongly conp-comlt, in Procdings of th 27th Euromicro Confrnc on Ral-Tim Systms (ECRTS), 2015, [10], Unirocssor fasibility of soradic tasks rmains conp-comlt undr boundd utilization, in Procdings of th 36th Ral-Tim Systms Symosium (RTSS), 2015, [11] S. Baruah and J. Goossns, Schduling ral-tim tasks: Algorithms and comlxity, [12] F. Eisnbrand and T. Rothvoß, Static-riority ral-tim schduling: Rsons tim comutation is NP-hard, in Procdings of th 29th Ral-Tim Systms Symosium (RTSS). IEEE Comutr Socity, 2008, [13] T. Rothvoß, On th comutational comlxity of riodic schduling, Ph.D. dissrtation, SB, Lausann, [14] E. E. Jacobsthal, Übr Squnzn ganzr Zahln, von dnn kin zu n tilrfrmd ist, 1-3, Norsk Vid. Slsk. Forh. (Trondhim), vol. 33, , [15] H. Iwanic, On th roblm of Jacobsthal, Dmonstratio Math., vol. 11, , 1978.