jitter Abstract Output jitter the variation in the inter-completion times of successive jobs of the same

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1 Schedulng perodc task systems to mnmze output jtter Sanjoy Baruah Gorgo Buttazzo y Sergey Gornsky z Guseppe Lpar y Abstract Output jtter the varaton n the nter-completon tmes of successve jobs of the same task s studed n the context of the preemptve unprocessor schedulng of perodc real-tme tasks. A formal quanttatve model for output jtter s proposed. A lst of propertes that are desrable n any jtter-mnmzaton schedule s enumerated. Algorthms are presented for generatng such schedules, and bounds proved for the maxmum jtter n schedules generated by these algorthms. Keywords. Schedulng: unprocessor, preemptve perodc tasks output jtter. 1 Introducton In many real-tme control applcatons, perodc actvtes represent the major computatonal demand on the system. Such perodc actvtes typcally arse from sensory data acquston, low-level servong, control loops, acton plannng, and system montorng, whch need to be cyclcally executed at specc rates (derved from the applcaton requrements). These perodc actvtes are often formally modelled as perodc tasks: each perodc task T =(e p) scharacterzed by two parameters { an executon requrement e and a perod p { wth the nterpretaton that T generates an nnte sequence of jobs arrvng at tme-nstants 0 p 2p 3p : : :, respectvely, wth the job arrvng at tme-nstant k p needng to execute for e tme unts over the nterval [k p (k +1)p) (.e., we assume that each job has a deadlne p tme unts after ts arrval tme). In ths paper, we restrct our attenton to the unprocessor, preemptve, schedulng of systems of such tasks by unprocessor, we mean that all the jobs generated by all the tasks n the system are to execute on a sngle shared processor by preemptve, we mean that a job executng on the shared processor may be nterrupted at any nstant n tme, and ts executon resumed later, wth no cost or penalty. A schedule for a system of perodc tasks speces whch task executes on the shared processor at each nstant n tme. Let ; = ft 1 T 2 ::: T n g denote a system of n perodc tasks T =(e p ) for 1 n. Wth respect to a partcular schedule S for;,thecompleton tme c (k) of the k'th job of T s the earlest tme-nstant k p such that T has executed for e tme unts n S over the nterval [k p c (k) ). Schedule S s a correct schedule for ; f and only f for all T 2 ; and for all nteger k 0, c (k) (k +1)p.e., the completon tme of the k'th job of T s no larger than the assocated deadlne. Department of Computer Scence, The Unversty of North Carolna, Chapel Hll, NC 27599{3175, USA. Emal: baruah@cs.unc.edu y Scuola Superore S. Anna, Va Carducc, 40, Psa, Italy. Emal: fgorgo, lparg@sssup.t z Department of Computer Scence, The Unversty of Texas, Austn, TX 78712, USA. Emal: gornsky@cs.utexas.edu 1

2 The prmary goal of a schedulng algorthm s to generate correct schedules whenever possble. That s, a schedulng algorthm for schedulng systems of perodc tasks should accept as nput a system ; of perodc tasks, and attempt to generate as output a correct schedule S for ;. (In ths respect, the Earlest Deadlne Frst schedulng algorthm (EDF) [5, 9] s known to be optmal, n the sense that f there exsts a correct schedule for a system ; of perodc tasks, then t s guaranteed that EDF wll generate a correct schedule for ;.) Assumng that ths prmary goal of meetng all deadlnes can be met, we may n general desre that our schedulng algorthms be able to optmze certan secondary objectves. One such secondary objectve s the focus of ths paper: mnmzng output jtter. Output jtter refers to the varaton between the nter-completon tmes of successve jobs of the same task. More formally, wth respect to a partcular (correct) schedule for ; let p (mn) p (max) def = mn k0 def = max k0 n c (k+1) n c (k+1) ; c (k) ; c (k).e., p (mn) and p (max) denote the mnmum and maxmum separaton between successve completons of T,1 n. An output jtter-free (henceforth, smply \jtter-free") schedule s one n whch p (mn) = p (max) (n whch case t follows that p (mn) = p (max) = p ). Schedules that have as ther secondary objectve the mnmzaton of output jtter attempt to mnmze the varaton of p (mn) and p (max) from p. In a sense, schedulng to mnmze output jtter n perodc task systems s a non-ssue. Assumng that the system under study s feasble (.e., there exst correct schedules for the system), aschedule can be rendered vrtually jtter-free by postponng executon of the last unts of each job's executon requrement untl ts deadlne, for some very small! 0 (equvalently, by postponng notcaton of the completon of a job untl ts deadlne f t completes executon pror to ts deadlne, t s smply buered for the tme remanng untl ts deadlne s reached). Indeed, ths s the approach adopted n [6] to handle the jtter problem n packet-swtched networks each packet s held back n a specal buer by eachnternal node of the network for the maxmum possble delay nthatnode. From a real-tme systems pont ofvew, however, such a soluton s far from deal. As stated above, EDF s an optmal schedulng algorthm for schedulng systems of perodc real-tme tasks. In addton to ths desrable property of optmalty, EDF possesses many other features that render t partcularly sutable for schedulng unprocessor real-tme systems. These nclude: lowschedulng overhead a reasonable bound on the number of preemptons (no more than twce the number of jobs) satsfyng the ntegral boundary constrant (IBC) f all the executon requrements and perods are ntegers, then all processor preemptons must take place at nteger boundares only etc. We would lke any jtter-mnmzaton algorthm to possess these propertes as well below, we attempt to dentfy some of the propertes that we would lke any jtter-mnmzaton algorthm to satsfy (n addton to the prmary goal of meetng all deadlnes): 1. The scheme s not allowed to buer a job that s ready to execute. That s, f a job s ready to be released at tme t, we are NOT allowed to delay ts release to some tme after t. 2. Perhaps equvalently, we wll not allow for nserted dle tme n the schedule f there s some job ready to execute n the system, we are NOT allowed to dle the processor. 3. The scheme s not allowed to buer a job that has completed. That s, f a job has completed executon at tme t, we are NOT allowed to hold back from notfyng the external world of 2 o o

3 ths fact. 4. The schedule generated should satsfy the ntegral boundary constrant (IBC). 5. The scheme should not requre too many preemptons. We would lke to see no ncrease n the number of preemptons over current schedulng schemes, n the worst-case. That s, the bound on the number of preemptons proved for the Earlest Deadlne Frst schedulng algorthm (EDF) [9] the total number of preemptons s no more than twce the number of jobs should contnue to hold. 6. The on-lne schedulng overhead should contnue to be as small as n the case of EDF. Informally speakng, what we would lke to do s contnue to use EDF as our schedulng algorthm, but to perhaps change the \control" nformaton that we use n dong ths schedulng. To ths end, we descrbe n ths paper a transformaton on a system of perodc tasks such that the schedule generated by EDF on the transformed system has better jtter performance than the schedule generated by EDF on the orgnal system, whle contnung to be a correct schedule for the orgnal system. Related work. Consderable research has prevously been performed on the topc of jtter n a dynamc-schedulng context. For example, Ln and Herkert [8] have studed a dstance constraned task model n whch the derence between any two consecutve nshng tmes of the same task s requred to be bounded by a speced value. Ths approach attempts to mnmze the outgong jtter caused by the scheduler. Stankovc and D Natale [11] have studed jtter-mnmzaton n the context of provdng end-to-end tmng guarantees n dstrbuted real-tme systems. Researchers n network communcatons have also devoted consderable attenton to the phenomenon of jtter. They have consdered the problem of provdng determnstc tmng guarantees n crcut-swtched networks where the transmsson of a vdeo stream takes a route consstng of several hops, each of whch may add to the jtter. Jtter-control schemes to mnmze the end-to-end jtter n such networks have been proposed, for example by Ferrar [7] and Verma, Zhang, and Ferrar [6]. The research presented n [1, 3, 4, 12] addresses the problem of accomodatng nput jtter.e., schedulng systems of perodc tasks n whch the ready-tmes of jobs cannot be predcted exactly a pror, but are subject to tmng uncertantes at runtme. Organzaton. The remander of ths paper s organzed as follows. In Secton 2, we formally state our model of perodc tasks, and provde a quanttatve denton of output jtter. In Secton 3, we descrbe a polynomal-tme algorthm for transformng a perodc task system, such that the transformed task system can be scheduled by EDF to generate a schedule that exhbts sgncantly less jtter than would be exhbted by EDF-schedulng the orgnal task system. In Secton 4, we descrbe a pseudopolynomal-tme algorthm for transformng a perodc task system nto another that can be scheduled by EDF wth less jtter although ths algorthm s computatonally more expensve than the one n Secton 3, t can n general guarantee better jtter behavour. We have mplemented both our jtter-control algorthms, and have performed experments testng ther eectveness we descrbe these results n Secton 5. In Secton 6, we llustrate our technques by tracng ther behavour on some smple perodc task systems. 2 System model As stated n the ntroducton, we are concerned here wth the problem of mnmzng output jtter n EDF-generated schedules of systems of perodc tasks. We wll assume that each perodc task T 3

4 s characterzed by ajtter tolerance factor, wth the nterpretaton that tasks wth large are more tolerant to jtter, n addton to the executon requrement e and the perod p. Let;denote def a perodc task system consstng of n tasks T 1 T 2 ::: T n.let = e =p, and def = P n =1. It has been proved [9] that a necessary an sucent condton for EDF to generate a correct schedule for ; s that be at most one. Wth respect to a partcular schedule for ;, let p (mn) and p (max) denote the mnmum and maxmum separaton between successve completons of T,1n. Thequantty AbsJtter(T ) def =max p (max) ; p p ; p (mn) s dened to be the (absolute) jtter of task T observe that ths can also be computed as max k0 j(c (k+1) ; c (k) ) ; p j.e.,absjtter(t ) measures the maxmum varaton of the ntercompleton tme from the task perod. In the remander of ths paper, we wll make use of the followng nequalty: AbsJtter(T ) maxfc (k) ; k p g ; mnfc (k) ; k p g (1) k0 k0.e., the derence between the largest response tme 1 and the smallest response tme of a job of T s an upper bound on the varaton of ntercomplton tmes of jobs of T. (In general, ths bound need not be tght: consder, for example, a task T wth perod 5 and executo requrement 1,and = 19, and c (k) =5k +5forall k 4. Whle ; k p g;mn k0 fc (k) ; k p g equals 5 ; 1 = 4.) The (absolute) jtter of task system ; s dened to be largest absolute jtter of any task n ;: aschedule n whch c (0) =1,c (1) =7,c (2) =13,c (3) AbsJtter(T ) for ths schedule s 1, the bound max k0 fc (k) The weghted jtter of T s dened to be The weghted jtter of ; s dened to be AbsJtter(;) def = max fabsjtter(t )g T 2; WtdJtter(T ) def = AbsJtter(T ) : WtdJtter(;) def =maxf AbsJtter(T ) g : (2) T 2; The goal of ths research s to devse schedulng algorthms that accept as nput perodc task systems ;, and generate schedules for ; that meet all deadlnes and attempt to mnmze WtdJtter(;). The general model descrbed above can represent several nterestng forms of jtter whch may be of relevance to the applcaton system desgner. For nstance, f = 1 for all, then WtdJtter s the same as AbsJtter. Settng = p for all n Equaton 2, we obtan the noton of relatve jtter the jtter of a task s measured as a fracton of ts perod (as an example, a schedule wth a relatve jtter of 30 % guarantees that the ntercompleton tmes between successve jobs of T s at least 0:7p, and at most 1:3p,forall). Smlarly, we can model systems n whch only a subset of the tasks are jtter-senstve by settng the jtter-tolerance factor for the other (non jtter-senstve) tasks to nnty. 1 The response tme of the k'th job of T s dened to be the derence between the completon tme of the job c (k) and ts release tme k p. 4

5 3 Jtter control n polynomal tme 3.1 Computng jtter bounds Snce a job of T completes no sooner than e tme unts after ts ready tme and no later than ts deadlne n any correct schedule for ;, t s easly seen that e and 2p ; e are \quck-and-drty" bounds on the maxmum and mnmum ntercompleton tmes respectvely of T.e., the jtter for task T s clearly (p ; e ) equvalently, WtdJtter(;) max T 2; f p ; e g =maxf e ( 1 ; 1)g (3) T 2; Wth a lttle more eort, we can compute tghter bounds for the case when { the system utlzaton { s less than 1: Theorem 1 In an EDF-generated schedule for ;, the separaton between the completon tmes of successve jobsoftaskt les n the nterval [p ; e (= ; 1) p + e (= ; 1)].e., AbsJtter(T ) e ( ; 1) equvalently, WtdJtter(T ) e ( ; 1). Proof: Consder a scenaro where the k'th job of T arrves at tme t o and completes at the earlest possble tme, the (k + 1)'th job completes at the latest possble tme, and the (k +2)'th job completes, once agan, at the earlest possble tme. The completon tme of the k'th job s at least t o + e. Let the completon tme of the (k + 1)'th job be t 1, and let [t o +2p ; ` t 1 ] denote the longest contguous nterval precedng t 1 durng whch the processor only executes jobs wth deadlnes at or before t o +2p. Snce all such jobshave arrval-tmes t o +2p ; `, and deadlnes t o +2p, and they all complete by t 1, t follows that t 1 ; (t o +2p ; `) s at most `. Hence, t 1 t o +2p ; `(1 ; ) snce ` s at least p, ths mples that t 1 t o +2p ; p (1 ; ) : The completon tme of the (k + 2)'th job s, once agan, at least t o +2p + e. The maxmum ntercompleton tme, as dened by the derence between the completon tmes of the (k + 1)'th and k'th jobs, s thus at most t o +2p ; p (1 ; ) ; (t o + e ) = 2p ; p (1 ; ) ; e = p (1 + ) ; e = p + p ; e = p + e ( p e ; 1) = p + e ( ; 1) Smlarly, the mnmum ntercompleton tme, as dened by the derence between the completon tmes of the (k + 2)th and the (k + 1)'th jobs, s at least t o +2p + e ; (t o +2p ; p (1 ; )) 5

6 = e + p (1 ; ) = p ; (p ; e ) = p ; e ( p e ; 1) = p ; e ( ; 1) Thus, gven a system ; of perodc tasks, we can determne an upper bound on ts WtdJtter by the followng relatonshp: e WtdJtter(;) max ( ; 1) (4) T 2; Is ths bound tght? Unfortunately, no. Tghter bounds can be obtaned by actually smulatng EDF on the gven task-set, or by makng use of the actual parameters the executon requrements and perods of all the tasks n the system. However, such approaches take exponental (or at least pseudo-polynomal) tme as opposed to the computaton suggested by the above relatonshp, whch takes tme lnear n the number of tasks. 3.2 Jtter-mnmzaton n polynomal tme In attemptng to decrease AbsJtter(;), we can transform task system ; by ncreasng the processorshare reserved for the use of each task T to (whle ensurng that P n =1 1 holds). Havng done so, we cannow \assgn" a new deadlne to the job of T, released at k p,tobekp +(e = ) snce, ths assgned deadlne wll be no larger than the job's actual deadlne at (k +1)p. These assgned deadlnes are the ones that wll be used by EDF s schedulng ;, and t s guaranteed that each job wll complete by ts new assgned deadlne: ths follows from the fact [5] that EDF s an optmal unprocessor schedulng algorthm n the sense that f there s a correct schedule for a set of jobs, then EDF wll generate a correct schedule for ths set of jobs. The exstence of a correct schedule for the transformed system follows from the observaton that the task T can be assgned a fracton of the processor durng each tme-slot over the nterval [kp kp + e ), therefore, T s assgned the processor for a total of e = e slots. Indeed, f the nput parameters the e 's and the p 's are ntegers, t s easly seen that the \assgned" deadlne of the k'th job of T can n fact be set equal to bk p +(e = )c. Wth ths new assgned deadlne, t can be shown (usng technques vrtually dentcal to the ones used n the proof of Theorem 1) that the mnmum and maxmum separaton between successve jobs s now bounded by p e (= ; 1) (5) where def = P n =1.We therefore conclude the followng: Theorem 2 Let ; be a perodc task system wth processor-shares ( 1 2 ::: n ). The maxmum output jtter of ; s bounded as follows: WtdJtter(;) max f e ( ; 1)g (6) T 2; In order to mnmze the weghted jtter experenced by ;,we wll attempt to choose the processor-shares ( 1 2 ::: n )such that max T 2;f(e = ) (= ; 1)g s mnmzed. Of course, 6

7 any such choce wll have to preserve 1 (the total processor-shares allocated does not exceed unty) and for each (the processor-share allocated s at least as much as the amount the task needs). In addton, the resultng schedule should satsfy the lst of desrable propertes for jtter-control schemes enumerated n the Introducton. The above formulaton represents the problem as an optmzaton problem. We wll now prove some propertes about optmal solutons. But rst, a denton: Denton 1 Processor shares ( 1 2 ::: n ) are dened to be optmal processor shares for task system ;=ft 1 T 2 ::: T n g f and only f the followng condtons hold: P1: For each, 1 n,. P2: 1, where def = P n =1. P3: For any processor shares ( ::: 0 n) such that 1. for each, 1 n, 0,and , where 0 def = P n =1 0, t s the case that max f(e = ) (= ; 1)g max (e = ) ( 0 = 0 ; 1) T 2; Lemma 1 If ( 1 2 ::: n ) are optmal processor shares for ;, then so are ( 1 = 2 = : : : n =), where def = P n j=1 j. (Observe that P n j=1 ( j =) equals one.) Proof: Let ^ def def =( =), and let ^ = P n =1 ^. We wll prove belowthat(^ 1 ^ 2 ::: ^ n ) satses the propertes enumerated above for optmal processor shares. T 2; P1: Snce 1, t follows that ^ =( =). Therefore, ^. P2: ^ =( P n =1 ^ )=( P n =1 ( =)) = (( P n =1 )) =(=) =1. P3: Snce ^ = 1, t follows that (^=^ )=(1=^ )=(1=( =)) = =. Therefore, for all, from whch tfollows that max T 2; e ( ^ ; 1) = e ( ; 1) ^ ( e ( ^ ^ ; 1) ) e =max ( ; 1) T 2; Recall that our goal was to determne optmal processor shares, and hence to be able to generate EDF schedules whch mnmze output jtter. Lemma 1 above suggests that we can narrow our search to only those processor shares that sum to one.e., we need only look for ( 1 2 ::: n ) for whch P n =1 =1. But suppose an addtonal goal was to determne optmal processor shares n whch the cumulatve processor shares assgned to all the tasks.e., the quantty were as small as possblle. Lemma 2 below suggests how we can obtan such optmal processor shares from one n whch = 1. 7

8 Lemma 2 Let ( 1 2 ::: n ) be optmal processor shares, wth P n =1 =1.Let def =max n =1 ( = ). Then ( 1 2 ::: n ) are optmal processor shares. Proof: Let ^ def def =( ), and let ^ = P n =1 ^. We wll prove belowthat(^ 1 ^ 2 ::: ^ n ) satses the propertes enumerated above for optmal processor shares. P1: Snce, t follows that ^ =( ). Therefore, ^. P2: Observe rst that 1. Now, ^ =( P n =1 ^ )=( P n =1 ( )) = ( ( P n =1 )) =. Therefore ^ 1. P3: (^=^ )=(=( )) = (1= )==. Therefore, for all, from whch tfollows that max T 2; e ( ^ ; 1) = e ( ; 1) ^ ( e ( ^ ^ ; 1) ) e =max ( ; 1) T 2; Lemma 1 suggests that we can look for an optmal soluton wth the processor-shares assgned the varous tasks summng to one. Should we be lookng for an optmal soluton that uses the mnmum possble processor-shares, Lemma 2 tells us how we can get to such a soluton from a soluton n whch the processor-shares assgned equals one. We now descrbe how we can use the above results to determne processor-shares for a gven task system ;, such that these processor shares yeld the best jtter bounds obtanable usng our method. Suppose that WtdJtter(;) J when the processor-shares are ( 1 2 ::: n ). By Equaton 6, we have for all, 1 n, e ( 1 1 ; 1) J J +1 e e e + J : Snce we also requre,wehave max e e + J : (7) Hence, to determne whether a gven J can be a value for WtdJtter(;), we can compute each accordng to Equaton 7 above, and ensure that P n =1 does not exceed one. If the 's sum to at most one, then ths value of WtdJtter s achevable f not, then a jtter bound of J cannot be acheved usng ths partcular method. We can use a bnary search to determne the tghtest jtter bound obtanable by ths method. Let J nt denote the ntal jtter bound yelded by Inequalty 4 for the system: 1. J hgh J nt 2. J low 0 3. repeat 4. J md (J hgh + J low )=2 8

9 5. Compute the shares,1n, accordng to Equaton f ( P n =1 1) J hgh J md 7. else J low J md 8. untl (J hgh ; J low thres) (where thres s dened to be the degree of accuracy desred.) 4 Jtter control n pseudopolynomal tme A further generalzaton to the perodc task model has each task T =(e d p) characterzed by a relatve deadlne d p n addton to the perod p and the executon requrement e. The ssue of feasblty determnaton for a system of such tasks has been studed [2] for bounded-utlzaton task systems (systems where the ratos e=p of the executon-requrement to perod of all the tasks sum to no more than a constant strctly less than one), ths can be ecently done n tme pseudopolynomal n the sze of the problem nstance. Consder perodc task system ; = ft =(e p )g n =1. If we assgn a relatve deadlne d = e + J to each task T, and the resultng task system ; 0 = ft 0 =(e e + J p )g n =1 s feasble, then the maxmum separaton between successve completons of T 0 s at most p + J whle the mnmum separaton s at least p ; J, for each 1 n. Hence to obtan a run-tme schedule of ; wth a maxmum weghted jtter of J, t suces to Step 1: determne whether ; 0 s feasble, Step 2: f so, then schedule ; 0 rather than ; at run-tme. Usng the method of [2], Step 1 can be done n tme O(log n 1 ; n max fp ; d g) : (8) =1 Ths bound s pseudo-polynomal n the sze of the problem nstance. However, we expect to apply jtter-mnmzaton prmarly to systems wth a relatvely small value of (ntutvely speakng, ths s because systems wth a large value of are qute constraned and we have less freedom to modfy parameters to obtan better jtter behavor) consequently, wewould expect the tme bound of (8) to be qute reasonable n practce. For example, a of 0:9 would permt a processor utlzaton of 90 %, yet allow feasblty-analyss to be completed n tme proportonal to nne tmes the largest perod, multpled by the bnary logarthm of the number of tasks. In general, the jtter-mnmzaton procedure of ths secton yelds a tghter bound than the procedure n Secton 3.2, n the sense that the smallest jtter bound whch ths method can guarantee s smaller than the smallest bound obtanable usng the processor-share method of Secton 3.2. The downsde s that, due to the pseudo-polynomalty of (8), the tme-complexty s no longer polynomal. A reasonable approach n practce s to rst execute the method of Secton 3.2, and use the bound ^J thus obtaned as an upper bound for performng bnary search between 0 and ^J to determne the smallest J whch passes the pseudo-polynomal feasblty test of ths secton. A note on mplementaton. The feasblty testfrom[2]essentally conssts of smulatng the behavor of EDF on the perodc task system startng at tme zero, and termnatng when ether () a deadlne s mssed, at whch pont the system s known to be nfeasble, or () the processor s dled, whch would mply that the system s feasble. (See [2] for a proof of correctness.) The bound of (8) follows from the fact (proved n [2]) that one of these two events a mssed deadlne or an dle slot must occur wthn the rst 1; maxn =1 fp ; d g tme unts. 9

10 Implementng ths jtter-mnmzaton procedure therefore reduces to mplementng EDF for perodc task systems, for whch very ecent technques are known [10]. A note on ntal osets. The feasblty of a perodc task system n whch each task s characterzed by an executon requrement and a perod (no relatve deadlnes) s unaected by the presence of ntal osets {.e., f each taskt s characterzed by an addtonal oset parameter a wth the nterpretaton that the k'th job of T arrves at tme a + k p, and must receve servce by a +(k +1) p, for all nteger k 0. The jtter-mnmzaton procedure of Secton 3.2 enjoys the same property jtter analyss s unaected by whether tasks have ntal osets or not. Wth respect to the jtter mnmzaton procedure of ths secton, a slghtly weaker statement can be made. If the method outlned above guarantees a jtter bound of J, then ths jtter bound s obtanable even n the presence of osets. However, a system wth osets can sometmes be scheduled wth far smaller jtter than can be shown usng ths method. Attemptng to ncorporate osets nto the methodology of ths secton nvolves feasblty-analyss of perodc task systems n whch each task s characterzed by an oset, an executon requrement, a relatve deadlne, and a perod unfortunately, ths feasblty-analyss problem has been shown [2] to be co-np-complete n the strong sense, ndcatng that we are unlkely to be able to perform t n pseudopolynomal tme. We have therefore not attempted to take advantage of ntal osets to reduce jtter even when these osets are present. In related ongong research, we are however attemptng to mnmze jtter n task-systems of the model n [9] (where each taskscharacterzed by executon requrements and perods only), by ntroducng osets to ad n jtter-control when applcaton system semantcs would permt us to do so. The basc dea s to use the methodology proposed n ths paper to obtan an ntal jtter bound, and then to use heurstcs to search for phasngs whch may further reduce the jtter. Although such an approach would lkely prove ntractable n the worst case and may sometmes result n no mprovement n jtter-behavour, we expect that substantal mprovement may be obtaned on average. 5 Expermental evaluaton Gven a system ; of perodc tasks that are to be scheduled usng EDF, Equaton 3 represents a nave bound on the jtter experenced by the system WtdJtter(;) max T 2;f e ( 1 ; 1)g. In Theorem 1 (Equaton 4), we derved a somewhat tghter bound for systems where the utlzaton s strctly less than one WtdJtter(;) max T 2;f e ( ; 1)g. Then n Sectons 3 and 4, we presented algorthms for generatng EDF-schedules that exhbt lower jtter than mpled by these bounds the technque n Secton 3 takes tme polynomal n the representaton of ;, whle the one n Secton 4 takes pseudopolynomal tme but may n general result n schedules wth better jtter behavour. We have conducted a seres of experments to evaluate the eectveness of our jtter-mnmzaton schemes. We have mplemented both our jtter-mnmzaton schemes and have used them to mnmze the relatve jtter 2 of randomly-generated systems of perodc tasks some of the results are tabulated n Fgures 1{5. In each gure, the system jtter WtdJtter(;) s plotted as a functon of the total system utlzaton (also known as system load). For each value of the system load, we generated 900 task sets. Each task set was generated by randomly choosng the tasks' computaton 2 Recall that the relatve jtter of a perodc task measures ts jtter as a fracton of ts perod, and s obtaned by settng = p. 10

11 tmes as ntegers between 1 and 10, and then randomly choosng the perods such that the total system load be approxmately equal to the desred load. For each set we calculated the bound obtaned by Equaton 4 (the lne labelled \Bound" n the gures) then we appled both the polynomal-tme transformaton of Secton 3 (labelled \Method 1" n the gures) and the pseudopolynomal-tme transformaton of Secton 4 (labelled \Method 2" n the gures). Only average values are depcted n the gures. (For each pont we also calculated the 90% condence nterval these are not shown n the gures, but were small enough to conrm the valdty of our results.) The ve gures Fgures 1{5 represent the results of ve derent sets of experments: Number of tasks: 5/5 Bound Method 1 Method 2 Rel. Jtter Load Fgure 1: Fve tasks { mnmze jtter of the entre task set Fgure 1: ; conssts of ve tasks, and the goal s to mnmze the relatve jtter over the complete task set. Ths s obtaned by settng = p for all. Fgure 2: ; conssts of ten tasks, and the goal s to mnmze the relatve jtter over the complete task set. Ths s obtaned by settng = p for all. Fgure 3: ; conssts of ten tasks, and the goal s to mnmze the relatve jtter of just one of the tasks. Ths s obtaned by settng 1 = p 1,and = 1 for all >1. Fgure 4: ; conssts of ten tasks, and the goal s to mnmze the relatve jtters of three of the tasks. Ths s obtaned by settng = p for =1 2 3, and = 1 for all >3. Fgure 5: ; conssts of ten tasks, and the goal s to mnmze the relatve jtters of ve ofthe tasks. Ths s obtaned by settng = p for =1 2 ::: 5, and = 1 for all >5. Among the conclusons we are able to draw from our smulatons: For low system loads, EDF schedules wth reasonably low relatve jtter can be obtaned by both Method 1 and Method 2. As system load ncreases, Method 2 generally outperforms Method 1. If the relatve jtters of all tasks s to be mnmzed (as n Fgures 1 and 2), the methods presented n ths paper are not very eectve. In fact, f the system load s low, the results 11

12 1 0.8 Number of tasks: 10/10 Bound Method 1 Method 2 Rel. Jtter Load Fgure 2: Ten tasks { mnmze jtter of the entre task set Number of tasks: 1/10 Bound Method 1 Method 2 Rel. Jtter Load Fgure 3: Ten tasks { mnmze jtter of one task 12

13 1 0.8 Number of tasks: 3/10 Bound Method 1 Method 2 Rel. Jtter Load Fgure 4: Ten tasks { mnmze jtter of three tasks Number of tasks: 5/10 Bound Method 1 Method 2 Rel. Jtter Load Fgure 5: Ten tasks { mnmze jtter of ve tasks 13

14 obtaned by both methods are very close to the ntal bound (whch salreadylow) f the system load s hgh, Method 2 can obtan some mprovement over the ntal bound however the jtter remans hgh. Ths s not surprsng ntutvely speakng, mnmzng the jtter of an entre task system s a hghly-constraned optmzaton problem. Indeed, decreasng the jtter of some task can sgncantly ncrease the jtter of other tasks n the system. As Fgures 3{5 llustrate, jtter mnmzaton s more eectve when just a few of the tasks are jtter-senstve. Intutvely speakng, ths s because, once the prmary goal of meetng deadlnes have been satsed, all the freedom remanng n the schedule can be devoted to mnmzng the jtters of a few tasks. Especally n the case n whch we wsh to mnmze the jtter of only one task, Method 2 seems to provde a very low relatve jtter (Fgure 3) regardless of the overall system load. Ths s a potentally useful lesson n applcaton system desgn jtter mnmzaton s a precous resource, whch should be attempted only upon those perodc tasks that really need t. Although not llustrated by Fgures 1{5, another noteworthy observaton s that t s, n general, easer to mnmze relatve jtter of tasks wth low utlzaton. Once agan, ths lends tself to an ntutve explanaton: ncreasng the processor share of T k tmes (.e., k ), whle keepng every other task's processor shares the same, decreases WtdJtter(T )by a factor of k. Snce spare capacty (.e., 1 ; ) can be used to ncrease the processor shares of lowerutlzaton tasks by a larger factor than t can be used to ncrease the processor shares of hgher-utlzaton tasks, t s more ecent to use the excess capacty to nuence the jtter of lower-utlzaton tasks. 6 Examples In ths secton, we llustrate the technques presented n Sectons 3 and 4 by means of some examples. Example 1 Consder a system of three tasks: ; = ft 1 =(2 10) T 2 =(3 15) T 3 =(2 20)g wth 1 = 2 = 3 = 1 observe that 1 = 2 =0:2, 3 =0:1, and =0:5. Theorem 1 yelds a jtter bound WtdJtter(;) = 8. Ths s the ntal value for J hgh the ntal value for J low s zero. Observe that, snce 1 = 2 = 3 = 1, the nteger boundary constrant (IBC) guarantees that all jtter values wll be ntegers. Hence we may use nteger bnary search n ths example J hgh J md,and J low all need take onnteger values only. (Equvalently, the quantty thres n the pseudocode at the end of Secton 3.2 can be set equal to one.) Choosng J md =(8+0)=2 = 4 and pluggng nto Equaton 7, we determne that 1 =0:333, 2 =0:428, and 3 =0:333. Snce > 1, ths s an nfeasble soluton. Hence, J low s set equal to 4. Next, choosng J md =(8+4)=2 = 6 and applyng the same technques yelds ( )= (0:250 0:333 0:250), whch s a feasble soluton snce 0: : : Hence, J hgh s set equal to 6. Next, choosng J md =(6+4)=2 = 5 and applyng the same technques yelds ( )= (0:286 0:375 0:286), whch s once agan a feasble soluton snce 0: : : Hence, J hgh s set equal to 5. Next, we compute J md =(5+4)=2 =4. Butwe've already checked that ths results n an nfeasble soluton hence, the tghtest jtter bound obtanable usng the processor-share method s 5. 14

15 We now try usng the pseudo-polynomal tme test of Secton 4, performng bnary search wth ntal \hgh" of 5 and ntal \low" of 0. The test fals for J =(5+0)=2 = 2, and succeeds for J =(5+2)=2 = 3. Snce (3 + 2)=2 = 2 and we've already determned that the test fals for J =2, we can stop now, havng determned that the tghtest bound obtanable usng ths method s J = 3. Example 2 Consder a system of three tasks: ; = ft 1 =(2 9) T 2 =(4 15) T 3 =(2 12)g wth 1 = 2 = 3 = 1. Usng essentally the same approach asabove, Theorem 1 yelds an ntal jtter bound of 7 the polynomal-tme method of Secton 3.2 renes ths to 6 and the pseudopolynomaltme method of Secton 4 further mproves ths to 4. Example 3 Consder a system of three tasks: ; = ft 1 =(2 10) T 2 =(3 15) T 3 = (20 200)g wth 1 = 2 = 3 = 1. Once agan gong through the same steps as above, Theorem 1 yelds an ntal jtter bound of 80 the polynomal-tme method of Secton 3.2 renes ths to 14 and the pseudopolynomal-tme method of Secton 4 further mproves ths to 9. Example 4 Consder the same system of tasks ; = ft 1 =(2 10) T 2 =(3 15) T 3 =(2 20)g. Suppose that task T 3 s the only jtter-senstve task. Ths can be modelled by settng 1 = 2 = 1, and 3 = 1. Several knds of questons can now beasked for example: Is t possble to schedule ; such that the maxmum jtter for task T 3 never exceeds 3? Ths s equvalent to determnng a 3 such that e 3 ( 1 3 ; 1) 3 2( 1 3 ; 1) Snce assgnng a share of 2=5 to T 3 does not cause the total processor-share assgned to exceed 1, we conclude that t s possble to schedule ; such that T 3 's maxmum jtter does not exceed 3. What s the mnmum value for the maxmum jtter of task T 3? Snce the largest value that 3 can have s1:0 ; (0:2+0:2), ths s equvalent to computng e 3 ( 1 0:6 3 ; 1) = 2 10 ; 6 6 = 1:33 Snce the schedule wll obey the IBC, ths mples that the maxmum jtter of task T 3 wll be at most 1.e., the ntercompleton tme between consecutve jobs of task T 3 wll be ether 19, 20, or 21. Usng the method of Secton 4, we are reduced to determnng the smallest J such that the perodc task system f( ) ( ) (2 2+ J 20)g s feasble. Snce the utlzaton of ths perodc system s 0:5, we expect the pseudo-polynomal test to be qute ecent. 15

16 Ths turns out to ndeed be the case, and the smallest J satsfyng the desred condton s J = 0 (.e., the system f( ) ( ) (2 2 20)g s feasble). Ths permts us to conclude that the system can be scheduled such that T 3 suers absolutely no jtter.e., the ntercompleton tme between consecutve jobs of task T 3 wll always be exactly Conclusons As observed n the Introducton, output-jtter mnmzaton s not really an ssue n the schedulngtheoretcal sense, n that notcaton of the completon of jobs can always be postponed untl the jobs' deadlnes. In ths paper, we have decded that such an artcal soluton s unacceptable from a practcal pont of vew, and have enumerated a seres of condtons we beleve should be satsed by any reasonable jtter-mnmzaton scheme. Wthn the context of these condtons, we have proposed twoschedulng algorthms for mnmzng jtter. One scheme s polynomal-tme whle the other provdes better jtter bounds but takes tme pseudo-polynomal n the sze of the problem nstance however, even the pseudo-polynomal algorthm s very ecently mplementable. In practce, a reasonable approach to jtter mnmzaton seems to be to use the polynomal-tme algorthm to ecently obtan an upper bound on jtter, and then use the pseudo-polynomal algorthm to further mprove ths bound. We have mplemented and tested our schemes the algorthms are smple and easy to code, and the run-tme behavor s very satsfactory. References [1] N. Audsley, A. Burns, M. Rchardson, K. Tndell, and A. Wellngs. Applyng new schedulng theory to statc prorty preemptve schedulng. Software Engneerng Journal, 8(5):285{292, [2] S. Baruah, R. Howell, and L. Roser. Algorthms and complexty concernng the preemptve schedulng of perodc, real-tme tasks on one processor. Real-Tme Systems: The Internatonal Journal of Tme- Crtcal Computng, 2:301{324, [3] Sanjoy Baruah, Dej Chen, and Aloysus Mok. Jtter concerns n perodc task systems. In Proceedngs of the Eghteenth Real-Tme Systems Symposum, pages 68{77, San Francsco, CA, December IEEE Computer Socety Press. [4] A. Burns, M. Ncholson, K. Tndell, and N. Zhang. Allocatng and schedulng hard real-tme tasks on a pont-to-pont dstrbuted system. In Proceedngs of the Workshop on Parallel and Dstrbuted Real-Tme Systems, pages 11{20, Aprl [5] M. Dertouzos. Control robotcs : the procedural control of physcal processors. In Proceedngs of the IFIP Congress, pages 807{813, [6] D.Verma, H. Zhang, and Domenco Ferrar. Delay jtter control for real-tme communcaton n a packet swtchng network. In Proceedngs of the IEEE Conference on Communcatons Software: Communcatons for Dstrbuted Applcatons and Systems, pages 35{43, Chapel Hll, NC, [7] D. Ferrar. Clent requrements for real-tme communcatons servces. IEEE Communcatons, 28(11), November [8] K.J. Ln and A. Herkert. Jtter control n tme-trggered systems. In Proceedngs of the 29th Hawa Internatonal Conference on System Scences, Mau, Hawa, January [9] C. Lu and J. Layland. Schedulng algorthms for multprogrammng n a hard real-tme envronment. Journal of the ACM, 20(1):46{61, [10] A. Mok. Task management technques for enforcng ED schedulng on a perodc task set. In Proc. 5th IEEE Workshop on Real-Tme Software and Operatng Systems, pages 42{46, Washngton D.C., May

17 [11] J. Stankovc and M. DNatale. Dynamc end-to-end guarantees n dstrbuted real-tme systems. In Proceedngs of the Real-Tme Systems Symposum, San Juan, Puerto Rco, December [12] K. W. Tndell, A. Burns, and A. J. Wellngs. An extendble approach for analysng xed prorty hard real-tme tasks. Real-Tme Systems: The Internatonal Journal of Tme-Crtcal Computng, 6:133{151,