A Virtual Deadline Scheduler for Window-Constrained Service Guarantees

Transcription

1 A Vrtual Deadlne Scheduler for Wndow-Constraned Servce Guarantees Yutng Zhang, Rchard West and Xn Q Computer Scence Department Boston Unversty Boston, MA {danazh,rchwest,xq}@cs.bu.edu Abstract Ths paper presents a new approach to wndowconstraned schedulng, sutable for multmeda and weakly-hard real-tme systems. We orgnally developed an algorthm, called Dynamc Wndow-Constraned Schedulng (DWCS), that attempts to guarantee no more than x out of y deadlnes are mssed for real-tme jobs such as perodc CPU tasks, or delay-constraned packet streams. Whle DWCS s capable of generatng a feasble wndowconstraned schedule that utlzes 100% of resources, t requres all jobs to have the same request perods (or ntervals between successve servce requests). We descrbe a new algorthm called Vrtual Deadlne Schedulng (VDS), that provdes wndow-constraned servce guarantees to jobs wth potentally dfferent request perods, whle stll maxmzng resource utlzaton. VDS attempts to servce m out of k job nstances by ther vrtual deadlnes, that may be some fnte tme after the correspondng real-tme deadlnes. Notwthstandng, VDS s capable of outperformng DWCS and smlar algorthms, when servcng jobs wth potentally dfferent request perods. Addtonally, VDS s able to lmt the extent to whch a fracton of all job nstances are servced late. Results from smulatons show that VDS can provde better wndow-constraned servce guarantees than other related algorthms, whle stll havng as good or better delay bounds for all scheduled jobs. Fnally, an mplementaton of VDS n the Lnux kernel compares favorably aganst DWCS for a range of schedulng loads. 1. Introducton The ubquty of the Internet has led to wdespread delvery of content to the desktop. Much of ths content s now stream-based, such as vdeo and audo, havng qualty of servce (QoS) constrants n terms of throughput, delay, jtter and loss. More recently, developments have focused on large-scale dstrbuted sensor networks and applcatons, to support the delvery of QoS-constraned data streams from sensors to specfc hosts [11], hand-held PDAs and even actuators. Many stream-based applcatons can tolerate late or lost data delvery as long as a mnmum fracton s guaranteed to reach the destnaton n a tmely fashon. However, there are constrants on whch peces of the data can be late or lost. For example, the loss of too many consecutve packets n a vdeo stream sent over a network, mght result n sgnfcant pcture breakup rather than a tolerable reducton n sgnal-to-nose rato. Whle stream-based applcatons are often tolerant of late or lost nformaton, other real-tme applcatons (e.g., n embedded systems) are sometmes capable of functonng at acceptable levels even when a number of tasks are executed late or not at all. For example, a CPU-bound task that must sample and process sensor data may skp some samples as long as the mnmum samplng rate s above a certan threshold. To deal wth the above classes of applcatons, we have developed a number of wndow-constraned schedulng algorthms. Wndow-constraned schedulng s a form of weakly-hard [3, 4] servce, n whch a mnmum number of consecutve job nstances (e.g., perodc tasks or consecutve packets n a real-tme stream) must be processed by ther deadlnes n every fnte wndow. Smlar to the assumpton n the Lu and Layland model [10], every job J s perodc, wth a constant request perod, T, for each job nstance J,j. One such algorthm we developed n pror work s Dynamc Wndow-Constraned Schedulng (DWCS) [16, 14, 15]. DWCS attempts to guarantee no more than x out of a fxed wndow of y deadlnes are mssed for consecutve job nstances. DWCS s capable of guaranteeng a feasble schedule for each job, J, such that no more than x out of y nstances of J are servced late, or skpped, as long as the total utlzaton of all requred job nstances does not exceed 100%. However, DWCS s only capable of guaran-

2 teeng a feasble schedule when all jobs have the same request perod. Although ths seems restrctve, a smlar constrant apples to pnwheel schedulers [7, 5, 1], and t can be shown by careful manpulaton of x and y that mnmum fractons of servce are guaranteed for each J n fnte and tunable wndows of tme. Mok and Wang extended our orgnal work by showng that the general wndow-constraned problem s NP-hard for arbtrary servce tme and request perods [12]. Whle they also developed a soluton to the wndow-constraned schedulng problem for unt servce tme and arbtrary request perods, t s only capable of guaranteeng a feasble schedule when resources are utlzed up to 50%. Ths has prompted us to devse a new algorthm, called Vrtual Deadlne Schedulng (VDS), that guarantees resource shares to a specfc fracton of all job nstances, even when resources are 100% utlzed and request perods dffer between jobs. That s, two jobs J and J j, may have dfferent request perods, T and T 1 j. In order to generate a feasble schedule for the wndowconstraned problem, both the request deadlnes and wndow-constrants of jobs must be consdered. Instead of consderng these two factors separately as n DWCS, VDS combnes them together to determne a vrtual deadlne that s used to order job nstances. Vrtual deadlnes are set at specfc ponts wthn a wndow of tme, to ensure each job s gven a proportonate share of servce. Unlke other approaches that attempt to provde proportonal sharng of resources, VDS dynamcally adjusts vrtual deadlnes as the urgency of servcng a job changes. From expermental results, VDS s able to outperform other algorthms that attempt to satsfy the wndowconstraned schedulng problem. However, VDS s specfcally desgned to satsfy a relaxed form of the wndowconstraned schedulng problem, n whch m out of k job nstances must be servced by ther vrtual (as opposed to real) deadlnes. In effect, ths guarantees a fracton of resource usage to each job over a fnte nterval of tme, whle boundng the delay of each job nstance. Although a job nstance may mss ts real deadlne, VDS s stll able to ensure a mnmum of m job nstances are servced n a specfc wndow of tme. Ths s sutable for applcatons that can tolerate some degree of delay up to some maxmum amount. In fact, VDS mposes the same delay bounds on jobs servced accordng to both the relaxed and orgnal wndowconstraned schedulng models. These propertes of VDS make t sutable for a number of multmeda applcatons and those supported by weakly-hard real-tme systems. The contrbutons of ths paper can now be summarzed as follows: 1 Note that we assume the request perod, T, for job J s a constant but ths s more for analyss reasons than any mplct restrcton on the algorthm. We present a relaxed (m, k) wndow-constraned model, that s approprate for many classes of applcatons, such as multmeda streamng and real-tme data samplng. We present a new algorthm, called vrtual deadlne schedulng (VDS), that combnes a job s perodbased deadlne and wndow-constrant to determne the schedulng order. We show how VDS can make full use of resources, whle stll managng to servce n jobs, such that each job J s servced at least m tmes every non-overlappng wndow of k T real-tme. Moreover, we ensure that each job nstance s servced n keepng wth ts delay constrants, defned by a vrtual deadlne. For jobs wth dfferent request perods, VDS outperforms DWCS and smlar algorthms for the orgnal wndow-constraned schedulng problem, that requres m out of k deadlnes to be met. Addtonally, VDS can also guarantee servce n the relaxed model up to 100% utlzaton, for jobs wth dfferent request perods. We compare the performance of VDS to algorthms such as DWCS, Elgblty-based Wndow-Deadlne- Frst (EWDF), and EDF based on Pfar scope [12], usng a seres of smulatons. In these smulatons, a set of random jobs are servced accordng to varous wndow-constrants and arbtrary request perods. The rest of the paper s organzed as follows. In the next secton, we defne the wndow-constraned schedulng problem, n both ts orgnal and relaxed forms. The VDS algorthm and an analyss of ts characterstcs are then descrbed n Secton 3. In Secton 4, we smulate the performance of VDS, and compare t wth other wndowconstraned schedulng algorthms. Addtonally, we show the performance of VDS for real-tme workloads when operatng as a CPU scheduler n the Lnux kernel. Ths s followed by a descrpton of related work n Secton 5. Fnally, conclusons and future work are descrbed n Secton Wndow-Constraned Schedulng We orgnally developed the DWCS algorthm to address the wndow-constraned schedulng problem. Gven a set of n perodc jobs, J 1,,J n, a vald wndow-constraned schedule requres at least m out of k nstances of a job J to be servced by ther deadlnes. Deadlnes of consecutve job nstances are assumed to be separated by request perods of sze T, for each job J, as n Rate Monotonc schedulng [10]. One can thnk of a job nstance s request perod as the nterval between when t s ready and when t must complete servce for a specfc amount of tme. Moreover, the ready tme of one job nstance s also the deadlne of prevous job nstance. Therefore, the request perod T s also the nterval between deadlnes of successve nstances of J. Thus, f the jth nstance of J s denoted by J,j, then

3 the deadlne of J,j s d,j = d,j 1 + T. We assume that every nstance of J has the same servce tme requrement, C 2. Ths mples that a wndowconstraned schedule must (a) ensure at least m nstances of J are servced by ther respectve deadlnes, and (b) the mnmum servce share for J s m C tme unts every wndow of k T tme. In pror work, the assumpton was that each wndow of k T tme, or k deadlnes spaced apart by T tme unts, was non-overlappng wth a prevous or successve wndow. Ths dffers from the pnwheel schedulng model that consders wndows to be sldng. Whle we can transform wndow-constrants (m, k) to ther equvalent values (m, 2k m) for sldng wndows, we wll assume wndows are non-overlappng throughout the rest of ths paper. Based on the above, a wndow-constraned job, J,s defned by a 4-tuple (C,T,m,k ). A mnmum of m out of k consecutve job nstances must be servced for C tme n every wndow tme of k T for each job J wth request perod T. Ths mples the mnmum utlzaton factor of each job J s U = mc k T. Addtonally, the mnmum requred utlzaton for a set of n perodc jobs s U mn = n m C =1 k T. When the system s overloaded, the total resource utlzaton U = n C =1 T > 1.0, and t s therefore mpossble to servce every nstance of all n jobs. However, f the mnmum requred utlzaton U mn 1.0, a feasble wndow-constraned schedule may exst. It can be shown that a feasble wndow-constraned schedule must exst f each and every job J meets m deadlnes every k T wndow of tme durng the hyper-perod of sze lcm(k T ). However, the general wndow-constraned problem wth arbtrary servce tmes and request perods has been shown to be NP-hard [12]. Wth arbtrary servce tmes, t s not possble to guarantee a feasble wndowconstraned schedule for all job sets even f the mnmum requred utlzaton U mn 1.0. Fgure 1 shows an example job set for whch a feasble wndow-constraned schedule cannot be produced. It should be clear that J 1 and J 3 cannot both satsfy ther wndow-constrants. However, f the servce tme of each and every job nstance s constant, and all request perods are a fxed multple of ths constant, then a feasble wndow-constraned schedule exsts even when U mn 1.0 [16]. Relaxng the wndow-constraned schedulng problem: If we consder a schedule that starts at tme, t =0, then J requres servce for at least m C unts of tme by t = k T. However, as stated earler, each job nstance must be servced for C tme unts n the current request perod. Ths prevents J from recevng a contnuous burst of servce of m C unts from t = k T m C to t = k T. In effect, 2 C can be thought of as the worst-case executon tme of any nstance of J. Job (C,T,m,k) J 1 (2,3,2,3) J 2 (1,3,1,3) J 3 (2,3,2,3) U mn 1 T 1 T 2 T 3 J 1 J 1 J 2 J 1 J 1 J 2 J 1 J 1 J 2 J 3 J 3 J 2 J 3 J 3 J 2 J 3 J 3 J J 3 volates J 1 volates tme Fgure 1. Example of an nfeasble wndowconstraned schedule when servce tmes are not all the same. a wndow-constraned schedule prevents large bursts of servce to one job at the cost of others. However, a relaxed verson of the problem, n whch job nstances may be servced after ther deadlnes as long as a job receves at least m C unts of servce every nterval k T may be acceptable for some real-tme applcatons. Ths s true for many multmeda applcatons, and those whch can tolerate a bounded delay, as long as they receve a mnmum fracton of servce n fxed tme ntervals. For example, packets carryng multmeda data streams can experence fnte bufferng delays before transmsson, or processng at a recever. Ths has prompted us to relax the orgnal wndowconstraned problem, to allow job nstances to be servced after ther deadlnes as long as we guarantee a mnmum fracton of servce to a job. As wll be seen later, Vrtual Deadlne Schedulng (VDS) attempts to guarantee a feasble schedule accordng to these relaxed constrants. However, VDS stll prevents a job beng servced entrely at the end of a wndow of sze k T tme unts, by spreadng out where the m nstances of a job must be servced n that nterval. In effect, VDS adopts a form of proportonal far schedulng of at least m nstances of each job, J, every nterval k T. For clarfcaton, Fgure 2 shows the dfference between the orgnal and relaxed wndow-constraned schedulng problems. Case (a) descrbes the orgnal wndowconstraned problem, n whch at most one nstance of a job, J, s servced every request perod. A feasble schedule results n servce for a J n at least m out of k perods, every adjacent wndow of k T tme slots. Case (b) shows the relaxed wndow-constraned schedulng problem. Up to α nstances of a gven job can be servced n a sngle perod of sze, T,fα 1nstances have mssed ther real-tme deadlnes n the current wndow of sze k T. In case (b) of Fgure 2, up to 2 nstances of J can be servced n perod T,5, accordng to the relaxed wndow-constraned problem. However, case (c) shows that wth the relaxed wndowconstraned schedulng model, only one job nstance can be servced n perod T,4, because no deadlnes have been mssed n the current wndow. In prevous work, we show how the DWCS algorthm can meet wndow-constrants for n jobs when the mn-

4 (a) (b) (c) T,1 k T C k T T,5 T,1 k T k T T,4 T,1 k T k T = servced Job J : C =1, T =4, m =2, k =3; T,j : jth request perod of job J Fgure 2. Orgnal versus relaxed versons of the wndow-constraned schedulng problem. mum requred utlzaton factor, U mn = n m C =1 k T 1.0, f all servce tmes are a constant, and request perods are a fxed multple of ths constant. That s, DWCS s capable of producng a feasble wndow-constraned schedule when resources are 100% utlzed, f schedulng s performed at dscrete tme ntervals,, when C = and T = q, for all, such that 1 n and q s a postve nteger [16]. However, when jobs have dfferent request perods, DWCS may not generate a feasble schedule even f U mn < 1.0. Ths has motvated us to develop the VDS algorthm, to provde servce guarantees to jobs wth potentally dfferent request perods, whle maxmzng resource utlzaton. 3. Vrtual Deadlne Schedulng Vrtual deadlne schedulng (VDS) s desgned to provde servce guarantees accordng to the relaxed form of the wndow-constraned schedulng problem. However, t s able to outperform algorthms such as DWCS for the orgnal wndow-constraned problem, when jobs have dfferent request perods. VDS derves vrtual deadlnes for each job nstance from the correspondng wndow-constrant and request perod, and the job nstance wth the earlest such deadlne s scheduled frst. In effect, a vrtual deadlne s used to loosely enforce proportonal farness on the servce granted to a job n a specfc wndow of tme. Ths means the amount of servce currently granted to a job n a specfc wndow of real-tme should be proportonal to the mnmum fracton of servce requred n the entre wndow Vrtual Deadlnes Earlest-deadlne-frst (EDF) schedulng has the property that t can guarantee all deadlnes wll be met f resource usage does not exceed 100%. However, when the total utlzaton exceeds 100%, t s mpossble for any schedule to meet every deadlne. Wth wndow-constraned schedulng, strategc deadlnes may be mssed, so that a mnmum of m out of k deadlnes are met every non-overlappng wndow of k T real-tme. As a result, t may be possble for the total utlzaton for n jobs, n C =1 T, to exceed 1.0, whle the mnmum utlzaton to guarantee a feasble schedule, U mn = n m C =1 k T s less than or equal to 1.0. To produce such a feasble schedule, job nstances must be prortzed usng a combnaton of deadlnes (based on ther correspondng request perods) and wndow-constrants. As stated above, VDS gves precedence to the job nstance wth earlest vrtual deadlne. A job s vrtual deadlne wth respect to real-tme, t, s shown n Equaton 1. T s the remanng tme n the current request perod for J. (t+t T ) s the start tme of current request perod, whch we denote as t r (t) for brevty. Smlarly, (m,k ) represent the current wndow-constrant at tme, t. Ths mples that wndow-constrants change dynamcally, dependng on whether or not a job nstance s servced by ts deadlne. The exact rules that control the dynamc adjustment of wndow-constrants wll be descrbed later. At ths pont, t s worth outlnng the ntuton behnd a job s vrtual deadlne. If at tme t, J s current wndow-constrant s (m,k ), then m m out of k k job nstances have been servced. There are stll m job nstances that need to be servced n the remanng tme n the current wndow, whch s k T. If one nstance of J s servced every nterval k T m, then m job nstances wll be servced n the current remanng wndow-tme, k T. A sde-effect of ths s that J s assured proportonal farness guarantees wth respect to other wndow-constraned jobs. Addtonally, the delay bound s mnmzed, by preventng at least m nstances of J beng servced n a sngle burst at the end of a gven real-tme wndow. Vd (t) = k T m + t r (t) (1) t r (t) = (t + T T ) Fgure 3 gves an example of the vrtual deadlne calculaton. We can see that, f a job s current wndow-constrant does not change wthn a request perod, ts vrtual deadlne wll not change ether. Ths example corresponds to the relaxed wndow-constraned model, where more than one job nstance can be served n one request perod The VDS Algorthm Although VDS gves precedence to the job wth the earlest vrtual deadlne, t wll only do so f that job s elgble for servce. There are several cases that preclude a job from beng scheduled, even when t has the lowest vrtual deadlne:

5 C =1, T =4, m =2, k =3 C = servced f ((C == 0) (m 0)) job J s nelgble for servce ; T T k T k T..... t=0 Current tme, t=16, Vd(16) = 20 Vd(t=16) = (k *T /m ) +( t + T -T) =(2*4/2) + (16+4-4)= 20 Vd(t=17) = (2*4/2) + (17+3-4) = 20 Vd(t=18) = (2*4/1) + (18+2-4) = 24 Vd(t=19) = (2*4/1) + (19+1-4) = 24 Vrtual deadlne, Vd(t), remans at 24 untl start of next wndow, at t=24, because m =0 at t=20 Fgure 3. Example showng how to calculate vrtual deadlnes. 1) A job nstance cannot be servced before the start of ts request perod, even f t arrves early for servce. It follows that f all currently avalable nstances of a job have been servced, the job s nelgble untl a new arrval s ready. 2) If J has been servced at least m tmes n ts current wndow, t s gven lower prorty than a job yet to meet ts wndow-constrant. Only f all jobs have acheved ther mnmum level of servce can they agan be consdered n ther current wndows. When a job s servced ts current wndow-constrant s adjusted. Job J has an orgnal wndow-constrant of (m,k ) that s set to a current value of (m,k ),toreflect how many more nstances requre servce n the remander of the actve wndow. Fgure 4 shows how current wndowconstrants are updated. Here, the assumpton s that schedulng decsons and servce constrant adjustments are made once every tmeslot,. Unless stated otherwse, we assume throughout the rest of the paper that represents a unt tme-slot. Addtonally, we assume the servce tme, C, of each and every job, J s the same as. In Fgure 4, C and T represent the remanng servce tme, and tme remanng n the current request perod of J, respectvely. Every tme job J s servced, ts remanng servce tme, C, s decremented by. At the start of a new request perod J s remanng servce tme, C, s reset to ts orgnal value, C.IfC decreases to 0, J s nelgble for servce untl the start of the next request perod. At every schedulng pont, J s remanng tme n ts request perod, T, s decreased by. If T reaches 0, ts reset to the orgnal value, T. Snce the current tme, t, ncreases by every tme the scheduler s actvated, we need only update the value of t r n Equaton 1 once every request perod, to determne J s new vrtual deadlne, Vd. The last few lnes of the pseudo-code n Fgure 4 show how constrant adjustments dffer between the relaxed and orgnal models. In the relaxed model, f there are outstandng job nstances n the prevous request perod of the current wndow, C s reset. In the orgnal model, C s reset Serve elgble job J wth lowest vrtual deadlne & update m, C : C = C ; f ( C == 0) m = m 1 ; For every job J j, check volatons and update constrants: f ((Vd j <= +t) &&(j! =)) Tag J j wth a volaton; T j = T j ; f (T j == 0) k j = k j 1; C j = C j; T j = T j; f (k j == 0) { m j = m j; k j = k j; Dscard the remanng job nstances n the prevous wndow } f (m > 0) Update Vd j accordng to Equaton 1 // Only for the relaxed model f (((k j k j ) (m j m j ))&&(C j == 0)) C j = C j; Fgure 4. Updatng servce constrants usng VDS. only at the begnnng of each request perod. Ths makes the relaxed wndow-constraned model more flexble, ncreasng the potental for more job nstances to be servced over tme. As stated above, the current wndow-constrant (m,k ) s dynamcally adjusted accordng to the servce receved by J. When an nstance of J s servced, m s decreased by 1, because fewer nstances need to be servced n current wndow. If m reaches 0 n the current wndow, J has met ts wndow-constrant and becomes nelgble for servce untl the start of the next wndow, unless all other jobs have met ther current wndow-constrants. The value of k s decreased by 1 every request perod, T, untl t reaches 0, whch ndcates the end of the current wndow. At ths pont, J s current wndow-constrant (m,k ) s reset to ts orgnal value, (m,k ). A wndow-constrant volaton s observed f any job nstance msses ts vrtual deadlne VDS versus Other Algorthms The Earlest-deadlne-frst (EDF) algorthm produces a schedule that meets all deadlnes, f such a schedule s known to theoretcally exst. For the wndow-constraned schedulng problem, f each job J requres that m = k, then every real-tme deadlne must be met. In ths case, the vrtual deadlnes of job nstances servced by VDS are the same as ther correspondng real-tme deadlnes. In effect, VDS and EDF behave the same when m = k for each J.

6 Ths mples that VDS shares the same optmal characterstcs of EDF, when t s possble to meet all deadlnes. Now, when m =1for each and every J, vrtual deadlnes usng VDS are at the end of the current request wndow of sze k T. Here, VDS behaves the same as an EDF scheduler for jobs wth request perods of length k T. Furthermore, when k s a multple of m for each and every J, the correspondng wndow-constrant can be reduced to (1, k m ). Once agan, ths s equvalent to servcng jobs usng EDF wth deadlnes at the ends of perods of length kt m. DWCS was our frst algorthm desgned explctly to support jobs wth wndow-constrants. In orderng jobs for servce, DWCS compares deadlnes and wndowconstrants separately. In one verson of the algorthm [16, 14], DWCS frst compares the deadlnes of jobs, gvng precedence to the one wth the earlest such deadlne. If two or more jobs have the earlest deadlne, ther current wndow-constrants are then compared. In ths case, the job, J, wth the hghest rato, m k, s gven precedence. It can be shown that f all jobs have the same request perods, DWCS can generate a feasble wndow-constraned schedule, even when U mn =1.0. Ths mples that a feasble wndow-constraned schedule s possble even when all resources (e.g., CPU cycles) are utlzed. Comparng VDS to DWCS, f all request perods are equal, then each job s vrtual deadlne only depends on ts current wndow-constrant. Moreover, f all jobs have the same request perods then ther current nstances have the same real-tme deadlnes. In ths case, DWCS wll gve precedence to the job wth the hghest value of m k. Lkewse, VDS wll select the job wth hghest rato m k, snce (from Equaton 1) t has the earlest vrtual deadlne. Consequently, VDS s also able to produce a feasble wndowconstraned schedule that utlzes 100% of resources when all job request perods are equal. Now, when jobs have dfferent request perods and wndow-constrants, DWCS may fal to produce a vald schedule. As an example, consder the job set n Fgure 5, wth a mnmum requred utlzaton, U mn = n m C =1 k T = 8 9 < 1.0. The fgure shows a number of schedules durng the hyper perod (0, 9], for four dfferent algorthms. As can be seen, J 3 cannot be scheduled n the frst wndow usng ether EDF or DWCS, so t volates ts wndow-constrant. Observe that EDF and DWCS both choose J 1 frst, because t has the earlest deadlne, rather than J 2 or J 3 that have tghter wndow-constrants. In contrast, VDS produces a schedule that satsfes the servce constrants of all jobs. Ths s because VDS combnes deadlnes and wndow-constrants to derve a vrtual deadlne and, hence, prorty for orderng jobs. We stated earler that VDS s desgned to explctly servce jobs accordng to the relaxed form of wndowconstraned schedulng. That s, t guarantees each job, J, receves at least m C servce tme every wndow k T, rather than meetng at least m out of k deadlnes. However, Fgure 5 shows that t can outperform algorthms such as DWCS, accordng to the orgnal wndow-constraned schedulng problem, when job request perods dffer. Now, wth the relaxed model, jobs may be servced late as long as a mnmum number of nstances are scheduled n a gven wndow. Thus, we can consder every job nstance scheduled n the same wndow as havng a common deadlne at the end of that wndow. By settng deadlnes at the ends of wndows, an alternatve to VDS s to use a deadlne-drven scheduler that we call Elgblty-based Wndow-Deadlne- Frst (EWDF). It behaves smlar to EDF but gves precedence to the job wth the earlest wndow deadlne that s elgble for servce. Secton 3.2 descrbes the two condtons preventng a job from beng elgble for servce. Wth EWDF, k nstances of J all have deadlnes at the end of the current wndow of sze k T, rather than each nstance havng a separate deadlne at the end of ts request perod. As can be seen from Fgure 5, EWDF s able to servce all three jobs accordng to ther wndow-constrants. In general, EWDF s able to guarantee m C unts of servce every k T for each job J,fU mn 1.0. However, t may delay the servce of a job untl the end of a wndow, k T. In the worst case, all m nstances of J may be servced n a sngle burst durng the last m tme unts n the current wndow. Hence, the worst-case delay of a job nstance wth EWDF s (k T m C + T C ). Ths compares to the maxmum delay wth VDS of (k m + 1)T C, as shown n the next secton. Job J 1 J 2 (C,T,m,k) (1,1,2,9) (1,3,1,1) J 3 (1,3,1,1) U mn J 1 J 1 J 2 J 2 J 3 J 1 J 2 J 3 J 1 J 3 volates J 1 J 1 J 2 J 2 J 3 J 1 J 2 J 3 J 1 J 3 volates EDF J 2 J 3 J 1 J 2 J 3 J 1 DWCS J 2 J 3 J 1 J 2 J 3 J 1 VDS J 2 J 3 J 1 J 2 J 3 J EWDF tme Fgure 5. A comparson of schedulng algorthms. Fgure 6 shows an example of the dfferences n delays experenced by jobs usng the VDS and EWDF algorthms. Usng EWDF, all three job nstances for J 1 are servced n the last request perod of the current wndow. The maxmum servce delay s 24, and only the last job nstance meets ts request deadlne. However, usng VDS, the maxmum delay s 13, and all 3 job nstances are servced n ther own request perods. EWDF does not consder m,

7 Job (C,T,m,k) J 1 (1,7,3,4) J 2 (1,1,24,27) U mn delay = 13 delay = 24 J 2 J 2 J 2 J 2 J 2 J 2 J 2 J 2 J 2 J 2 J 2 J 2 J 2 J 2 J 2 J 2 J 2 J 2 J 2 J 2 J 2 J 2 J 2 J 2 J 1 J 1 J 1 J J 2 J 2 J 2 J 2 J 2 J 2 J 2 J 2 J 2 J 2 J 2 J 2 J 2 J 1 J 2 J 2 J 2 J 2 J 2 J 2 J 1 J 2 J 2 J 2 J 2 J 2 J 1 J 2 EWDF tme VDS Fgure 6. Example servce delays for VDS versus EWDF. but only wndow-sze, k T, to decde the schedulng prorty. Though EWDF s sutable for the relaxed wndowconstraned model, t s not sutable for the orgnal model VDS Propertes Ths secton descrbes some of the key propertes of VDS. These are summarzed as follows: If a feasble schedule exsts, such that at any tme no vrtual deadlnes are mssed, then VDS ensures that the maxmum delay of each job s bounded. If a feasble schedule exsts, t follows that VDS guarantees each job a mnmum share of servce n a fnte nterval. Ths s based on the assumpton that each job s servced at the granularty of a fxed-szed tme slot, (.e.,, C = ), and all request perods are multples of such a tme slot (.e.,, T = q q Z + ). If the mnmum requred utlzaton, U mn, s less than or equal to 1.0, and servce tmes are all constant, then a feasble schedule s guaranteed usng VDS. The algorthmc complexty of the VDS algorthm s a lnear functon of the number of jobs needng servce, n the worst case. Lemma 1. If a feasble VDS schedule exsts, the current wndow-constrant (m,k ) of job J always satsfes the condton that k m. Proof. The proof s by contradcton. We wll show that f there exsts a job J, whose current wndow-constrant s such that k <m, then there s a servce volaton n the VDS schedule. If at some tme there exsts the condton k = m 1, then n the prevous request perod, k = m, and J was not servced. If we let t be the tme at the begnnng of the last tme unts of the prevous request perod, then T = and J s vrtual deadlne s: Vd (t) = k m T +(t + T T ) = T + t + T T = t + T = t + ; We know that J was not servced n the nterval [t, t+ ],so there must be a volaton accordng to the VDS algorthm. Hence, by contradcton, f a feasble VDS schedule exsts, the current wndow-constrant (m,k ) of job J always satsfes the condton that k m. Delay Bound Theorem 1. If a feasble schedule exsts, the maxmum delay of servce to a job, J 1 n,s(k m +1)T C. Proof. From Lemma 1, we know that f a feasble VDS schedule exsts, the current wndow-constrant (m,k ) of job J at any tme satsfes the condton k m. Hence, f no nstance of J has been servced by the (k m +1)th perod of the current wndow, then k = m = m. An nstance must be served durng ths perod, otherwse k < m n next perod. Ths mples the worst case delay for J s (k m +1)T C n a feasble VDS schedule. Servce Share Theorem 2. If there s a feasble VDS schedule, every job has at least m nstances servced n each k T wndow of real-tme. Hence, the mnmum servce share of each job s m C k T n every request wndow. Proof. Agan from Lemma 1, we know that f a feasble VDS schedule exsts, the condton k m must hold. Now, n the last request perod of a gven wndow, k =1 and m 1 s true. If m =1=k, then an nstance of J must be servced n ths last perod of the wndow. If m 0 n the last perod of a gven wndow, then J has already been be served at least m out of k tmes before the wndow has ended. Hence, each job, J, receves at least servce n every request wndow. m C k T Feasblty Test Theorem 3. If U mn = n m C =1 k T 1.0, C = and T = q, q Z + then VDS guarantees a feasble schedule accordng to the relaxed wndow-constraned model. Proof. For brevty we do not provde a rgorous proof. However, t nvolves a reducton to an equvalent EDF schedulng problem. Note that EDF s optmal n the sense that f t s possble to produce a schedule n whch all deadlnes are met, such a schedule can be produced usng EDF.

8 In the equvalent EDF schedule, we must guarantee that n perodc jobs are each servced for C unts of tme, every perod kt m.now,f n C =1 (k T )/m 1.0 then EDF guarantees all jobs wll be servced for C tme unts every perod, k T m. Wth VDS, we requre a feasble schedule to have a mnmum utlzaton of mc k T. Ths s the same utlzaton as that n the equvalent EDF schedule. In meetng the utlzaton requrement, VDS must guarantee every servced nstance of J (of whch there must be at least m such nstances) meets ts vrtual deadlne wth respect to the current tme, t. Let us assume that t =0ntally. At the begnnng of the frst request wndow, J s vrtual deadlne s set to kt m. Ths s the same as the deadlne of the frst nstance of J n the equvalent EDF schedulng problem. Now, wth VDS, vrtual deadlnes ncrease over tme. Hence, f EDF can guarantee servce to the frst nstance of J by tme t = kt m then the frst nstance servced by VDS must have a vrtual deadlne greater than or equal to ths tme when t s actually servced. The worst-case vrtual deadlne of each servced job nstance wll not be earler than the equvalent deadlne n an EDF schedule. Wth the relaxed wndow-constraned schedulng model, job nstances are not dscarded after ther request perods, so we need only select a mnmum of m such nstances for each J by the correspondng vrtual deadlnes. That s, at least one nstance of J s servced n a request wndow by the vrtual deadlne wth respect to the current tme. The requrement that C = and T = q, q Z + s mposed because we assume VDS makes schedulng decsons at the granularty of -szed tme-slots. Ths allows VDS to emulate the preemptve nature of EDF. Algorthmc Complexty Theorem 4. The complexty of the VDS algorthm s O(n), where n s the number of jobs requrng servce. Proof. The VDS algorthm s based on vrtual deadlne orderng. The cost of orderng such deadlnes can be O(log(n)) f a heap structure s used. However, when VDS ether servces a job or swtches to a new request perod, t must update the correspondng vrtual deadlne. In the worst-case all n jobs requre ther vrtual deadlnes to be recalculated at the same tme. Ths s an O(1) operaton on a per-job bass, mplyng that the schedulng complexty s O(n) for n jobs. 4. Expermental Evaluaton 4.1. Smulatons Ths secton evaluates the performance of VDS, va a seres of smulatons comprsng a total of 1, 300, 000 randomly generated job sets. We assume that all jobs n each set are perodc wth unt processng tme, =1, although they may have dfferent request perods, q q 1. Each job J has a new nstance arrve every request perod, T, and a schedulng decson s made once every unt-length tme slot,. A range of mnmum utlzaton factors, U mn, up to 1.3 are derved by randomly choosng the number of jobs n a set, as well as values for job request perods and wndow-constrants. Utlzaton factors are ncremented n steps of 0.1, resultng n 13 such values wth 100, 000 jobs sets n each case. Schedulng s performed for each job set over ts hyper-perod, to capture all possble wndowconstrant volatons. In each test case, VDS s compared to several other algorthms, and volatons are determned for both the orgnal and relaxed wndow-constraned schedulng problems. Performance Metrcs: The followng metrcs are defned to measure the performance of each algorthm: Vtest s : Ths s the number of smulaton tests that volate the servce requrements of each job, accordng to the relaxed wndow-constraned schedulng problem. That s, f there s any job J that has less than m nstances servced n any wndow of k T real-tme, the correspondng test volates the servce requrements. It should be noted that one test conssts of a schedule for all jobs n a sngle set, servced over ther entre hyperperod. Vtest d : Ths s the number of smulaton tests that volate the servce requrements of each job, accordng to the orgnal wndow-constraned schedulng problem. That s, f there s any job J that has less than m job nstances meetng ther request deadlnes n any wndow of k T real-tme, the correspondng test volates the servce requrements. V s : Ths s the total volaton rate of all jobs, n all tests, that fal to be servced at least m tmes n any wndow of k T real-tme. V d : Ths s the total volaton rate of all jobs, n all tests, that fal to meet at least m deadlnes n any wndow of k T real-tme. The volaton rate of each job J s calculated as the rato of the number of wndows wth volatons n the hyperperod, to the number of wndows n the hyper-perod. For each J, the number of real-tme wndows n the hyperperod s lcm(k T, ) /k T. Orgnal Wndow-Constraned Schedulng Problem: In the orgnal wndow-constraned schedulng problem, each job nstance must be servced n ts current request perod, otherwse t wll be late. If we assume late job nstances are smply dscarded, the number of nstances that meet deadlnes must be the same as the number that are servced. In ths case, a wndow-based servce constrant s equvalent to a wndow-based deadlne constrant. Therefore, V d = V s and Vtest d = Vtest s.

9 Vtest d,vtest s V d, V s VDS EWDF VDS EWDF U mn DWCS EDF-Pfar VDS DWCS EDF-Pfar VDS U mn Vtest s Vtest d Vtest s Vtest d V s V d V s V d ( ] ( ] ( ] ( ] ( ] ( ] ( ] ( ] ( ] ( ] ( ] ( ] ( ] ( ] ( ] ( ] ( ] ( ] ( ] ( ] ( ] ( ] ( ] ( ] ( ] ( ] Fgure 7. Comparsons of servce volatons for (a) the orgnal, and (b) the relaxed wndowconstraned schedulng problem. Fgure 7(a) shows results for VDS versus DWCS and the EDF-Pfar algorthm, wth respect to the orgnal wndowconstraned schedulng problem. The latter EDF-Pfar algorthm s a form of EDF-based pfar schedulng, as descrbed by Mok and Wang [12]. It can be seen that, n underload cases, VDS results n fewer volatons than the other schedulng algorthms. Moreover, VDS only starts to show volatons when the utlzaton s up to 0.9, when there are only 14 out of 100, 000 tests whch fal. Smlarly, the volaton rate for VDS s very small. Although the EDF-Pfar algorthm performs well, t s not as good as VDS. However, DWCS results n volatons when the mnmum utlzaton factor s only 0.6. Lkewse, the number of volatng test cases, and the volaton rate are much larger wth DWCS than VDS. Relaxed Wndow-Constraned Schedulng Problem: For the relaxed wndow-constraned schedulng problem, each nstance of job J can legtmately be servced n the current wndow of sze k T, even f a correspondng request deadlne has passed. Ths means there can be less job nstances meetng deadlnes than are actually servced. Therefore, V s V d and Vtest s Vtest d. Fgure 7(b) shows results for VDS versus EWDF, wth respect to the relaxed wndow-constraned schedulng problem. In ths case, VDS and EWDF are able to guarantee no servce volatons up to 100% utlzaton. In the overload cases, VDS has more volatons than EWDF, because t tres to provde (proportonally) far servce to every job. That s, VDS attempts to provde each job wth at least m C unts of servce tme every k T, even though ths s not possble. However, compared to EWDF, VDS has (1) better delay propertes, as t attempts to servce job nstances earler, and (2) has fewer deadlne volatons CPU Schedulng Experments n Lnux We have mplemented VDS as part of a CPU scheduler n the Lnux kernel, to evaluate ts performance n a workng system. A Dell precson 330 workstaton, wth a sngle 1.4Ghz Pentum 4 processor, 256KB L2 cache and 512MB RDRAM s used to compare VDS and DWCS schedulers. The expermental setup s smlar to that n pror studes nvolvng DWCS n the Lnux kernel [13]. In the results that follow, we used the Pentum tmestamp counter to accurately measure elapsed clock cycles and, hence, schedulng performance. Average Volatons per Task VDS DWCS Utlzaton Fgure 8. Volatons usng VDS versus DWCS CPU schedulers n the Lnux kernel. Fgure 8 compares the performance of VDS and DWCS n a real system, n terms of average volatons per task 3. In these experments, a volaton occurs when fewer than m out of k consecutve deadlnes are met for perodc, preemptve CPU-bound tasks. Each task runs n an nfnte loop but can be preempted every clock tck, or jffy, to allow the scheduler to execute. In effect, one can thnk of a task as an nfnte sequence of sub-tasks, each requrng one jffy s 3 For schedulng purposes, Lnux treats both threads and processes as tasks.

10 worth (about 10mS on an Intel x86) of servce every request perod. It should be noted that the x-axs of Fgure 8 does not represent a lnear scale. Rather, each data pont represents the utlzaton, U mn = n m C =1 k T. These values are derved from a combnaton of up to n =8tasks, wth randomly generated schedulng parameters m, k and T for each task. Snce each task executes for one jffy between schedulng ponts (dscountng any system-processng overheads), we can assume that servce tmes are all untlength. As can be seen, when the utlzaton s less than 1.0, there are almost no wndow-constrant volatons usng VDS compared to DWCS. As expected, volatons occur for both algorthms when U mn exceeds Related Work Wndow-constraned schedulng s a form of weaklyhard servce [3, 4], that s smlar to the (m, k)-frm schedulng [6]. Hamdaou and Ramanathan [6] were the frst to ntroduce the noton of (m, k)-frm deadlnes, n whch statstcal servce guarantees are appled to jobs. Ther algorthm uses a dstance-based prorty scheme to ncrease the prorty of a job n danger of mssng more than m deadlnes over a wndow of k requests for servce. In contrast, VDS uses a vrtual deadlne scheme to derve a job s prorty and ensure determnstc servce guarantees. Smlar to (m, k)- frm schedulng s the work by Koren and Shasha on skp over schedulng [9]. Skp over schedulng allows certan job nstances to be skpped, but may unnecessarly mss servcng a job nstance when there are resources avalable. There are also examples of (m, k)-hard schedulers [2] but most such approaches requre off-lne feasblty tests, to ensure predctable servce. Addtonally, our VDS algorthm s targeted at a specfc wndow-constraned problem that requres explct servce of a mnmum number (m )of nstances of each job J n a wndow of k T tme unts, such that strong delay bounds are met. Other related research ncludes pnwheel schedulng [7, 5, 1] but all tme ntervals, and hence request perods, are of a fxed sze. In essence, the generalzed pnwheel schedulng problem s equvalent to determnng a schedule for a set of n jobs {J 1 n}, each requrng at least m deadlnes are met n any wndow of k deadlnes, gven that the tme between consecutve deadlnes s a multple of some fxed-sze tme slot, and resources are allocated at the granularty of one tme slot. Both our prevous DWCS algorthm, and VDS, can be thought of as specal cases of pnwheel schedulng. Wth VDS, servce guarantees are provded over non-overlappng wndows of k deadlnes spaced apart by T tme unts. However, VDS guarantees feasblty when resources are 100% utlzed, even when k s fnte and dfferent jobs have arbtrary request perods. Fnally, Jeffay and Goddard s Rate-Based Executon (RBE) model [8] attempts to servce jobs at an average rate of x tmes every y tme unts. In essence, ths s smlar to the wndow-constraned guarantee usng VDS, that ensures a mnmum servce tme of m C every wndow of k T unts of real-tme (gven that C s a constant for each and every job, J ). However, RBE does not consder that a certan fracton of job nstances can be late or dscarded. 6. Conclusons and Future Work We orgnally addressed the wndow-constraned schedulng problem n our prevous work on the DWCS algorthm. DWCS attempts to guarantee that no more than x out of y deadlnes are mssed for real-tme perodc CPU tasks, or consecutve packets n delay-constraned streams, when all request perods are dentcal and resources are 100% utlzed. Ths paper descrbes an extenson to our prevous work on wndow-constraned schedulng. We propose a relaxed verson of the wndow-constraned problem, n whch m out of k job nstances must be servced wthn a fnte wndow of tme, possbly after ther real-tme deadlnes. Wthout loss of generalty and applcablty, feasble schedules are possble for a wder range of jobs under the relaxed wndow-constraned schedulng problem. Ths s because m out of k jobs may be servced late, as long as they are servced n a bounded, and specfc, wndow of tme. The Vrtual Deadlne Schedulng (VDS) algorthm descrbed n ths paper s specfcally amed at provdng servce to real-tme jobs, based on the relaxed wndowconstraned schedulng problem. That s, rather than guaranteeng at least m out of k real-tme deadlnes are met for consecutve nstances of each job, t s only necessary to provde servce to m nstances every wndow of k request perods. Wth VDS, at least m out of k job nstances are servced by ther vrtual deadlnes, even when resources are 100% utlzed. Although such deadlnes may be after ther correspondng real-tme deadlnes, VDS s able to lmt the extent to whch job nstances are servced late. As a result, VDS lmts the delay mposed on each nstance of a job. Addtonally, VDS s capable of outperformng DWCS and other smlar algorthms for the orgnal wndow-constraned schedulng problem when jobs have arbtrary request perods. In many cases, ths makes VDS a more flexble algorthm, snce t places fewer restrctons on the servce specfcatons of wndow-constraned jobs. Future work nvolves the development and analyss of algorthms such as VDS and DWCS, to provde end-to-end servce guarantees across mult-hop networks. Such algorthms wll feature as part of our work on the constructon of a scalable dstrbuted system of end-hosts for processng and delverng lve data streams n real-tme.

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