The Schedulability Region of Two-Level Mixed-Criticality Systems based on EDF-VD

Transcription

1 The Schedulablty Regon of Two-Level Mxed-Crtcalty Systems based on EDF-VD Drk Müller and Alejandro Masrur Department of Computer Scence TU Chemntz, Germany Abstract The algorthm Earlest Deadlne Frst wth Vrtual Deadlnes (EDF-VD was recently proposed to schedule mxedcrtcalty task sets consstng of hgh-crtcalty ( and lowcrtcalty ( tasks. EDF-VD dstngushes between and mode. In mode, the tasks may requre executng for longer than n mode. As a result, n mode, EDF- VD assgns vrtual deadlnes to tasks (.e., t unformly downscales deadlnes of tasks to account for an ncrease of workload n mode. Dfferent schedulablty condtons have been proposed n the lterature; however, the schedulablty regon to fully characterze EDF-VD has not been nvestgated so far. In ths paper, we revew EDF-VD s schedulablty crtera and determne ts schedulablty regon to better understand and desgn mxed-crtcalty systems. Based on ths result, we show that EDF-VD has a schedulablty regon beng around 85% larger than that of the Worst-Case Reservatons (WCR approach. Keywords-real-tme schedulng; mxed crtcalty; EDF-VD; resource effcency I. INTRODUCTION In real-tme systems, deadlnes need to be guaranteed. To ths end, a task model s used based on whch a schedulablty analyss s performed. Usually, ths task model conssts of a mnmum nter-arrval tme and a worst-case executon tme (WCET []. However, t s a challenge to determne the WCET of a task [2]. For the known two basc approaches, there s a trade-off between safety and resource utlzaton. Statc methods yeld safe results [2]. However, they are often pessmstc n terms of resource utlzaton. On the other hand, measurement-based methods allow for a better resource utlzaton, but do not provde any guarantees on the safety of results snce t s nearly mpossble to cover all executon paths by measurement. Recently, technques have been proposed combnng statc and measurement-based approaches [3]. Nevertheless, the problem of fndng a proper trade-off between safety and resource utlzaton remans an open challenge. Mxed-crtcalty (MC systems are characterzed by schedulng tasks of dfferent mportance (.e., dfferent crtcalty. In ths paper, we consder two such levels: lowcrtcalty ( level, and hgh-crtcalty ( level. All jobs of tasks must be certfed to meet ther deadlnes under all possble crcumstances, whereas those of tasks are not subject to such a certfcaton [4]. Clearly, to certfy a task, statc methods should be used to determne the safe WCETs of tasks. However, as mentoned before, ths leads to a conservatve estmaton and tends to overdesgn. To allow for an ncreased resource effcency, optmstc (measurement-based WCET estmatons can be used. In the latter case, addtonal provsons have to be made n order to take potental WCET overruns of jobs of tasks nto account. Towards ths, the semnal algorthm Earlest Deadlne Frst wth Vrtual Deadlnes (EDF-VD was recently proposed [5]. EDF-VD dstngushes between two modes: and mode. In mode, tasks are scheduled together wth tasks, where optmstc WCETs are used for the tasks. In mode, only the tasks are allowed to run usng ther conservatve WCETs. Here, the algorthm EDF-VD dscards all tasks n order to accommodate an ncrease n the executon demands of tasks. Dfferent schedulablty condtons have been publshed for EDF-VD [5] [6]. However, a full characterzaton of ts schedulablty regon has not been proposed so far. In ths paper, we revew EDF-VD s schedulablty crtera and determne ts schedulablty regon to fully characterze ts behavor. Schedulablty regons have been already used n the lterature n order to better understand and compare schedulablty crtera [7] [8]. Smlarly, n ths work, we compare EDF-VD wth the well-known Worst-Case Reservatons (WCR approach both graphcally and analytcally based on schedulablty regons. A. Contrbutons In ths secton, we state the contrbutons of ths paper. These can be summarzed n the followng manner: We revew the schedulablty crtera of EDF-VD from the lterature and derve a new, more ntutve, schedulablty condton. We determne EDF-VD s schedulablty regon whch allows us to better understand MC systems. We compare schedulablty regons of EDF-VD and WCR both n an analytcal and graphcal manner. We show that EDF-VD s schedulablty regon s around 85% larger than that of WCR /DATE4/ c 204 EDAA

2 B. Structure of the Paper The rest of ths paper s structured as follows. Related work s dscussed n Secton II. Next, Secton III explans the task and system model. We shortly revst WCR n Secton IV. Secton V ntroduces EDF-VD and proposes a new upper bound on the desgn parameter deadlne scalng factor. Next, EDF-VD s necessary and suffcent schedulablty condtons are dscussed n Sectons VI and VII. Based on ths, we study the EDF-VD schedulablty regon n Secton VIII and compare t to the one of WCR. Fnally, some examples and concludng remarks wll be gven n Sectons IX and X. II. RELATED WORK Mxed-crtcalty schedulng was frst proposed under ths name by Vestal [4]. Baruah et al. later analyzed per-task prorty assgnments and the resultng response tmes for sporadc task sets [9]. In [5], Baruah et al. proposed the EDF-VD algorthm to schedule MC sporadc task sets wth prorty promoton by scalng deadlnes of tasks. The speed-up factor of.69 (by roundng up the golden rato 5+ 2 proposed n [5] was later mproved to 4/3 n [6]. For multprocessor MC schedulng, a parttoned and a global schedulng approach both based on EDF-VD are proposed n [0]. Accordng to ths work, parttoned schedulng behaves better than the global approach for MC task sets. Further, n [], Pathan studes global MC schedulng wth task-level fxed prortes and gves a schedulablty test based on response tme analyss for > 2 crtcalty levels. A more flexble approach wth per-task deadlne scalng factors was presented by Ekberg and Y [2]. However, ths approach makes t hard to derve utlzaton and speed-up bounds due to the bg number of scalng factors,.e., tunng parameters, nvolved. Recently, other mprovements compared to EDF-VD were proposed. In [3], Su and Zhu used an elastc task model [4] [5] to mprove resource utlzaton n MC systems. Fnally, n [6], Zhao et al. appled preempton thresholds [7] n MC schedulng n order to better utlze the processor. III. TASK AND SYSTEM MODEL We consder n ndependent sporadc tasks τ wth ther nstances or jobs separated by a mnmum nter-arrval tme T under preemptve, mplct-deadlne (.e., : D T where D s a task s relatve deadlne, unprocessor schedulng. There s no self-suspenson, and context-swtch overheads are assumed to be zero. In ths paper, only dual-crtcalty systems wth two levels of crtcalty are examned. In ths MC settng, an addtonal task parameter s requred: crtcalty denoted by χ {, }. For tasks, there s one WCET C. Opposed to ths, tasks are provded wth an optmstc WCET C and a conservatve WCET C beng C C. As mentoned before, the system operates n two modes: and mode. Intally, we assume the mode m of the system to be m where all tasks are scheduled. As soon as a job of a task executes for longer than ts C, the system swtches to mode where only the subset of tasks are subject to ther deadlnes. All actve jobs of tasks are mmedately dscarded and no further such jobs wll be allowed to run. In a short notaton, a task s then specfed as (T, C whle a task s expressed as (T, (C, C. We defne the utlzaton parameters wth χ, m {, } where χ denotes task crtcalty and m mode. u m χ : χ χ C m T ( Note that among the four potental crtcalty-to-mode combnatons, only u, u and u are defned. u does not exst snce there s no C specfed for tasks. Snce C C holds for all for whch χ,.e., the optmstc WCET s always less than or equal to the conservatve WCET of all tasks, we obtan the followng constrant on utlzatons u and u. u u (2 IV. WORST-CASE RESERVATIONS In the smple strategy of WCR, the optmstc C of tasks n mode s replaced by the conservatve C. It then apples the well-known EDF schedulng strategy. Ths way, we obtan (3 as an exact schedulablty condton for the WCR approach usng []. u + u (3 The WCR approach assumes that tasks always requre ther maxmum WCETs. Ths s pessmstc (wth respect to EDF-VD snce t gnores the fact that, n mode, the optmstc WCETs could be used. V. THE EDF-VD ALGORITHM The algorthm EDF-VD [5] s an adapton of the classc EDF [] to MC schedulng. Its basc dea s to promote jobs n mode by shortenng ther relatve deadlnes n order to reserve processor tme for the mode. The swtch from to mode s trggered by a job exceedng ts C. The EDF-VD approach s to scale these relatve deadlnes homogeneously n mode. That s, D x T wth x [0, ] for all. D s referred to as vrtual deadlne and s used to schedule tasks n mode. Opposed to ths, there s no deadlne scalng for tasks, they keep ther orgnal deadlnes D. In mode, agan the orgnal deadlnes D are used to schedule tasks accordng to EDF. As mentoned above, tasks are dscarded n mode. Note that EDF reles on absolute deadlnes such that usng vrtual deadlnes n mode mght lead to nconsstences.

3 Based on these consderatons, Baruah et al. [6] found a utlzaton bound of 3/4, see (4 below. max ( u + u, u 3/ (4 A lower bound (5 of the scalng factor x can be used as a suffcent condton for meetng all deadlnes n mode. u u x (5 An upper bound (6 of x can be found by consderng the transton between and mode [6]. Ths serves as a suffcent schedulablty condton for the mode. x u u A. A Straghtforward Upper Bound on the Scalng Factor Let us assume that the system swtches from to mode at a tme t recall ths happens when one of the tasks executes for C tme wthout sgnalng end of executon. As a result, no tasks wll be executed after t. In addton, snce (5 holds, u + u x also holds and t can be guaranteed that all tasks execute for C tme wthn ther correspondng vrtual deadlnes (.e., D x T. Let us assume that only one task causes the change to mode,.e., t has executed for C at t wthout sgnalng ts end. Now, snce (5 holds guaranteeng the schedulablty of all tasks n mode, t has to be less than or equal to the vrtual deadlne of the job trggerng the mode swtch. Otherwse (5 would not hold. In worst case, t s exactly equal to the deadlne of that job. For ths job to fnsh before ts deadlne n mode, t has to execute C tme wthn T x T,.e., wthn an nterval equal to the dfference between ts orgnal and vrtual deadlne. As a result, the followng nequaltes must hold. C T xt C ( xt The worst-case stuaton happens when all tasks swtch to mode smultaneously 2 at tme t and hence the followng must hold for all of them to be schedulable, cf. [5]. u x (6 x u (7 On the other hand, a job has already executed for C t and, hence, t only has to execute for addtonal C tme wthn T x T (nstead of C at C as assumed n [5]. As a result, the followng must hold. 2 Even f only one task trggers the mode at a tme, other tasks mght also be arbtrarly close to trggerng t. C C C T xt C ( xt Agan, the worst-case stuaton happens when all tasks swtch to mode smultaneously at tme t. As a result, we obtan the followng expresson. u u x x ( u u VI. NECESSARY SCHEDULABLITY CRITERIA FOR EDF-VD Snce EDF-VD apples tradtonal EDF n both and mode, two necessary EDF-VD schedulablty condtons (9 and (0 can be drectly derved usng the exact EDF unprocessor utlzaton bound []. (8 u + u (9 u (0 VII. SUFFICIENT SCHEDULABLITY CRITERIA FOR EDF-VD Unfortunately, condtons (9 and (0 are not suffcent for EDF-VD to be schedulable snce these dsregard the schedulablty of mode transtons. The exstence of a lower and an upper bound on the scalng factor x allows us to derve a thrd condton to guarantee schedulablty of mode transtons. Note that ths s possble snce the upper bound on x see (8 s obtaned takng mode transtons nto account. If a vald x exsts, ( must hold from (5 and (8. u u u + u ( Condton ( can be easly shown to be equvalent (.e., f one of them holds, then all of them hold to (2 as gven n [6] and to (3 as proposed n [0] see Appendx A. u u u u (2 u u (u u (3 Note that (2 s based on (5 and (6, whle (3 results from solvng (2 for u. Another suffcent schedulablty condton (4 results from combnng the more conservatve upper bound of (7 wth (5 as proposed n [5]. u u u (4

4 A. EDF-VD VIII. THE SCHEDULABILITY REGION Fg. llustrates EDF-VD s schedulablty regon. The abscssas represent values of u whereas the ordnates represent values of u. Clearly, snce (9 must hold for EDF- VD to be schedulable, the schedulablty regon of EDF-VD must be below the lne crossng by (u, u 0 and (u 0, u. As dscussed prevously, the utlzaton bound of EDF-VD s 3/4 as shown n (4. Ths s represented by the lower lne crossng by (u 3/4, u 0 and (u 0, u 3/4 n Fg.. That s, utlzaton values that are below ths lower lne are schedulable wthout needng further tests. Clearly, from (4, nothng can be concluded f the utlzaton s between 3/4 and,.e., for the secton between the lower and the upper lne n Fg.. To characterze the schedulablty regon between these 3/4. Agan, ths s the maxmum value of u by whch schedulablty s guaranteed by (4. Now, replacng ths value nto ( and solvng for u, we obtan (5 note that ths can also be obtaned by replacng 3/4 and solvng for u n (2 or (3. two lnes, let us set u u u u 4u (5 Inequalty (5 expresses how u vares wth respect to u under a fxed u 3/4. Ths s a hyperbolc functon of u as shown n Fg.. All pars of u and u below ths curve are schedulable under EDF-VD, whereas values of u and u above t cannot be guaranteed to be schedulable. Snce condton (2 must hold, the hyperbolc curve s truncated at u u 3/4. EDF-VD s schedulablty regon s gven by the area below the truncated hyperbolc curve. It should be notced that the area below the lne gven by (4 s fully contaned below the truncated hyperbolc curve. Now, consderng u 3/4 and solvng for u n (4 we obtan (6 llustrated n Fg.. B. WCR /4 u 4 u (6 Let us now draw the schedulablty regon for WCR. To ths end, we set u 3/4 and solvng for u n (3 we obtan (7, whch s a constant as llustrated n Fg.. u 3/4 (7 Ths means that all pars (u, u are schedulable under WCR f u s not greater than /4. Ths results n the area to the left of u /4 as llustrated n Fg.. Agan, snce condton (2 must hold, WCR s schedulablty regon s lmted from above by u u 3/4. u u Fgure. u WCR's schedulablty regon Border of EDF-VD's schedulablty regon Border of conservatve EDF-VD schedulablty regon Model constrant (2 +56% Suffcent condton (3 Suffcent cond. (4, Utlzaton bound Necessary condton (9 u u lne Suffcent condton (20, especally (5 Suffcent condton (6 0.5 u u Schedulablty Regons of EDF-VD & WCR for u 3/4 C. Comparng EDF-VD and WCR Clearly, EDF-VD outperforms WCR. That s, f a set of MC tasks s schedulable under WCR, t wll also be schedulable under EDF-VD; cf. Fg., where WCR s schedulablty regon s fully contaned by EDF-VD s one. In ths secton, we further make a quanttatve comparson between these two algorthms. To ths end, let us assume that u u holds. Replacng ths n (5 and reshapng we obtan a quadratc expresson n u. 4(u 2 + u 0 (8 Equalzng (8 to zero, we can compute ts roots and take the postve one as gven by (9. u (9 8 Hence, for u u and u 3/4, a set of MC tasks s schedulable under EDF-VD f u s at most On the other hand, for WCR, we obtan from (3 that u Note that ths s more than 56% more allowable utlzaton for EDF-VD compared to WCR. The above comparson s vald for the partcular case where u 3/4. To obtan a more general comparson of EDF-VD wth WCR, let us reshape (2 as follows note that ( or (3 can also be reshaped to ths form. u ( u ( u u (20 Inequalty (20 descrbes the general case for u. There, we have a scalng by u nstead of /4 as n (5. By decreasng u from 3/4 towards 0 note that must stll hold, the boundary of EDF-VD s u u

5 Eff WCR to EDF VD Fgure u Effcency of WCR vs. EDF-VD dependng upon u schedulablty regon (20 approaches the necessary condton ( (9. The corner cases at (u 0, u and u u, u u reman. The advantage of EDF-VD compared to WCR worsens wth a decreasng u snce, n ths case, u approaches u and, hence, makes reservatons less wasteful. Clearly, ncreasng u from 3/4 towards mproves the advantage of EDF-VD over WCR 3. In general, the area of a schedulablty regon s a measure of performance for a gven schedulng algorthm 4. Thus, the area rato of two schedulablty regons allows quantfyng the advantage or effcency of one algorthm wth respect to another, see also [8] and [8]. Here, we determne the area rato of WCR over EDF-VD. From Fg., WCR s regon (a rectangle has an area of (2. u ( u (2 The EDF-VD schedulablty regon comprses the one of WCR plus the area under a hyperbolc curve gven by (20. Accordng to Fg. and (20, the dfference n the area can be calculated as the defnte ntegral (22. A EDF-VD u ( ( u u u du ( u ( ln( u + u (22 Fnally, (2 and (22 gve the area rato of WCR over EDF- VD as a functon of u (23, cf. Fg. 2, see Appendx B. Eff WCR-to-EDF-VD For u A EDF-VD u ln( u (23 0, WCR s as effcent as EDF-VD. Wth 3 Increasng u beyond 3/4 makes the hyperbolc curve (20 cross the utlzaton bound lne gven by (4. Hence, condtons (2, ( and (3 outperform (4. That s, a u greater than 3/4 s rendered unfeasble by (4; however, ths mght stll be feasble as per (, (3 or (4. 4 Assocated wth a schedulablty condton. ncreasng u, WCR s effcency decreases. Up to u 0.9, ths decrease s almost lnear wth a slope of 2/3. From 0.9 onwards, WCR s relatve performance rapdly falls to zero at u. Whle WCR mght be acceptable for small u values, hgh u values call for EDF-VD. For u 3/4, EDF-VD s advantage to WCR s obtaned from (23 by takng the nverse, gvng ca..85. EDF-VD has a schedulablty area ca. 85% greater than WCR s. IX. EXAMPLES Frst, let us consder, from [3], the task set {(8, 2, (30, 3, (0, (2, 4, (25, (4, 0}. Usng (, we obtan u 0.35, u 0.36 and u 0.8. The necessary condtons (9 and (0 are met wth 0.7 and 0.8. In ths example, u + u.5 > holds whch calls for EDF-VD. (5 and (6 gve an nterval of [36/65, 4/7] or ca. [0.5538, 0.574] for EDF-VD s scalng factor x. Our approach (8 easly gves the sound result of 0.56 whch s n the mddle of the nterval and, thus, ncreases tolerance to arrval tme jtter. We take {(6, 2, (0, (, 2, (20, (2, 0} as a 2nd example. Based on (, u /3, u 0.2 and u 0.7 are obtaned. The necessary condtons (9 and (0 are met wth 8/ and 0.7. But, u + u 3/ > holds, agan callng for EDF-VD. We obtan a mn. scalng factor (5 of x 0.3. The max. scalng factor (6 s x 0.9. Our new approach (8 returns 0.5 for x (n the mddle of the nterval, mprovng jtter tolerance. X. CONCLUDING REMARKS In ths paper, we obtaned the schedulablty regon of EDF-VD. Based on ths, we showed how dfferent schedulablty condtons for EDF-VD relate to each other and whch among them are the most accurate. In addton, we compared the schedulablty regons of EDF-VD and WCR n a graphcal and analytcal manner. We have shown analytcally that the area of the schedulablty regon of EDF- VD s ca. 85% larger than that of WCR. Our technque to compare schedulablty regons s general enough and can be used to analyze the performance of schedulng algorthms and schedulablty crtera n other domans. REFERENCES [] C. L. Lu and J. W. Layland, Schedulng algorthms for multprogrammng n a hard-real-tme envronment, J. ACM, vol. 20, no., pp. 46 6, Jan [2] R. Wlhelm, J. Engblom, A. Ermedahl, N. Holst, S. Thesng, D. Whalley, G. Bernat, C. Ferdnand, R. Heckmann, T. Mtra, F. Müller, I. Puaut, P. Puschner, J. Staschulat, and P. Stenström, The worst-case executon-tme problem - overvew of methods and survey of tools, ACM Trans. Emb. Comp. Sys., vol. 7, pp. 36: 36:53, May [3] M. Lv, W. Y, N. Guan, and G. Yu, Combnng abstract nterpretaton wth model checkng for tmng analyss of multcore software, n RTSS, IEEE 3st, 200, pp

6 [4] S. Vestal, Preemptve schedulng of mult-crtcalty systems wth varyng degrees of executon tme assurance, n RTSS, 28th IEEE, 2007, pp [5] S. K. Baruah, V. Bonfac, G. D Angelo, A. Marchett- Spaccamela, S. Van Der Ster, and L. Stouge, Mxedcrtcalty schedulng of sporadc task systems, n 9th Eur. conf. on Alg., ser. ESA. Sprnger, 20, pp [6] S. Baruah, V. Bonfac, G. D Angelo, H. L, A. Marchett- Spaccamela, S. van der Ster, and L. Stouge, The preemptve unprocessor schedulng of mxed-crtcalty mplct-deadlne sporadc task systems, n ECRTS, 24th, 202, pp [7] E. Bn and G. C. Buttazzo, Schedulablty analyss of perodc fxed prorty systems, IEEE Trans. Comput., vol. 53, no., pp , Nov [8] D. Müller and M. Werner, Quantfyng the Advantage of EDF vs. RMS Schedulablty on a Unprocessor Usng a Dfferental Analyss and a Power-law Total Utlzaton Dstrbuton, n ISORC, 6th, Jun [9] S. Baruah, A. Burns, and R. Davs, Response-tme analyss for mxed crtcalty systems, n RTSS, 20, pp [0] S. Baruah, B. Chattopadhyay, H. L, and I. Shn, Mxedcrtcalty schedulng on multprocessors, Real-Tme Systems, pp. 36, 203. [] R. Pathan, Schedulablty analyss of mxed-crtcalty systems on multprocessors, n ECRTS, 202, pp [2] P. Ekberg and W. Y, Boundng and shapng the demand of mxed-crtcalty sporadc tasks, n ECRTS, 202, pp [3] H. Su and D. Zhu, An elastc mxed-crtcalty task model and ts schedulng algorthm, n DATE. EDA Consortum, 203, pp [4] T.-W. Kuo and A. K. Mok, Load adjustment n adaptve real-tme systems, n RTSS, Dec.. 99, pp [5] G. Buttazzo, G. Lpar, and L. Aben, Elastc task model for adaptve rate control, n RTSS, 9th, 998, pp [6] Q. Zhao, Z. Gu, and H. Zeng, PT-AMC: Integratng Preempton Thresholds nto Mxed-Crtcalty Schedulng, n DATE, 203, pp [7] Y. Wang and M. Saksena, Schedulng fxed-prorty tasks wth preempton threshold, n RTCSA, 999, pp [8] A. Baston, B. B. Brandenburg, and J. H. Anderson, Cacherelated preempton and mgraton delays: Emprcal approxmaton and mpact on schedulablty, n 6th Int l Workshop on OS Platforms for Emb. RT Appl., Jul. 200, pp APPENDIX A. Proof of Equvalence of Inequaltes (, (2 and (3 Frst we show by equvalent transformatons that ( mples (2 and vce versa. u u u u u u + u ( u u u u u u u : u u u u u Next, t s shown that ( and (3 are equvalent. u u u u u + u u u u u u + u u u + u ( u + u u + u + u u + u ( u ( u u u + u Fnally, by transtvty also (2 and (3 are equvalent. B. Dervaton of the WCR-to-EDF-VD-Effcency Formula Frst, the defnte ntegral of (22 s calculated usng the antdervatve. ( ( u u A EDF-VD u u ( u u u + u du du ln u + ( u u ] u [( u ( u (( u ( ln u + ( u ( u ( u ( ln( u + u Fnally, ths result and (2 lead to (23. Eff A EDF-VD + A ( EDF-VD u u ( ( u u + u ( ln( u + u ( u u ( + + ln( u ( u ln( u +u u ( u u u u ln( u