AS real-time embedded systems become more complex,

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1 1766 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 29, NO. 8, AUGUST 2018 Non-Preeptve Schedulng for Mxed-Crtcalty Real-Te Multprocessor Systes Hyeongboo Baek, Nayong Jung, Hoon Sung Chwa, Meber, IEEE, Insk Shn, Meber, IEEE, and Jnkyu Lee, Meber, IEEE Abstract Real-te schedulng for Mxed-Crtcalty (MC) systes has receved a growng attenton as real-te ebedded systes accoodate varous tasks wth dfferent levels of crtcalty. Whle any studes have addressed how to guarantee tng requreents for MC systes wth unprocessor and ultprocessors, ost of the have focused on supportng preeptve tasks. On the other hand, there have been few studes to address non-preeptve schedulng especally for MC ultprocessor platfors, n whch the obs under executon cannot be preepted by other obs. In ths paper, we develop schedulablty tests for non-preeptve schedulng, whch s the frst attept for MC ultprocessor systes. To ths end, we frst generalze an exstng NP-EDF (Non-Preeptve Earlest Deadlne Frst) schedulablty test developed for sngle-crtcalty ultprocessor systes, towards for MC ultprocessor systes. For the generalzaton, we ntroduce new tng guarantee technques for the syste transton between two dfferent crtcaltes, whch s one of the key features n MC systes. We next extend the proposed NP-EDF schedulablty test towards NP-EDFVD (NP-EDF wth Vrtual Deadlnes) that s specalzed for MC systes, and pose a vrtual deadlne assgnent proble. We develop an optal vrtual deadlne assgnent polcy usng a control knob of the syste-level deadlne-reducton paraeter and then a suboptal one for the task-level paraeter. Our sulaton results deonstrate that the NP-EDFVD schedulablty test wth the proposed vrtual deadlne assgnent polces fnds a nuber of addtonal schedulable task sets, whch are not schedulable by the NP-EDF schedulablty test. Index Ters Real-te schedulng, schedulablty analyss, xed-crtcalty, non-preeptve tasks, real-te ultprocessor systes Ç 1 INTRODUCTION AS real-te ebedded systes becoe ore coplex, they need varous functonaltes often wth dfferent levels of crtcalty. Ths necesstates tng guarantees of xed-crtcalty (MC) systes at dfferent levels of assurance. Startng fro Vestal [1], any studes have ntroduced schedulng algorths and analyss talored to MC systes, developng technques that gve tng guarantees n the presence of the syste transton between dfferent crtcaltes, wth a focus on unprocessor platfors [2], [3], [4], [5], [6], [7], [8], [9], [10]. To follow the popularty of the ult-core archtecture, soe of those studes have been extended to a ultprocessor platfor [11], [12], [13], [14], [15], [16], [17]. However, such progress has been based to preeptve schedulng, n whch a hgher-prorty ob can preept a lower-prorty ob at any te. H. Baek, N. Jung, and J. Lee s wth the Departent of Coputer Scence and Engneerng, Sungkyunkwan Unversty (SKKU), Suwon-s, Gyeongg-do 16419, Republc of Korea. E-al {hbbaek359, goswlsqkqh1}@gal.co, nkyu.lee@skku.edu. H. S. Chwa s wth the Departent of Electrcal Engneerng and Coputer Scence, Unversty of Mchgan, Ann Arbor, MI E-al hchwa@uch.edu. I. Shn s wth the School of Coputng, KAIST, Daeeon 34051, Republc of Korea. E-al nsk.shn@cs.kast.ac.kr. Manuscrpt receved 10 July 2017; revsed 4 Jan. 2018; accepted 8 Feb Date of publcaton 15 Feb. 2018; date of current verson 13 July (Correspondng author Jnkyu Lee.) Recoended for acceptance by B. Ravndran. For nforaton on obtanng reprnts of ths artcle, please send e-al to reprnts@eee.org, and reference the Dgtal Obect Identfer below. Dgtal Obect Identfer no /TPDS On the other hand, n non-preeptve schedulng, a runnng ob s executed tll copleton wthout beng preepted. It ust be used when the ob s of nherently non-preeptve nature and/or t s subect to extreely large or unpredctable preepton/graton overhead (e.g., nterrupts and transactonal operatons) [18]. Despte ts necessty, there exst only a few studes that have addressed non-preeptve schedulng on MC systes [19], [20], [21], [22], [23], whch target unprocessor or dstrbuted platfors. In partcular, few studes have addressed tng guarantees for non-preeptve schedulng on MC ultprocessor systes. In ths paper, we a at developng the frst schedulablty test for non-preeptve schedulng on MC ultprocessor systes. We target NP-EDF (Non-Preeptve Earlest Deadlne Frst), one of the ost fundaental, popular nonpreeptve schedulng algorths, and then extend the results to NP-EDFVD (NP-EDF wth Vrtual Deadlnes) that s desgned for MC systes. To ths end, we nvestgate an exstng schedulablty test of NP-EDF for Sngle-Crtcalty (SC) systes (denoted by Bar [24]), addressng the followng ssues. Q1. How does Bar gve tng guarantees under NP- EDF for SC systes? Q2. How can we generalze the tng guarantee technques towards MC systes, dealng wth the ost portant feature of MC systes, whch s the syste transton fro the low to hgh crtcalty? To address Q1 and Q2, we frst carefully nvestgate how Bar operates. Bar consders two executon envronents 1) ß 2018 IEEE. Personal use s pertted, but republcaton/redstrbuton requres IEEE persson. See ht_tp// for ore nforaton.

2 BAEK ET AL. NON-PREEMPTIVE SCHEDULING FOR MIED-CRITICALITY REAL-TIME MULTIPROCESSOR SYSTEMS 1767 n tasks executng under NP-EDF schedulng on processors and 2) n tasks executng on n agnary fractonal processors ndvdually such that each task t runs on a fractonal processor n wth a gven constant rate V and fnshes ts executon no later than Y aount of te ahead of ts deadlne. Then, t calculates the total aount of executon of an entre task set untl t for the forer and untl t for the latter, respectvely. Once we fnd a condton where the forer has no saller aount than the latter does, we can use the condton to establsh tng guarantees. Buldng upon the nvestgaton on Bar, we propose a ethod to deterne the executon rate of V and two shftng values and Y so as to accoodate the syste transton between two dfferent crtcaltes, yeldng an NP-EDF schedulablty test for MC ultprocessor systes. Then, the next step s to extend the above results to an MCaware schedulng algorth, NP-EDFVD. EDFVD [4], [5] (orgnally developed for preeptve schedulng) s ntroduced as an extenson of EDF that proves schedulablty sgnfcantly by the use of dfferent deadlnes (.e., vrtual deadlnes) n dfferent crtcaltes. Hence, we seek to develop technques for NP-EDFVD by answerng the followng ssues. Q3. How can we adapt the proposed NP-EDF schedulablty test to NP-EDFVD? Q4. Gven that deternng rght vrtual deadlnes s crtcal to provng the schedulablty, how can we assgn the vrtual deadlne of every task so as to gve tng guarantees? As to Q3, we observe that the use of vrtual deadlnes ntroduces new propertes. For exaple, under NP-EDF schedulng even for MC systes, the executon rate assgned to a task n the hgh-crtcalty ode s always no saller than that to the sae task n the low-crtcalty ode; ths s because they share the sae relatve deadlne but the worstcase executon te of a task n the hgh-crtcalty ode s no saller than that n the low-crtcalty ode. However, the sae cannot hold f we assgn vrtual deadlnes to tasks n the low crtcalty ode. Therefore, we ncorporate ths property to the schedulablty of NP-EDFVD when we deterne the executon rate. When t coes to Q4, t s crtcal to schedulablty to deterne rght vrtual deadlnes. We frst consder a vrtual deadlne assgnent proble usng a control knob of the syste-level deadlne-reducton paraeter, and develop an optal assgnent polcy. Then, we pose a ore general proble of assgnng vrtual deadlnes usng a task-level deadlne-reducton paraeter, and propose a suboptal assgnent polcy. We evaluate the schedulablty perforance of the proposed schedulablty tests wth/wthout our vrtual deadlne assgnent polces. Our sulaton results deonstrate that our schedulablty test for NP-EDFVD wth an optal vrtual deadlne assgnent polcy usng the syste-level paraeter sgnfcantly proves the schedulablty over that for NP-EDF. Also, f we addtonally apply our suboptal vrtual deadlne assgnent polcy usng the task-level paraeter, we can fnd a nuber of addtonal schedulable task sets. In suary, ths paper akes the followng contrbutons. We generalze an exstng NP-EDF schedulablty test orgnally developed for SC systes towards MC systes, whch yelds the frst schedulablty test for non-preeptve schedulng on MC ultprocessor systes; We extend the proposed NP-EDF schedulablty test towards NP-EDFVD, and pose a vrtual deadlne assgnent proble; We develop an optal vrtual deadlne assgnent polcy usng the syste-level deadlne-reducton paraeter, and a suboptal one usng the task-level paraeter; and We deonstrate the sgnfcant schedulablty proveent of NP-EDFVD wth the proposed vrtual deadlne assgnent, over NP-EDF. The rest of ths paper s structured as follows. Secton 2 explans our syste odel. Secton 3 develops a new schedulablty test of NP-EDF for MC ultprocessor systes. Secton 4 extends the proposed schedulablty test to NP- EDFVD, and develops (sub)optal vrtual deadlne assgnent polces. Secton 5 evaluates the schedulablty perforance of the proposed schedulablty tests wth our vrtual deadlne assgnent polces. Secton 6 dscusses related work, and Secton 7 fnally concludes ths paper. 2 SYSTEM MODEL We consder an MC ultprocessor syste wth dentcal processors. We consder a sporadc task odel, and specfy a task t n a task set t by 5-tuple, t ¼ðL ;T ;C ;C HI ;D Þ, each of whch s descrbed as follows. L 2fHI; g denotes the crtcalty level of a task t ; HI and present hgh- and low-crtcalty levels, respectvely. A task t wth L ¼ HI, referred to as a HI-crtcalty task, should be certfed by the certfcaton authortes (CAs) whle a task t wth L ¼, referred to as a -crtcalty task, s not assued to be certfed by CAs. T s the nu separaton between release tes of two consecutve obs nvoked by t. C and C HI denote the worst-case executon tes (WCETs) of t requred for the hgh- and low-crtcalty levels, respectvely. D denotes the relatve deadlne of t. We assue that C C HI D T holds for every t 2 t. Let n denote the nuber of tasks n t. Usng the above task paraeters, each task t nvokes a seres of obs. Schedulng algorth. When t coes to schedulng algorths, we consder global non-preeptve and workconservng schedulng, n whch a ob of a task can be executed on a dfferent core fro another ob of the task, a ob cannot be preepted by any ob durng ts executon, and a processor cannot be left dle f there s at least one ob ready to execute. Syste behavors. We assue two dfferent odes of a syste run, each referred to as -ode and HI-ode. Every ob of a task t n -ode executes for no ore than ts C whle the ob n HI-ode executes up to ts C HI ; also, all -crtcalty tasks are not consdered to be scheduled n HI-ode. The syste starts at t ¼ 0 and ntally runs n -ode, and the syste transton fro -ode to HI-ode s trggered when any ob s executon perfored for ore than C (but not larger than C HI ) s observed. We

3 1768 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 29, NO. 8, AUGUST 2018 let t TR denote the te nstant when the syste transton occurs. We assue that all obs nvoked by -crtcalty tasks are dropped edately at t TR even f they are executng; snce we focus on non-preeptve schedulng, we ay consder another odel where the currently-executng obs nvoked by -crtcalty tasks are not dropped even n the presence of the syste transton, whch we can address f we extend ths paper n the future. Note that both odels ake sense as dscussed n [21]. Schedulablty. We are nterested n udgng f a task set t s schedulable by a schedulng algorth on an MC ultprocessor platfor, eanng that every ob of every task t 2 t n -ode copletes ts executon (aountng to at ost C ) wthn D te unts, and every ob of every HI-crtcalty task t 2 tl ¼ HI n HI-ode copletes ts executon (aountng to at ost C HI ) wthn D te unts. We now ntroduce soe notatons to be used n the later sectons. Cax ¼def ax t 2t C ; Cax HI ¼def ax t 2tL ¼HIC HI ; C def ax ¼ axðcax ;CHI ax Þ; C!0 def ¼ l axð; D C ax HI ;V Þ C HI!0 def ¼ l axð; D Cax HI Þ ; Vax ¼def ax t 2t ; Vax HI ¼def ax t 2tL ¼HIV HI ; V su ¼def ; V HI t 2t su ¼def t 2tL ¼HI V HI 3 SCHEDULABILITY ANALYSIS FOR NP-EDF ON MC MULTIPROCESSOR PLATFORMS In ths secton, we develop an NP-EDF schedulablty test for MC ultprocessor systes. To ths end, we frst present a schedulablty condton for a case where the syste s under -ode, usng the exstng NP-EDF schedulablty test for SC ultprocessor systes. We then pose questons how to desgn schedulablty condtons for MC systes, and derve portant paraeters for the condtons. Based on the desgn, we propose an NP-EDF schedulablty test for MC ultprocessor systes, and prove ts correctness. 3.1 Schedulablty Condton Under -Mode by Recaptulaton of Bar As a frst step towards an NP-EDF schedulablty test for MC ultprocessor systes, we nvestgate Bar [24], an exstng NP-EDF schedulablty test for SC ultprocessor systes, and present how Bar can be used to derve a schedulablty condton before the syste transton. Let PS ðtþ denote a process-sharng schedule such that a ob of t 2 t executes wth V rate, thereby fnshng ts executon at Cax aount of te ahead of ts deadlne (f 1). Also, let NP-EDFðtÞ denote a schedule such that obs nvoked by tasks n t are scheduled by NP-EDF on an -processor platfor. Let W AðtÞ; d; ½t a ;t b Þ denote the aount of executon of d n ½t a ;t b Þ,underascheduleAðtÞ; the second paraeter d can be a sngle task, a sngle ob, a set of tasks, or a set of obs as long as t belongs to or s nvoked by t. Then,there s a relatonshp between the aount of executon by NP-EDFðtÞ and PS ðtþ as follows. Lea 1 (fro Bar [24]). Suppose t satsfes Eq. (1). Vsu þð 1ÞVax (1) Then, the followng condton holds. W NP-EDFðtÞ;Jðt;qÞ; ½0;tþ Cax Þ W PS ðtþ;jðt;qþ; ½0;tÞ ; where Jðt;qÞ denotes a set of q obs nvoked by tasks n t wth the q earlest absolute deadlnes. Proof. We prove t by contradcton, and the proof dea s slar to [24]. Suppose that t 0 denotes the frst te nstant n whch Eq. (2) s volated. By the defnton of t 0, there should be at least one ob J of t satsfyng the followng condton. W NP-EDFðtÞ;J ; ½0;t 0 þ C ax Þ <W PS ðtþ;j ; ½0;t 0 Þ Let r be the release te of J. Snce r <t 0 holds and t 0 s the frst te nstant volatng Eq. (2), the followng condton holds. W NP-EDFðtÞ;Jðt;qÞ; ½0;r þ Cax Þ W PS ðtþ;jðt;qþ; ½0;r Þ Now, for J and Jðt;qÞ, we copare the aount of executon under NP-EDFðtÞ n ½r þ Cax ;t 0 þ Cax Þ, wth that under PS n ½r ;t 0 Þ. Let x and y respectvely denote the cuulatve length of ntervals n ½r þ Cax ;t 0 þ Cax Þ where all processors are busy under NP-EDFðtÞ, and that where at least one processor s dle under the schedule; therefore y equals to t 0 r x. Note that n ½r þ Cax ;t 0 þ Cax Þ, no ob n Jðt;qÞ under NP-EDFðtÞ s blocked or nterfered by other obs than Jðt;qÞ because every lower-prorty ob whch starts ts executon before r, fnshes ts executon before r þ Cax and obs n Jðt;qÞ have a hgher prorty than other obs by the defnton of Jðt;qÞ; ths eans that we only focus on Jðt;qÞ n ½r þ Cax ;t 0 þ Cax Þ under NP-EDFðtÞ. We now present two propertes. J under NP-EDFðtÞ does not coplete ts executon untl t 0 þ Cax (otherwse t volates Eq. (3)), and thus executes for at least y aount of te snce NP-EDF s work-conservng. Snce J cannot execute for ore than ðx þ yþvax under PS ðtþ, Eqs. (3) and (4) for J yeld the followng condton (2) (3) (4) y<ðx þ yþ ax (5) The aount of total executon of Jðt;qÞ under NP-EDFðtÞ s at least x þ y, and that under PS s at ost ðx þ yþ P t 2t V. Eq. (3) for Jðt;qÞ and Eq. (4) derve the followng condton

4 BAEK ET AL. NON-PREEMPTIVE SCHEDULING FOR MIED-CRITICALITY REAL-TIME MULTIPROCESSOR SYSTEMS 1769 x þ y<ðx þ yþ su (6) By addng ( 1) ultpled by Eqs. (5) to (6), we have Vsu þð 1ÞVax >; (7) whch contradcts the supposton of ths lea. tu Usng Lea 1, we can guarantee schedulablty by NP- EDF under MC ultprocessor systes as follows. Lea 2 (fro Bar [24]). Suppose that t s scheduled by NP- EDF on an processor platfor and satsfes Eq. (1). Then, there s no ob deadlne ss untl the syste transton t TR. Proof. We frst check that the su of executon rates under PS ðtþ s always no larger than, snce Vsu holds by Eq. (1). Then, the proof s by nducton of obs generated by t, sorted n the order of ther absolute deadlnes. For the base case wth a sngle ob of t, NP-EDF trvally schedules t wthout ts deadlne ss snce C D holds. Then, we assue that q 1 obs wth the q 1 earlest absolute deadlnes eet ther deadlnes and prove that a ob wth the q th earlest absolute deadlne (denoted by J ) also eets ts deadlne, for q>1. J under PS ðtþ copletes ts executon no later than d Cax, where d denotes the deadlne of J. By Eq. (2) of Lea 1, ths ples that J under NP-EDFðtÞ copletes ts executon no later than d, whch proves the lea. tu 3.2 Desgn of Schedulablty Condtons for MC Multprocessor Systes In the prevous secton, there are two portant paraeters for PS ðtþ () each ob of t executes wth V rate, and () each ob of t fnshes ts executon at Cax aount of te ahead of ts deadlne. Deternng rght values of the two portant paraeters akes t possble to gve tng guarantees to non-preeptve obs under NP-EDF before the syste transton. For tng guarantee after the syste transton, we need to properly deterne the two paraeters for a new process-sharng schedule that generalzes PS ðtþ, stated as follows. How can we deterne the executon rate of t for HI-ode? (.e., V TR ) How can we deterne the fnshng te of a ob of t n HI-ode? (.e., Y TR aount of te ahead of ts deadlne) In addton, we need to deterne a rght te nstant when the new processor-sharng schedule swtches the executon rate of t fro V to V TR, stated as follows. How can we deterne the rght te nstant at whch the executon rate s changed fro - to HI-ode? (.e., t TR TR, eanng TR aount of te ahead of the syste transton) Here, we would lke to ephasze that the executon rate under a process-sharng schedule (correspondng to PSðtÞ for -ode) s changed before t TR. In ths secton, we wll detal how to calculate the three portant values V TR, Y TR and TR. Fg. 1. How PSðtÞ works a ob of t executng wth (a) V rate, and (c) and (d) V and then V TR rate. Wth the three unknown varables V TR, TR and Y TR addressed n each ssue, we now defne a new processsharng schedule. Let PSðtÞ denote a process-sharng schedule such that a ob of t 2 t executes wth V rate before t TR TR, and wth V TR and zero rate after t TR TR f L ¼ HI and L ¼, respectvely. We let the actual executon te of each ob under PSðtÞ be the sae as that under NP-EDFðtÞ. Under PSðtÞ, a ob of t 2 t fnshes ts executon no later than Cax aount of te ahead of ts deadlne, f t fnshes ts executon earler than t TR under NP-EDFðtÞ. On the other hand, a ob of t 2 t wth L ¼ HI fnshes ts executon no later than Y TR aount of te ahead of ts deadlne under PSðtÞ, f t fnshes ts executon no earler than t TR under NP-EDFðtÞ. Fg. 1 presents how PSðtÞ works for a ob of t, wth four exaples wth dfferent release tes. A ob can execute wth () rate n Fg. 1a, () V TR rate n Fgs. 1c and 1d. rate, (b) V TR rate n Fg. 1b, and () V and then V HI Then, we would lke to prove a condton for addressng the aount of executon of PSðtÞ copared to that of NP-EDFðtÞ, whch corresponds to Eq. (2), as follows W NP-EDFðtÞ;Jðt;qÞ; ½0;tþ Y TR Þ W PSðtÞ;Jðt;qÞ; ½0;tÞ Recall that Jðt;qÞ denotes a set of q obs nvoked by tasks n t wth the q earlest absolute deadlnes. For addressng the executon rate of PSðtÞ n - and HI-ode, we consder task sets satsfyng Eq. (1) and the followng condton, respectvely (8) Vsu TR TR þð 1ÞVax ; (9) where Vsu TR ¼ P t 2tL ¼HI V TR and Vax TR ¼ ax t 2tL ¼HIV TR.

5 1770 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 29, NO. 8, AUGUST 2018 Then, the reanng ssues are to deterne the unknown varables V TR, TR and Y TR such that we can prove that f Eqs. (1) and (9) hold for t, t s schedulable by NP-EDF on MC ultprocessor systes. We now dscuss qualfcaton of the unknown varables, startng fro V TR. We consder a ob of nterest J nvoked by t, such that the syste transton occurs n the ddle of executon of J or before J s executon (therefore J executes C HI aount). Let TR denote an nterval length fro ts release te to the earler te nstant between t TR and the te nstant when J fnshes ts executon under PSðtÞ; fj s release te s after t TR, TR s zero. Then, the aount of executon of J under PSðtÞ s calculated by axð0; TR þ V TR TR Þ ax 0;D Y TR axð0; TR TR Þ ; (10) whch should be equal to C HI for J to fnsh ts executon. That s, as shown n Fgs. 1c and 1d, under PSðtÞ, a ob of t executes wth V rate fro ts release te to t TR TR (whose length s axð0; TR TR Þ), whle the ob executes wth V TR rate fro t TR TR to Y TR aount of te ahead of ts deadlne (whose length s D Y TR axð0; TR TR Þ). Ths derves V TR and ts propertes as follows Lea 3. V TR s calculated by Eq. (11), and V TR decrease. stays) as TR ncreases and Y TR and TR V TR decreases (or C HI V axð; TR TR Þ ¼ l!0 ax ; D Y TR axð; TR TR Þ (11) Proof. Fro the condton where Eq. (10) s equal to C HI, Eq. (11) s edately derved. In Eq. (11), t s straghtforward that as Y TR ncreases, V TR ncreases. Also, f we ncrease TR or decrease TR, we can reduce (or do not change) the te nterval when a ob of t executes wth V rate, yeldng a decrease n V TR, e.g., copare Fgs. 1c and 1d. tu Next, the followng lea dscusses the qualfcaton of TR. Lea 4. TR C ax. Proof. Suppose that TR s strctly larger than Cax. Then, we cannot guarantee the schedulablty of a ob of t wth L ¼ whosedeadlnesnolaterthant TR. Ths s because the aount of executon of a ob of t whosedeadlneslaterthant TR TR þ Cax (but no later than t TR ) under PSðtÞ s strctly less than C, whch akes t possble to say that f Eq. (2) holds, all the obs are schedulable (as we do that n Lea 2). tu When t coes to Y TR, we need to address the blockng by lower-prorty obs n both - and HI-odes. For the forer and the latter, we need at least Cax and at least Cax HI, respectvely; therefore, Y TR should be at least C ax. Consderng Lea 3, we set Y TR to the sallest possble value C ax. As to TR, the value depends on each ob s release pattern and the syste transton nstant, plyng that we cannot control t. Therefore, we need to calculate an upper-bound. Once we target a task set satsfyng Eq. (1), we can use the property of Eq. (2) and calculate an upper bound of the aount of executon of obs whose prorty s hgher than the ob of nterest. Ths allows to calculate the axu nterval between the release and fnshng te of the ob of nterest, whch s also of the value n order to upper-bound V TR an upper-bound of TR. The followng lea records ths. s upper- Lea 5. Suppose that t satsfes Eq. (1). Then, TR bounded by R where R P ¼ C þ Cax þ l ax ; D Cax!0 t 2tnft g V (12) Proof. We now prove r þ TR f respectvely denote the release te of J and the te f r þ R, where r and nstant when J fnshes ts executon of C under NP-EDFðtÞ. Recall that we focus on a ob of nterest of J nvoked by t, such that the syste transton occurs n the ddle of executon of J or before J s executon (therefore J executes C HI aount). We also recall that TR denotes an nterval length fro r to the earler te nstant between t TR and the te nstant when J fnshes ts executon under PSðtÞ. Once we prove the two nequaltes, we can derve TR R. Fro now on, we prove the two nequaltes separately. Frst, we prove the frst nequalty r þ TR f. Suppose that the nequalty does not hold. By defnton of TR, the syste transton occurs no earler than r þ TR. Therefore, r þ TR >f ples that the syste transton occurs after J fnshes C aount of executon, whch contradcts the fact that the syste transton occurs n the ddle of executon of J or before J s executon (whch s the defnton of J ). Second, we prove the second nequalty f r þ R. We consder two cases () J under NP-EDFðtÞ starts ts executon before r þ Cax, and () otherwse. Case () ples that f s no later than r þ Cax þ C. Snce the latter s no later than r þ R by Eq. (12), the nequalty holds for Case (). When t coes to Case (), snce every ob released before r fnshes ts executon before r þ Cax, we now calculate the aount of executon of every ob whose prorty s hgher than J n ½r þ Cax ; r þ D Š n whch any lower-prorty ob cannot block J.If the aount s no larger than l!0 ax ; D Cax Pt 2tnft g V, the lea holds, whch s true as follows. Let r þ x denote the earlest deadlne of a ob of t (6¼ t ), after r.ifx >D, no ob of t has a hgher prorty than J n ½r þ Cax ;r þ D Þ. Otherwse, we consder two ntervals ½r þ Cax ;r þ x Þ and ½r þ x ;r þ D Þ; note that the frst nterval ay not exst f x Cax.In ½r þ Cax ;r þ x Þ, a ob of t has a hgher prorty than J, and the aount of ts executon s W NP-EDFðtÞ; t ; ½0;r þ TR Þ W NP-EDFðtÞ; t ; ½0;r þ Cax Þ, whch s the sae as W PSðtÞ; t ; ½0;r þ TR Cax Þ W NP-EDFðtÞ; t ; ½0;r þ Cax Þ, because a ob under

6 BAEK ET AL. NON-PREEMPTIVE SCHEDULING FOR MIED-CRITICALITY REAL-TIME MULTIPROCESSOR SYSTEMS 1771 PSðtÞ fnshes ts executon no later than Cax aount of te ahead of ts deadlne. The aount of executon of obs of t wth a hgher prorty than J s at ost b D x T cc. Then, the aount of executon of obs whose prorty s hgher than J n ½r þ Cax ;r þ D Þ s calculated by t 2tnft g þ W PSðtÞ; t ; ½0;r þ x C ax Þ t 2tnft g t 2tnft g W NP-EDFðtÞ; t ; ½0;r þ C ax Þ D x T k C (13) Snce J cannot execute untl r þ C ax, W NP-EDFðtÞ; t; ½0;r þ C ax Þ W PSðtÞ; t; ½0;r Þ ples that W NP-EDF ðtþ; t nft g; ½0;r þ C ax Þ W PSðtÞ; t nft g; ½0;r Þ, whch derves the followng condton fro Eq. (13). Eq ð13þ t 2tnft g þ t 2tnft g t 2tnft g t 2tnft g þ t 2tnft g t 2tnft g þ t 2tnft g W PSðtÞ; t ; ½0;r þ x C ax Þ W PSðtÞ; t ; ½0;r Þ D x T k C W PSðtÞ; t ; ½r ;r þ x C ax Þ D x D C ax ðx Cax Þ ¼ðD C ax Þ D x D Cax t 2tnft g C C D C ax C Therefore, f r þ R for Case () holds. Fnally, r þ TR f r þ R holds, plyng TR R. Ths proves the lea. tu Fnally, f we cobne all the results fro Leas 3, 4 and 5 and apply TR ¼ Cax, Y TR ¼ C ax, and TR ¼ R, we have the followng V TR that yelds a tghter schedulablty condton. V TR C HI V axð; R Cax ¼ l Þ!0 ax ; D C ax axð; R (14) Cax Þ 3.3 Fnal Schedulablty Test for NP-EDF So far, we deterned the unknown varables V TR, TR and Y TR so as to ake a correct and tght schedulablty condton n the presence of the syste transton. In ths secton, we forally present and prove our NP-EDF schedulablty test for MC ultprocessor platfors, startng fro the followng lea that corresponds to Lea 1. Lea 6. Suppose that t satsfes Eq. (1) as well as Eq. (9) wth V TR n Eq. (14). Then, Eq. (8) wth Y TR ¼ C ax holds for all t>0. Proof. We follow the basc proof dea of Lea 1, usng the contradcton. Suppose that t 0 denotes the frst te nstant n whch Eq. (8) s volated. By the defnton of t 0, there should be at least one ob J of t satsfyng the followng condton. W NP-EDFðtÞ;J ; ½0;t 0 þ C ax Þ <W PSðtÞ;J ; ½0;t 0 Þ (15) Let r be the release te of J. Snce r <t 0 holds and t 0 s the frst te nstant volatng Eq. (8), the followng condton holds. W NP-EDFðtÞ;Jðt;qÞ; ½0;r þ C ax Þ W PSðtÞ;Jðt;qÞ; ½0;r Þ (16) Dependng on the te nstants t TR, r and t 0, we consder three cases () t TR Cax t 0, () t TR Cax 2½r ;t 0 Þ, and () t TR Cax <r. Whle the executon rate of t under PSðtÞ n ½r ;t 0 Þ s V and V TR, respectvely for () and (), () exhbts both executon rates. Now, for J and Jðt;qÞ, we copare the aount of executon under NP-EDFðtÞ n ½r þ C ax ;t 0 þ C ax Þ, wth that under PS n ½r ;t 0 Þ. Let x and y denote the cuulatve length of ntervals n ½r þ C ax ;t 0 þ C ax Þ where all processors are busy under NP-EDFðtÞ, and that where at least one processor s dle under the schedule; therefore y equals to t 0 r x. Note that n ½r þ C ax ;t 0 þ C ax Þ, no ob n Jðt;qÞ under NP-EDFðtÞ s blocked or nterfered by other obs than Jðt;qÞ because every lower-prorty ob whch starts ts executon before r fnshes ts executon before r þ C ax and obs n Jðt;qÞ have a hgher prorty than other obs by the defnton of Jðt;qÞ; ths eans that we only focus on Jðt;qÞ n ½r þ C ax ;t 0 þ C ax Þ under NP-EDFðtÞ. We now present two propertes. J under NP-EDFðtÞ does not coplete ts executon untl t 0 þ C ax (otherwse t volates Eq. (15)), and thus executes for at least y te unts snce NP-EDF s work-conservng. Snce J cannot execute for ore than axð0; nðt TR Cax ;t 0Þ r Þ Vax þ axð0;t 0 axðt TR Cax ;r ÞÞ Vax TR under PSðtÞ, Eqs. (15) and (16) for J yeld the followng condton 1 y<axð0; nðt TR Cax ;t 0Þ r ÞVax þ axð0;t 0 axðt TR Cax ;r ÞÞ Vax TR (17) The aount of total executon of Jðt;qÞ under NP-EDFðtÞ s at least x þ y, and that under 1. The ax and n functons are needed to address Cases (), () and ().

7 1772 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 29, NO. 8, AUGUST 2018 s at ost axð0; nðt TR Cax ;t 0Þ r Þ Vsu þ axð0;t 0 axðt TR Cax ;r ÞÞ Vsu TR under PSðtÞ. Eq. (15) for Jðt;qÞ and Eq. (16) derve the followng condton PS x þ y<axð0; nðt TR Cax ;t 0Þ r ÞVsu þ axð0;t 0 axðt TR Cax ;r ÞÞ Vsu TR (18) By addng ( 1) ultpled by Eqs. (17) to (18), we can show the contradcton of Eq. (1) for Case () and the contradcton of Eq. (9) wth V TR n Eq. (14) for Case (). Also, f we further reove Eq. (1) ultpled by axð0; nðt TR Cax ;t 0Þ r Þ fro addng ( 1) ultpled by Eqs. (17) to (18), we can also show the contradcton of Eq. (9) wth V TR for Case (). Snce all the three cases yeld contradcton, the lea holds. tu Usng the lea, we fnally present our schedulablty test. Theore 1. Suppose that t satsfes Eq. (1) as well as Eq. (9) wth V TR n Eq. (14). Then, t s schedulable by NP-EDF under MC ultprocessor systes. Proof. Once Lea 6 derves that Eq. (8) wth Y TR ¼ C ax holds for all t>0, the proof of ths theore s the sae as that of Lea 2. tu We can sply calculate the te-coplexty of the proposed NP-EDF schedulablty test for MC ultprocessor systes n Theore 1, whch s OðnÞ. 4 VIRTUAL DEADLINE ASSIGNMENT FOR NP-EDFVD In ths secton, we adapt the proposed schedulablty test for NP-EDF to NP-EDFVD, and pose a vrtual deadlne assgnent proble. We then present an optal vrtual deadlne assgnent polcy for the syste-level deadlne reducton paraeter, and a suboptal one for the task-level paraeter. Note that NP-EDFVD can prove schedulablty of NP-EDF by adustng the deadlne of each task t wth L ¼ HI n -ode as the sae concept works n preeptve schedulng [4], [5], [14], [15]. Ths requres developent of prncples how executon deand for HI-ode (ncludng the syste transton) and -ode vary wth vrtual deadlnes, whch s a atter of ths secton. 4.1 Schedulablty Analyss for NP-EDFVD on MC Multprocessor Platfors In ths paper, we consder NP-EDFVD, and shorten the relatve deadlne of t wth L ¼ HI n -ode, fro D to a new relatve deadlne D. Snce the new relatve deadlne of t (.e., D ) can be n ½Cax ;D Š as seen n the denonator of V, we let a for t to adust the value of D fro C ax (by a ¼ 0) tod (by a ¼ 1). That s, we replace the ter of axð0;d Cax Þ wth axð0;d Cax Þa. Then, once we deterne a, D can be calculated by Cax þðd Cax Þa ; therefore, ths secton dscusses how to deterne a for every t 2 tl ¼ HI, assung a s set to 1 and never changed for every t 2 tl ¼. Fg. 2. How PSðtÞ works wth and wthout vrtual deadlnes (a) a ¼ 1 (no vrtual deadlne); (b) sall a ; and (c) saller a. Usng a, ða Þ and R and R ða Þ for NP-EDFVD, as follows. for NP-EDF can be expressed by ða Þ¼l ax ¼ V ; D C 1 ; (19) a a!0 C R ða Þ¼C þ Cax ax þ ax D C ax a Pt 2tnft g ða Þ (20) Dfferently fro V ða Þ and R ða Þ, we need to carefully construct V TR ða Þ, because of the followng observaton. Observaton 1. Under NP-EDF (wthout the use of deadlne scalng), V TR ncreases as TR ncreases as dscussed n Lea 3. However, under NP-EDFVD (wth the use of the proposed deadlne scalng), the sae does not always hold. As shown n Fg. 2a, wthout vrtual deadlnes, V holds. However, f we decrease a fro 1 to a partcular V TR value less than 1, V ða Þ ncreases whle V TR ða Þ decreases as shown n Fg. 2b. If we further decrease a, t s possble to satsfy V ða Þ >V TR ða Þ as shown n Fg. 2c. In the last case shown n Fg. 2c, f TR ncreases, V TR ða Þ does not ncrease. That s, V s always no larger than V TR wth TR ¼ 0 under NP-EDF, whle t s possble for V ða Þ for soe a to be larger than V TR ða Þ wth TR ¼ 0 under NP-EDFVD. In ths case, V TR ða Þ s axzed when TR ¼ 0. Therefore, we need to consder both cases where TR s the sallest (.e., 0) and largest (.e., R ) for calculatng V TR ða Þ, whch are addressed n the frst and second ters, respectvely n the ax functon of the RHS of the followng condton.

8 BAEK ET AL. NON-PREEMPTIVE SCHEDULING FOR MIED-CRITICALITY REAL-TIME MULTIPROCESSOR SYSTEMS 1773 V TR C HI ða Þ¼l ax!0 axð; D C ax Þ ; C HI ða Þaxð; R ða Þ C ax ; D C ax axð; R ax Þ ða Þ C ax Þ (21) Then, once we substtute V ða Þ and V TR ða Þ for V n Eqs. (1) and (9), for every t L ¼ HI 2 t, we can and V TR develop the followng NP-EDFVD schedulablty test for MC ultprocessor systes, correspondng to Theore 1. Theore 2. Suppose that t satsfes Eqs. (22) and (23). V ða Þþð 1Þax t 2t ða Þ (22) t 2t t 2tL ¼HI V TR ða Þþð 1Þax t 2tL ¼HIV TR ða Þ (23) Then, t s schedulable by NP-EDFVD under MC ultprocessor systes. Proof. The proof s the sae as that for NP-EDF n Lea 6 and Theore 1. tu 4.2 Optal Vrtual Deadlne Assgnent for the Syste-Level Deadlne Reducton Paraeter a Now, we consder a vrtual-deadlne assgnent proble for the syste-level deadlne reducton paraeter a ¼ a for every t 2 tl ¼ HI. We wsh to fnd the value of a that satsfes the condtons Eqs. (22) and (23). To ths end, we need to derve propertes on how the LHS of the condtons vares wth a. Observaton 2. As a decreases, the LHS of Eq. (22) ncreases but that of Eq. (23) does not ncrease. Ths s because, V ðaþ and V TR ðaþ are a decreasng and a non-decreasng functon of a, respectvely; the forer s obvous, whle the latter s not. Thus, let us explan t now. In V TR ðaþ for t 2 tl ¼ HI, ðaþ and R ðaþ are two ðaþ ters that depend on a. Snce we can conclude that V TR s a non-decreasng functon of a f we know that V ðaþ and R ðaþ are a non-ncreasng and non-decreasng functon of a, respectvely. And what reans to be explaned s the latter,.e., R ðaþ. The thrd ter of R ðaþ for t 2 tl ¼ HI n the RHS of Eq. (20) can be expressed as follows. ax D C ax a Pt 2tnft g V ða Þ ¼ ax P D Cax t 2tnft gl ¼HI V þ ax P D Cax a t 2tnft gl ¼ V (24) Therefore, R ðaþ s a non-decreasng functon of a. Whle we focus on a task set whch volates at least one of Eqs. (22) and (23) (otherwse, Theore 1 dees the task set schedulable by NP-EDF wthout vrtual deadlnes), Observaton 2 ndcates that we need to focus on a task whch satsfes Eq. (22) but volates Eq. (23), and decrease a as uch as possble untl the LHS of Eq. (22) s equal to, whch reduces the LHS of Eq. (23) as uch as possble. We now propose an optal vrtual deadlne assgnent polcy for the syste-level paraeter a n Algorth 1. Due to the exstence of the ax functon n the LHS of Eq. (1), we need to consder two cases dependng on whether the task whch exhbts the ax value of V s a HI-crtcalty task (the frst case) or a -crtcalty task (the second case). Algorth 1. Optal Vrtual Deadlne Assgnent for the Syste-Level Paraeter 1 arg ax t 2t ; 2 f L ¼ HI then 3 a 4 else 5 a P t 2tL P ¼HI V þð 1ÞV t 2tL ¼ V P t 2tL ¼HI V P t 2tL ¼ ; ð 1Þ 6 2 arg ax t 2tV ða ¼ aþ; 7 f 6¼ 2 then P t 8 a 2tL ¼HI V þð 1Þax t 2tL ¼HI V P t 2tL ¼ 9 end f 10 end f 11 f Eqs. (22) and (23) hold wth a ¼ a for every t 2 tl ¼ HI and a ¼ 1 for every t 2 tl ¼ then 12 return Schedulable; 13 end f 14 return Unschedulable; In the frst case, snce the task whch exhbts the ax value of V (denoted by t ) stll exhbts the ax value of V ðaþ after changng a, we can sply solve the followng equaton (Lnes 1 3) þ 1 a V þ 1 ð 1ÞV ¼ a t 2tL ¼ t 2tL ¼HI Pt 2tL ¼HI V ) a ¼ þð 1Þ P t 2tL ¼ ;. (25) On the other hand, for the second case, t s possble to change the task whch exhbts the ax value of V (fro t to t 2 ) f we reduce a. Therefore, we consder two subcases dependng on whether the task whch exhbts the ax value of V s () unchanged or () changed by the change of a. For (), we can solve the slar equaton as follows (Lnes 4-5). þ 1 a V þð 1ÞV ¼ t 2tL ¼ ) a ¼ t 2tL ¼HI Pt 2tL ¼HI V P t 2tL ¼ ð 1Þ (26) In order to know whether the current stuaton belongs to () or (), we frst solve the above equaton. If the task whch exhbts the ax value of V stll exhbts the ax value of

9 1774 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 29, NO. 8, AUGUST 2018 V ðaþ after applyng a calculated by the above equaton, the resultng a s the fnal value; otherwse, we should consder (). For (), the task whch exhbts the ax value of V ðaþ after applyng a new a becoes the task wth the largest V aong every task t 2 tl ¼ HI, yeldng the followng equaton (Lnes 6 9). t 2tL ¼ þ 1 a t 2tL ¼HI þ 1 a ð 1Þax t 2tL ¼HI ¼ ) a ¼ Pt 2tL ¼HI þð 1Þax t 2tL ¼HI P t 2tL ¼ (27) Usng the calculated a for the three cases, Algorth 1 fnally checks the schedulablty (Lnes 11 14). Then, the followng theore presents the optalty of the vrtual-deadlne assgnent for a n Algorth 1. Theore 3. If Algorth 1 returns unschedulable, there exsts no a that akes Eqs. (22) and (23) hold wth a ¼ a for every t 2 tl ¼ HI and a ¼ 1 for every t 2 tl ¼. Proof. Suppose that Algorth 1 returns unschedulable (but t yelds a denoted by a 0 ), but there exsts a 00 that akes Eqs. (22) and (23) hold wth a ¼ a 00 for every t 2 tl ¼ HI and a ¼ 1 for every t 2 tl ¼. If the LHS of Eq. (22) s exactly, a 00 should be the sae as a 0. If the LHS of Eq. (22) s strctly less than, a 00 should be larger than a 0.Snce ðaþ s a non-decreasng functon of a,tspossbleto eet Eq. (23) wth a 00 but not to do that wth a 0. tu V TR We can easly calculate the te-coplexty of Algorth 1, whch s OðnÞ. 4.3 Suboptal Vrtual Deadlne Assgnent for the Task-Level Deadlne Reducton Paraeter a In ths secton, we solve a vrtual deadlne assgnent proble for the task-level deadlne reducton paraeter a.to ths end, we frst dscuss the vald range of a and analyze the effect of ts change on schedulablty. Whle the defntonal range of a s ½0; 1Š as we dscussed n Secton 4.1, a should be larger than V for schedulablty. Ths s because, f a, then V ða Þ1, resultng n the LHS of Eq. (22) no saller than only wth a sngle task t. Therefore, the vald range of a s ½ ; 1Š. Observaton 3. If a sngle a of t decreases wthout changng a for every t 2 t nft g, V TR ða Þ non-ncreases and V TR ða Þ for every t 2 t nft g non-decreases. Ths s because, f a of t s reduced, the followng propertes hold ða Þ and R ða Þ for every t 2 t nft g ncrease; R ða Þ decreases; and V ða Þ for every t 2 t nft g stays. Hence, wth decreent of a, whle V TR ða Þ non-ncreases, V TR ða Þ for every t 2 t nft g non-decreases. Ths ples t s not true that the LHS of Eq. (23) s a non-decreasng functon of a for every t 2 tl ¼ HI; therefore, we cannot sply fnd a for every t 2 tl ¼ HI that satsfes Eqs. (22) and (23), wthout exhaustve search. Therefore, we propose a suboptal algorth that fnds a proper ndvdual a for every t 2 tl ¼ HI wth a sall nuber of trals. The algorth targets task sets that are schedulable by nether NP-EDF (by Theore 1) nor NP- EDFVD wth the syste-level deadlne-reducton paraeter a (by Algorth 1). Starng fro a ¼ 1 for every t 2 tl ¼ HI, we try to reduce each a by and copute the dfference between the LHS of Eq. (22) and that of Eq. (23). If t yelds the largest decreent of the latter noralzed by the ncreent of the forer, we decde to reduce a by. We repeat ths process untl there s no task to reduce ts a or the task set s schedulable wth the current settng of a. We present ths suboptal algorth n Algorth 2 for a gven. We frst set a of every task t 2 t to 1 (Lnes 1 3). Then, for every t 2 tl ¼ HI and a >V (.e., every HI-crtcalty task t that can reduce ts a ), we calculate the decreent of the LHS of Eq. (23) n case that a s reduced by, noralzed by the ncreent of the LHS of Eq. (22) n case that a s reduced by (denoted by ~ n Lne 8); we then select the largest ~ (Lnes 9 12). If the value of ~ s not changed for every t 2 tl ¼ HI, we dee t unschedulable (Lnes 14 16). Otherwse, we decrease a by, for t 2 tl ¼ HI wth the greatest value of ~ (Lne 17). Thereafter, we repeat to fnd an a to be reduced, and we stop the repetton f the LHS of Eq. (22) s larger than (unschedulable, Lne 4) or Eqs. (22) and (23) hold (schedulable, Lnes 18 20). Algorth 2. Vrtual Deadlne Assgnent for the Task- Level Paraeter () 1 for t 2 t do 2 a ¼ 1; 3 end for 4 whle the LHS of Eq. (22) <do 5 ndex ¼ 1; 6 ~ ax ¼ 0; 7 for t 2 tl ¼ HI and a >V do 8 ~the LHS of Eq. (23) Copute ~ ¼ ~the LHS of Eq. (22) 9 f ~ > ~ ax then 10 ndex ; 11 ~ ax ¼ ~ 12 end f 13 end for 14 f ~ ax ¼ 0 then 15 return Unschedulable; 16 end f 17 a ndex a ndex ; 18 f Eqs. (22) and (23) hold then 19 return Schedulable; 20 end f 21 end whle 22 return Unschedulable; for. There are at ost ð1=þ ponts to be checked for a sngle a, n total up to Oðn=Þ ponts, and each pont needs to calculated Eqs. (22) and (23). Therefore, the te-coplexty of Algorth 2 s Oðn 2 =Þ.

10 BAEK ET AL. NON-PREEMPTIVE SCHEDULING FOR MIED-CRITICALITY REAL-TIME MULTIPROCESSOR SYSTEMS 1775 Fg. 3. Schedulable rato of the three ndvdual schedulablty tests over varyng n, CP, and CF. 5 EVALUATION In ths secton, we evaluate the schedulablty perforance of the proposed schedulablty tests for NP-EDF and NP- EDFVD on MC ultprocessor platfors. We frst llustrate our sulaton envronents and then dscuss varous factors nfluencng schedulablty of the proposed schedulablty tests wth eprcal sulaton results. 5.1 Task Set Generaton For task set generaton, we eploy UUnfast-dscard [25], whch s a popular task set generaton ethod for ultprocessors, orgnated fro UUnfast for unprocessor platfors [26]. Under UUnfast-dscard, we have three nput paraeters () the nuber of processors, () the nuber of tasks n, and () task set utlzaton U ¼ P t 2t C =T.In addton, we have two addtonal nput paraeters for MC schedulng [13] (v) the rato between each task t s WCETs for the hgh- and low-crtcalty level,.e., CHI (denoted by C CF), and (v) the probablty of each task t havng L ¼ HI (denoted by CP). We apply a slar paraeter settng to [13], whch s a ultprocessor MC schedulng study, as follows () = 2, 4, 8, () n = +1, 2, 3, 4, () U = 0005, 0010, 0015;...; 1000, (v) CF = 2, 3, 4, and (v) CP = 0.1, 0.3, 0.5, 0.7, 0.9. For a 5-tuple ð; n; U; CF; CPÞ, each task paraeter s deterned as follows. Based on a 3-tuple ð; n; UÞ, UUnfast-dscard [25] generates every task s utlzaton (.e., u ¼ C =T ). Each task s perod T s unforly selected n ½1s; 1000sŠ; C s selected based on the chosen task utlzaton (.e., C ¼ u T ); and L s selected as HI wth probablty CP (and as wth probablty ð10 CPÞ). If L ¼ HI, C HI otherwse, C HI s set to C. s set to ðcf C Þ; 2 For gven a lst of fve paraeters, we generate 1000 task sets, yeldng 1000 * 3 * 4 * 200 * 3 * 5 = 36,000,000 task sets n total. 5.2 Evaluaton Results Wth the generated task sets, we copare the followng three schedulablty tests NP-EDF the schedulablty test n Theore 1 for NP- EDF; NP-EDFVD-S the schedulablty test n Theore 2 for NP-EDFVD wth the syste-level paraeter (Algorth 1); and NP-EDFVD-T the schedulablty test n Theore 2 for NP-EDFVD wth the task-level paraeter (Algorth 2 wth ¼ 001), after applyng NP-EDFVD-S. We show results of plct-deadlne task sets (D ¼ T for every t 2 t); the results of constraned-deadlne task sets (D T for every t 2 t) exhbt a slar trend but wth low schedulablty. For perforance etrc, we use schedulable rato defned as the rato of the nuber of tasks that are deeed schedulable by each ndvdual schedulablty test, to the nuber of generated task sets under the gven nput paraeters. For dfferent cobnatons of nput paraeters (.e., n, CP, and CF), Fg. 3 shows the schedulable ratos for the three schedulablty tests, and Fg. 4 shows the relatve rato 2. If C HI >T (eanng an nfeasble task set), we dscard ths task set and re-generate another task set.

11 1776 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 29, NO. 8, AUGUST 2018 Fg. 4. Relatve rato between the schedulable ratos of NP-EDFVD-S and NP-EDF (denoted by NP-EDFVD-S/NP-EDF), and that between the schedulable ratos of NP-EDFVD-T and NP-EDF (denoted by NP-EDFVD-T/NP-EDF) over varyng n, CP, and CF. between the schedulable ratos of NP-EDFVD-S and NP-EDF, and that between the schedulable ratos of NP-EDFVD-T and NP-EDF, accordng to varyng value of U=. 3 Now we dscuss how each nput paraeter nfluences the overall schedulable rato of the three schedulablty tests (wth Fg. 3) and pact of the syste-level and tasklevel vrtual-deadlne assgnent schees (wth Fg. 4). The nuber of tasks (n subfgures (a), (b), (c) and (d) of Fgs. 3 and 4). As seen n Fgs. 3a, 3b, 3c and 3d, schedulable ratos of the three schedulablty tests for a gven task set utlzaton (U) becoe proved as the nuber of tasks n a task set (n) ncreases. Ths s because, a large nuber of tasks n a task set for a gven U reduces average utlzaton of each task, whch tends to yeld low Vax TR and Vax n Eqs. (1) and (9), respectvely. Also, Fgs. 4a, 4b, 4c and 4d show that schedulablty proveent acheved by NP-EDFVD-S (as well as NP-EDFVD-T) copared to NP-EDF becoes agnfed as the nuber of tasks n a task set s ncreased. For exaple, NP-EDFVD-T exhbts up to percent schedulablty proveent over NP-EDF (wth n = 4 and U= = 0.27 n Fg. 4d). If we copare schedulablty proveent by NP-EDFVD-T, wth that by NP-EDFVD-S, the forer outperfors the latter up to percent (wth the sae paraeters). These observatons ndcate that a lower average utlzaton of tasks (due to a greater value of n) allows a 3. Indvdual subfgures n Fg. 3 plot up to U= = 0.3 for x-axs snce ost lnes converge to 0 after 0.3. When t coes to Fg. 4, each x-axs of ndvdual subfgures plots up to at ost 0.3 (but dfferent values) n order to avod a sall saple sze where the nuber of task sets deeed schedulable by NP-EDF s very sall. better chance for task sets deeed unschedulable by NP-EDF to becoe schedulable by NP-EDFVD-S (or NP-EDFVD-T); ths s because, a lower average utlzaton of tasks tends to yeld a saller Vax TR, whch allows unschedulable task sets by NP-EDF to be schedulable by slghtly adustng the vrtual deadlne of HI-tasks. The probablty of CP (n subfgures (e), (f), (g), (h) and (b) of Fgs. 3 and 4). It s observed fro Fg. 3 that a larger CP (.e., a hgher nuber of HI-tasks n a task set) degrades schedulablty perforance of the three schedulablty tests. Ths s due to our proposed task set generaton ethod, n whch U =T of all tasks n a task set. A hgher value of CP ncreases the nuber of HI-tasks n a task set for a gven U, there by decreasng schedulablty. On the other hand, such an ncreased nuber of HI-tasks allows a larger roo for schedulablty perforance proveent acheved by vrtual deadlne assgnent as shown n Fg. 4. NP-EDFVD-T proves schedulable rato of NP-EDF up to percent for CP = 0.9 (Fg. 4b for U= = 0.19) whle NP-EDFVD-T does t up to 7.1 percent for CP = 0.1 (Fg. 4e for U= = 0.225). The nuber of processors (n subfgures (b), () and () of Fgs. 3 and 4). As ncreases, schedulable ratos of the three schedulablty tests decrease for a gven n= and U= as shown n Fgs. 3b, 3 and 3. To nterpret ths observaton, s deterned by the suaton of C we dvde Eq. (1) by, thereby resultng n V su þ 1 V ax 1.0. Whle the frst ter tends to have a slar expected value for a fxed value of U=, Vax sply depends on the largest V ; a larger tends to have a larger Vax. In addton, 1 also becoes larger as ncreases 0.5, 0.75, and

12 BAEK ET AL. NON-PREEMPTIVE SCHEDULING FOR MIED-CRITICALITY REAL-TIME MULTIPROCESSOR SYSTEMS for ¼ 2, 4, and 8. On the other hand, our vrtual deadlne assgnent schees sgnfcantly ake up such degradaton steng fro ncreasng. For exaple, the schedulablty proveent by NP-EDFVD-T s percent for ¼ 2 (n Fg. 4b), whle t s ncreased to percent and percent for ¼ 4 and 8 (n Fgs. 4 and 4). If we focus on the gap between schedulablty proveent by NP-EDFVD-T and NP-EDFVD-S, the gap ncreases as ncreases. That s, the gap s = 17.7 percent for ¼ 2, but t s ncreased to = 39.1 percent, and = percent for ¼ 4 and 8, respectvely. Ths ndcates that the task-level vrtual deadlne assgnent schee (rather than the syste-level one) has hgher opportuntes to prove perforance of NP-EDF by adustng ndvdual task s vrtual deadlne. The rato of CF (n subfgures (b), (k) and (l) of Fgs. 3 and 4). We observe that overall schedulablty perforance of the three schedulablty tests degrades as CF ncreases (as shown n Fgs. 3b, 3k and 3l); the reason s the sae as that for CP. Note that schedulablty proveent by NP-EDFVD-T and NP-EDFVD-S does not ncrease as CF ncreases (as shown n Fgs. 4b, 4k and 4l). Ths s because, the roo for proveent s lted wth a large value of where t s dffcult for unschedulable task sets to becoe schedulable wth any vrtual deadlne assgnent. C HI 6 RELATED WORK Snce Vestal s senal work [1], there have been a body of research on MC schedulng for a unprocessor platfor. Baruah et al. proposed RTA schedulablty tests for statc and adaptve xed-crtcalty prorty assgnent schees, and deonstrated a donance relatonshp between the two [2], [3]. Also, they proposed a new schedulng algorth referred to as preeptve EDFVD (Earlest Deadlne Frst wth Vrtual Deadlnes) [4], [5]. In preeptve EDFVD, saller (vrtual) relatve deadlnes are assgned n -ode for HI-crtcalty tasks by usng a sngle syste-level scalng factor so as to guarantee schedulablty across ode changes. Ekberg and Y [27] proved upon preeptve EDFVD by enablng task-level deadlne scalng factors. L et al. ntroduced an OCBP (Own Crtcalty Based Prorty) schedulng algorth and ts schedulablty analyss for general task sets [6], [7]. Based on OCBP schedulng, Guan et al. proposed a ore effcent algorth called PLRS [8]. As to new odels, Su et al. proposed an E-MC (Elastc Mxed-Crtcalty) odel [9]. Also, Baruah ntroduced a general odel of xed-crtcalty recurrent real-te tasks consderng dfferent estates on WCET, relatve deadlne, and perod dependng on crtcalty levels [10]. The frst work dscussng MC ultprocessor schedulng was by Anderson et al. [11] and extended n 2010 [12]. They consdered fve levels of crtcalty and suggested an pleentaton schee called MC 2, eployng dfferent schedulng algorths accordng to crtcalty level. Pathan proposed a response te analyss (RTA) for global preeptve FP (Fxed-Prorty) schedulng, whch s applcable to Ausley s optal prorty assgnent [13]. L et al. extended preeptve EDFVD to ultprocessors wth respect to both global and parttonng schedulng for MC systes, and copared ther effectveness [14], [15]. Su et al. studed the E-MC odel n ultcore systes consderng the systes wth or wthout task gratons [16]. Lee et al. ncorporated the concept of a flud schedulng odel nto the MC doan and ntroduced a new schedulng algorth, called MC- Flud, whch executes each task accordng to ts crtcaltydependent executon rate [17]. Whle a lot of studes on preeptve schedulng for MC systes have been ade, research on non-preeptve scheudulng have not atured, and have been lted to unprocessor systes and dstrbuted systes. Baruah s RTA approach for adaptve xed-crtcalty prorty assgnent schee was extended by Zhao et al. to ncorporate preepton thresholds nto the odel [19], [20]. Burns and Davs consdered deferred preepton that explots the noton of fnal non-preeptve regon (FNPR) [22]. Baruah and Guo studed non-preeptve schedulng on unrelable processors and proved that the polynoal-te optal schedulng strateges cannot exst for non-preeptve MC schedulng [21]. Hanzalek et al. addressed the non-preeptve xed-crtcalty atch-up schedulng proble arsng fro the areas of the councaton protocols, and they also proved the NP-hardness of the proble [23]. As far as we know, there has been no work consderng non-preeptve schedulng for MC ultprocessor systes, and ths paper s the frst work to develop schedulablty tests for NP-EDF as well as NP-EDFVD. 7 CONCLUSION In ths paper, we developed schedulablty tests of NP-EDF and NP-EDFVD for MC ultprocessor systes, whch s the frst attept for non-preeptve schedulng on MC ultprocessor systes. To ths end, we frst nvestgated the schedulablty analyss technques of an exstng NP-EDF schedulablty test for SC ultprocessor systes, and generalzed the technques to address the syste transton. We then extended the proposed NP-EDF schedulablty test to NP-EDFVD. After posng the vrtual deadlne assgnent proble for NP-EDFVD, we developed an optal assgnent polcy for the syste-level deadlne-reducton paraeter and a suboptal polcy for the task-level paraeter. Our sulaton results deonstrated that the NP-EDFVD schedulablty analyss wth the proposed assgnent polces exhbts up to percent schedulablty proveent, copared to the NP-EDF schedulablty test. In future, we plan to target other exstng schedulablty tests of non-preeptve schedulng for SC ultprocessor systes, and develop tghter schedulablty tests for MC ultprocessor systes. ACKNOWLEDGMENTS Ths research was supported by the Natonal Research Foundaton of Korea (NRF) funded by the Mnstry of Scence and ICT (2017R1A2B , 2017H1D8A , 2017K2A9A1A ) and the Mnstry of Educaton (2016R1A6A3A ). Ths research was also supported by the IITP (Insttute for Inforaton & councatons Technology Prooton) funded by the MSIT (Mnstry of Scence and ICT) ( , IITP , , Reslent Cyber-Physcal Systes Research; 2017M3C4A ). Jnkyu Lee s the correspondng author.

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Scoredos, Mxed crtcalty real-te schedulng for ultcore systes, n Proc. 7th IEEE Int. Conf. Ebedded Softw. Syst., 2010, pp [13] R. Pathan, Schedulablty analyss of xed-crtcalty systes on ultprocessors, n Proc. Eurocro Conf. Real-Te Syst., 2012, pp [14] H. L and S. Baruah, Global xed-crtcalty schedulng on ultprocessors, n Proc. Eurocro Conf. Real-Te Syst., 2012, pp [15] S. Baruah, B. Chattopadhyay, H. L, and I. Shn, Mxed-crtcalty schedulng on ultprocessors, Real-Te Syst., vol. 50, no. 10, pp , [16] H. Su, D. Zhu, and D. Mosse, Schedulng algorths for elastc xed-crtcalty tasks n ultcore systes, n Proc. IEEE Int. Conf. Ebedded Real-Te Coput. Syst. Appl., 2013, pp [17] J. Lee, K. Phan,. Gu, J. Lee, A. Easwaran, I. Shn, and I. Lee, Mc-flud Flud odel-based xed-crtcalty schedulng on ultprocessorss, n Proc. IEEE Real-Te Syst. Syp., 2014, pp [18] G. Buttazzo, M. Bertogna, and G. Yao, Lted preeptve schedulng for real-te systes A survey, IEEE Trans. Ind. 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Burns, Prorty assgnent for global fxed prorty pre-eptve schedulng n ultprocessor real-te systes, n Proc. IEEE Real-Te Syst. Syp., 2009, pp [26] E. Bn and G. Buttazzo, Measurng the perforance of schedulablty tests, Real-Te Syst., vol. 30, no. 1 2, pp , May [27] P. Ekberg and W. Y, Boundng and shapng the deand of xed-crtcalty sporadc tasks, n Proc. Eurocro Conf. Real- Te Syst., 2012, pp Hyeongboo Baek receved the BS degree n coputer scence and engneerng fro Konkuk Unversty, South Korea, the MS and PhD degrees fro the Departent of Coputer Scence at Korea Advanced Insttute of Scence and Technology (KAIST), South Korea, n 2010 and He s currently a postdoctoral researcher wth Sungkyunkwan Unversty, South Korea. Hs research nterests nclude real-te ebedded systes, cyber-physcal systes and securty. He won the best paper award fro the 33rd IEEE Real-Te Systes Syposu (RTSS) n Nayong Jung receved the BS degree fro Sungkyunkwan Unversty, n He s workng toward the MS degree wth Sungkyunkwan Unversty. Hs research nterests nclude tng guarantees of real-te ebedded systes. Hoon Sung Chwa receved the BS, MS, and PhD degrees all n coputer scence, fro the Korea Advanced Insttute of Scence and Technology, n 2009, 2011, and 2016, respectvely. He s currently a research fellow wth the Unversty of Mchgan, Ann Arbor, Mchgan. Hs research nterests nclude syste desgn and analyss wth tng guarantees and resource anageent n real-te ebedded systes and cyber-physcal systes. He won two best paper awards fro the 33rd IEEE Real-Te Systes Syposu (RTSS) n 2012 and fro the IEEE Internatonal Conference on Cyber-Physcal Systes, Networks, and Applcatons (CPSNA) n He s a eber of IEEE.

14 BAEK ET AL. NON-PREEMPTIVE SCHEDULING FOR MIED-CRITICALITY REAL-TIME MULTIPROCESSOR SYSTEMS 1779 Insk Shn receved the BS degree fro Korea Unversty, the MS degree fro Stanford Unversty, and the PhD degree fro the Unversty of Pennsylvana, all n coputer scence, n 1994, 1998, and 2006, respectvely. He s currently an assocate professor n the Departent of Coputer Scence, KAIST, South Korea, where he oned n He has been a postdoctoral research fellow wth Malardalen Unversty, Sweden, and a vstng scholar wth the Unversty of Illnos, Urbana-Chapagn untl Hs research nterests nclude cyber-physcal systes and real-te ebedded systes. He s currently a eber of the edtoral board of the Journal of Coputng Scence and Engneerng. He has been co-char of varous workshops ncludng satellte workshops of RTSS, CPSWeek, and RTCSA and has served varous progra cottees n real-te ebedded systes, ncludng RTSS, RTAS, ECRTS, and EMSOFT. He receved best paper awards, ncludng Best Paper Awards fro RTSS, n 2003 and 2012, Best Student Paper Award fro RTAS, n 2011, and Best Paper runner-ups at ECRTS and RTSS, n He s a eber of the IEEE. Jnkyu Lee receved the BS, MS, and PhD degrees n coputer scence fro the Korea Advanced Insttute of Scence and Technology, South Korea, n 2004, 2006, and 2011, respectvely. He s an assstant professor n the Departent of Coputer Scence and Engneerng, Sungkyunkwan Unversty, South Korea, where he oned n He has been a research fellow/ vstng scholar n the Departent of Electrcal Engneerng and Coputer Scence, Unversty of Mchgan untl Hs research nterests nclude syste desgn and analyss wth tng guarantees, QoS support, and resource anageent n real-te ebedded systes and cyber-physcal systes. He won the best student paper award fro the 17th IEEE Real-Te and Ebedded Technology and Applcatons Syposu (RTAS), n 2011, and the Best Paper Award fro the 33rd IEEE Real- Te Systes Syposu (RTSS), n He s a eber of the IEEE. " For ore nforaton on ths or any other coputng topc, please vst our Dgtal Lbrary at