Spare CASH: Reclaiming Holes to Minimize Aperiodic Response Times in a Firm Real-Time Environment

Transcription

1 Spare : Reclaimig Holes to Miimize Aperiodic Respose Times i a Firm Real-Time Eviromet Deepu C. Thomas Sathish Gopalakrisha Marco Caccamo Chag-Gu Lee Abstract Schedulig periodic tasks that allow some istaces to be skipped produces spare capacity i the schedule. Oly a fractio of this spare capacity is uiformly distributed ad ca easily be reclaimed for servicig aperiodic requests. The remaiig fractio of the spare capacity is o-uiformly distributed, ad o existig techique has bee able to reclaim it. We preset a method for improvig the respose times of aperiodic tasks by idetifyig the o-uiform holes i the schedule ad addig these holes as extra capacity to the capacity queue of the mechaism. The o-uiform holes ca accout for a sigificat portio of spare capacity, ad reclaimig this capacity results i cosiderable improvemets to aperiodic respose times. 1. Itroductio Real-time systems execute periodic ad aperiodic tasks, ad each of these tasks has a deadlie. Periodic tasks are recurrig, ad each istace of such a task is called a job. A periodic task τ i is typically characterized by computatio time c i ad period ; the relative deadlie of a istace of a periodic task is assumed to be equal to the period of the task. Aperiodic tasks are executed oly occasioally but ofte require short respose times. The terms aperiodic task ad aperiodic job are used iterchageably i this discussio. Real-time tasks ca be classified based o the cosequeces of a missed deadlie as follows: Hard. If a hard real-time task misses its deadlie, it is assumed that cosequeces for the system are catastrophic. It is therefore imperative that a priori guaratees of ot missig the deadlies be provided for all hard real-time tasks. This work is supported i part by the NSF grat CCR , ad i part by NSF grat CCR Microsoft Corporatio, Redmod, WA 98052, USA; formerly with the Uiversity of Illiois at Urbaa-Champaig Departmet of Computer Sciece, Uiversity of Illiois at Urbaa-Champaig, Urbaa, IL 618, USA Departmet of Electrical ad Computer Egieerig, The Ohio State Uiversity, Columbus OH 432, USA Soft. A soft real-time task may miss deadlies, ad missed deadlies lead to degraded performace or lower quality of service. Firm. Firm real-time tasks are allowed to miss deadlies or eve skip some istaces occasioally. All other istaces should complete before their deadlies. Whe a job of a firm task is skipped, the processor gais some capacity for executig some other job. Liu ad Laylad [14] were the first to address the problem of schedulig periodic hard real-time tasks; they developed simple schedulability tests for periodic task sets scheduled by the Rate Mootoic algorithm or the Earliest Deadlie First algorithm. The RM algorithm assigs higher priorities to tasks with higher rates (lower periods) ad the EDF algorithm assigs higher priorities to tasks with earlier absolute deadlies. Systems that assig the same priority to every job of a task are called fixed priority real-time systems; systems that might assig differet priorities to differet istaces of the same task are called dyamic priority systems. The RM assigmet is fixed priority ad EDF is a dyamic priority assigmet. Later research o real-time systems exteded Liu ad Laylad s aalysis to derive schedulability coditios for hard real-time task sets uder more geeral settigs. Feasibility aalysis with resource sharig amog periodic tasks [18, 4, 3] ad i the presece of aperiodic tasks [13, 19, 12, 20, 9, 21] are importat geeralizatios that have bee studied. While it is true that there are some safety-critical systems that caot tolerate a sigle deadlie miss, may systems (e.g. multimedia systems) are capable of toleratig some missed deadlies. Moreover, eve i safety-critical systems, ot all tasks are hard tasks; some are soft ad others are firm real-time tasks. For optimal resource allocatio, soft ad firm real-time tasks eed to be hadled differetly. Skippig a few istaces of a firm real-time task allows a scheduler to utilize resources better ad schedule task sets that would otherwise overload the system. Hamdaoui ad Ramaatha [17] proposed the (m, k)-model for represetig a firm real-time task where at least m out of k cosecutive jobs must meet their deadlies. They described a heuristic priority assigmet scheme for such tasks but did ot develop a exact schedulability aalysis. Berat

2 ad Burs [5] described a techique for utilizig the (m, k)-model i the presece of aperiodic tasks alog with a offlie guaratee test usig a worst-case formulatio for fixed priority schedulig. Kore ad Shasha [11] made importat cotributios whe they proved that makig optimal use of skips is NP-hard ad described two (efficiet, but o-optimal) skipover algorithms for exploitig skips ad icreasig the feasible periodic load ad schedule task sets that are slightly overloaded. Oe skip-over algorithm is fixed priority ad exteds RM schedulig; the other algorithm is dyamic priority ad is based o the EDF algorithm. Kore ad Shasha modeled a firm real-time task, τ i, usig a skip factor, s i, which idicates that oe istace of τ i ca be skipped every s i istaces. Liu et al. [15] itroduced a ovel QoS model for etworked feedback cotrol systems: the authors showed that their QoS costrait ca directly be related to the cotrol system s performace. Buttazzo ad Caccamo [6] proposed a techique for miimizig aperiodic respose times i a firm real-time eviromet usig the model proposed by Kore ad Shasha; the uderlyig scheduler was the EDF scheduler. Buttazzo ad Caccamo reclaimed a portio of the spare time created by skippig jobs to improve the respose time for aperiodic tasks. They, however, were uable to reclaim all the spare time ad observed that a sigificat fractio of the spare time created by skippig jobs has a graular distributio across the schedule. They called these o-uiformly distributed capacities holes. Reclaimig those holes has bee a ope issue. Marchad ad Silly-Chetto [16] developed two ew algorithms, amed EDL-RTO ad EDL-BWP, which are able to exploit the skip model to ehace the respose time of soft aperiodic requests. Sice these algorithms are based o a optimal server like EDL, their rutime overhead icreases o the order of O(N 2 ) where N is the umber of tasks i the system. I this paper, we propose a ovel techique for reclaimig all the spare time, icludig both uiformly distributed ad o-uiformly distributed fractios, created whe jobs are skipped i a firm real-time eviromet; jobs are prioritized usig EDF. The o-uiformly distributed holes are idetified offlie, ad they are utilized for servicig aperiodic jobs olie by the Spare mechaism that provides a aggressive resource reclamatio techique, buildig upo the mechaism [8]. Experimetal results idicate that the reclamatio of the ouiformly distributed holes leads to sigificat improvemets i the respose time of aperiodic tasks. 2. Prelimiaries 2.1. Termiology ad Assumptios Each firm periodic task, τ i, is characterized by its worst-case computatio time, c i, its period,, a relative deadlie that is equal to the period, ad a skip parameter, s i,2 s i. The skip parameter specifies the miimum distace betwee two cosecutive skips. For example, if s i = 6, 1 i every 6 istaces of task τ i ca be skipped. Whe s i =, o skips are allowed ad the task is a hard periodic task. The skip parameter ca be viewed as a quality of service measure; higher the s, the better the QoS. τ i, j is used to deote the jth istace of task τ i. Usig the termiology itroduced by Kore ad Shasha [11], every istace of a firm periodic task ca either be red or blue. A red istace must be completed before its deadlie; a blue istace ca be aborted at ay time. Whe a blue istace is aborted, we say that it is skipped. If a blue istace is skipped, the the ext s 1 istaces must be red. O the other had, if a blue istace completes successfully, the ext task istace is also blue Firm Periodic Task Schedulig I the hard periodic model, where all task istaces are red (o skips are permitted), the schedulability of a periodic task set ca be tested usig a simple ecessary ad sufficiet coditio based upo cumulative processor utilizatio. Liu ad Laylad [14] showed that a periodic task set is schedulable by EDF if ad oly if its cumulative processor utilizatio is o greater tha 1. That is, U p = c i 1. (1) Aalyzig the feasibility of firm periodic tasks is ot equally easy. Kore ad Shasha [11] proved that determiig whether a set of skippable periodic tasks is schedulable is NP-hard. They also foud that, give a set Γ = {T i (,c i,s i )} of firm periodic tasks that allow skips, the U f irm = c i (s i 1) s i 1 (2) is a ecessary coditio for the feasibility of Γ, sice it represets the utilizatio based o the computatio that must take place. The cocepts metioed above ca be clarified with a example. Cosider the task set show i Table 1 ad the correspodig feasible schedule, obtaied by EDF, illustrated i Figure 1. Notice that the cumulative processor utilizatio, U p, is greater tha 1 (U p = 1.25), but coditio (2) is satisfied. Usig the processor demad criterio, Jeffay ad Stoe [] showed that a set of hard periodic tasks is schedulable by EDF if ad oly if, for ay iterval L 0, L L c i. (3) Based o this result, Kore ad Shasha [11] proved the followig theorem, which provides a sufficiet

3 Task Task1 Task2 Task3 Computatio Period Skip Parameter 4 3 U p 1.25 Table 1. A schedulable set of firm periodic tasks. coditio for the schedulability of a set of skippable periodic tasks uder EDF. τ1 τ 2 τ skip skip skip 12 skip Figure 1. Feasible schedule for tasks i Table 1. Theorem 1 A set of firm (i.e., skippable) periodic tasks is schedulable if L 0 L 24 D(i,[0,L]) (4) ( ) L L where D(i,[0, L]) = c i. (5) s i I their theorem, D(i,[0, L]) represets the effective time demaded by the periodic task set over the iterval [0, L]. Kore ad Shasha [11] also proposed two olie schedulig algorithms, Red Tasks Oly ad Blue Whe Possible, to hadle tasks with skips uder EDF. Red Tasks Oly RTO always skips blue istaces, whereas red oes are scheduled accordig to EDF. Blue Whe Possible BWP is more flexible tha RTO ad schedules blue istaces wheever there are o ready red jobs to execute. Red istaces are scheduled accordig to EDF. It is easy to fid examples to demostrate that BWP improves upo RTO i the sese that it is able to schedule task sets that RTO caot schedule. I the geeral case, the above algorithms are ot optimal, but they are optimal uder a special task model, called the deeply-red model. Defiitio 1 A system is deeply-red if all tasks are sychroously activated ad the first s i 1 istaces of every task τ i are red. I the same paper, Kore ad Shasha showed that the worst case for a periodic skippable task set occurs whe tasks are deeply-red. This meas that, if a task set is schedulable uder the deeply-red model, it is also schedulable without this assumptio. For this reaso, all results i this paper will be proved usig the deeply-red assumptio. Buttazzo ad Caccamo [6] defied the equivalet processor utilizatio, Up, for a set of firm periodic tasks to be U p = max L 0 { i D(i,[0,L]) L }. (6) They the used the remaiig (uiformly distributed) capacity, 1 Up, to schedule aperiodic tasks. However, the equivalet processor utilizatio over-estimates the system utilizatio, ad there is some processor capacity that is ot reclaimed because it has a graular distributio [6]. The total spare capacity i the system ca be calculated ad it is give by U spare = 1 U f irm = 1 U p + c i s i. (7) This spare capacity ca be categorized ito two portios U sa ad U sh. A portio of this capacity U sa = 1 Up is uiformly distributed ad is assiged to the aperiodic server. The remaiig portio of the spare capacity is o-uiformly distributed amog may holes [6], ad ca be calculated as U sh =U spare U sa. Table 2 shows a set of skippable tasks that ca be feasibly scheduled uder the RTO model with Up = Notice that the capacity distributed amog the holes i the schedule accouts for 27 percet of the processor utilizatio. Beig able to reclaim more tha a quarter of the processor capacity ca result i marked reductios i respose times for aperiodic tasks. I this paper, we provide a mechaism for idetifyig the spare capacity that is irregularly spaced ad for usig this capacity to improve respose times of aperiodic tasks. We cocetrate o the RTO schedulig approach ad defer work o BWP schedulig to a future paper The Mechaism Usig the basic results o firm periodic task schedulig, we address the feasibility aalysis of hybrid task sets, cosistig of firm periodic tasks ad soft aperiodic requests. I order to miimize aperiodic respose times, aperiodic tasks are hadled by the Spare mechaism. Spare builds upo the algorithm [8]; o-uiformly distributed spare capacities (holes) for a give firm periodic task set are calculated offlie ad placed i the global capacity queue of the server. Before proceedig further, we provide a outlie of the mechaism.

4 Task Task1 Task2 Computatio 2 2 Period 3 5 Skip Parameter 2 2 U p 1.07 Up 0.8 U sa = 1 Up 0.2 U spare 0.47 U sh 0.27 Table 2. Illustratig the existece of holes. The capacity sharig mechaism () works i cojuctio with the Costat Badwidth Server (CBS) [2]. CBS provides isolatio betwee tasks i a system; each task is allocated a badwidth ad a server to esure that it does ot use more tha the allotted badwidth. was proposed as a approach to hadlig overrus i systems executig periodic tasks while preservig isolatio. The primary motivatio for capacity sharig was the observatio that oly a few istaces of a task execute for the worst-case duratio ad reservig resources usig the worst-case cosumptio is expesive. advocates a resource budget based o the badwidth allocated to each task; whe a task exceeds the allocated budget, residual capacities from jobs that fiished before their budgets expired ca be utilized to hadle the overru. was proposed for periodic task sets with hard deadlies; if U s the processor utilizatio, the uused badwidth, 1 U p, ca be assiged to a aperiodic task server. A global capacity queue, or a queue, is used to keep track of the available excess capacity. The algorithm is specified by the followig rules: 1. Each CBS server S i is characterized by the curret remaiig budget c i ad by a ordered pair (Q i,t i ), where Q i is the maximum budget ad T i is the period. The ratio U i = Q i /T i is the server badwidth. At each istat, a fixed deadlie d i,k is associated with the server. At the begiig d i,0 = 0, i. 2. Each task istace, τ i, j with release time r i, j, hadled by server S i is assiged a dyamic deadlie equal to the curret server deadlie d i,k. 3. A server S i is said to be active at time t if there are pedig istaces. A server is said to be idle at time t if it is ot active. 4. Whe a task istace τ i, j arrives ad the server is idle, the server geerates a ew deadlie d i,k = max(r i, j,d i,k 1 ) + T i ad c i is recharged to the maximum value Q i. 5. Whe a task istace, τ i, j, arrives ad the server is active the request is queued with other pedig jobs accordig to a give (arbitrary) disciplie. 6. Wheever istace τ i, j is scheduled for executio, the server S i uses the capacity c q i the queue (if there is oe) with the earliest deadlie d q, such that d q d i,k, otherwise its ow capacity c i is used. 7. Wheever job τ i, j executes for δ time uits, the used budget c q or c i is decreased by δ. Whe c q becomes zero, it is deleted from the queue ad the ext capacity i the queue with deadlie less tha or equal to d i,k ca be used. 8. Whe the server is active ad c i becomes zero, the server budget is recharged at the maximum value Q i ad a ew server deadlie is geerated as d i,k = d i,k 1 + T i. 9. Whe a task istace fiishes, the ext pedig istace, if ay, is served usig the curret budget ad deadlie. If there are o pedig jobs, the server becomes idle, the residual capacity c i > 0 (if ay) is iserted i the queue with deadlie equal to the server deadlie, ad c i is set equal to zero.. Wheever the processor becomes idle for a iterval of time, the capacity c q (if it exists) with the earliest deadlie i the queue is decreased by util the queue becomes empty. was origially developed for hard realtime task sets; our ew work pushes the evelope further by dealig with firm real-time tasks. The holes that occur i a schedule are idetified ad added (at the appropriate time) to the queue ad ca be utilized by all tasks, especially aperiodic tasks. 3. Spare I this sectio, we formally describe the Spare techique assumig that each task, τ i, is hadled by a dedicated CBS server, S i, ruig o a uiprocessor system. Spare capacities for a give task set are idetified offlie ad added to the global capacity queue olie. Holes are idetified over the meta-hyperperiod for the give task set. Defiitio 2 Give a set Γ = {T i (,c i,s i )} of periodic tasks that allow skips, the meta hyper-period, H = lcm(p 1 s 1, p 2 s 2,..., p s ), is defied as the period after which the task schedule repeats itself. As a example, the meta hyper-period of the task set i Table 2 is 30.

5 Algorithm 1 LOCATE HOLES AND DETERMINE CAPACITIES Require: A set Γ = {T i (,c i,s i )} of firm periodic tasks with a equivalet processor utilizatio Up. k 0 d 1 0 H lcm(c 1 s 2,c 2 s 2,...,c s ) {meta hyper-period} for all tasks T i do c i = c i /Up ; iflate c i to c i to iclude uiformly distributed portio of spare capacity ed for Schedule the task set Γ = {T i (,c i,s i)} usig the EDF-RTO scheduler for all time t where (t is a task skip-deadlie) ad (t H) do E k (t A[0,t]) Up k 1 j=0 E j ; amout of a hole r k d k 1 ; hole release time d k t ; hole deadlie Add hole (E k,r k,d k ) to the hole capacity list k k+ 1 ed for 3.1. A Algorithm to Locate Holes Defiitio 3 The total activity duratio i a iterval [t 1,t 2 ] is defied as t2 A[t 1,t 2 ] = t 1 f(t)dt where { 1 processor is busy at t f(t) =. 0 otherwise Remark. It is trivial to observe that A[t 1,t 2 ] t 2 t 1. Moreover, sice D(i,[0, L]) is the effective time demad for a firm periodic task T i, whe a task set is schedulable, we must have the activity duratio over ay time iterval greater tha the effective time demad over that iterval. I effect, we ca restate Theorem 1 as: a set of firm periodic tasks is schedulable if L 0 L A[0,L] D(i,[0,L]) (8) Defiitio 4 A time istat t is called a skip deadlie if it is the deadlie for a task istace that is skipped. The algorithm 1 to locate holes i the schedule first iflates the utilizatio of the task set by the factor 1/U p. Note that a fractio, U sa = 1 U p, of the processor capacity is uiformly distributed ad ca be reclaimed simply by usig a aperiodic task server of badwidth U sa. Thus, iflatig the executio times accouts for the kow uiformly distributed spare capacity U sa = 1 U p. The task schedule after this iflatio gives us oly the o-uiformly distributed spare capacities, i.e., holes, which are idetified i the secod for loop. Spare capacities (the holes) are calculated at every skip deadlie i Algorithm 1 ad are characterized by the three-tuple (E k,r k,d k ) with E k beig the capacity, r k the release time ad d k the hole deadlie. Capacities ca also be calculated ad placed at every task deadlie. The algorithm has a complexity of O(H) where H is the meta hyperperiod ad represets the umber of tasks. I the ext sectio, we will first describe how the idetified holes ca be reused i the RTO model. Followig the descriptio, we will formally prove that the capacities idetified by Algorithm 1 are ideed holes ad that the schedulability of the periodic task set is preserved Schedulig with the RTO model I the RTO model, all the blue istaces are rejected. Sice all blue istaces are skipped uiformly the task schedule repeats every meta hyper-period. The extra capacities are calculated offlie accordig to Algorithm 1. τ skip skip skip skip skip τ 2 Hole Capacities Periodic executio skip skip skip Aperiodic badwidth (iflatio amout) Figure 2. Task set (Table 2) scheduled usig the RTO model with iflated computatio times. Figure 2 shows the hole capacities for the task set i Table 2. Holes are idetified at every skip deadlie. The hole capacity at t = 6 is calculated as E 0 = (L A[0,L]) Up = (6 5) 0.8 = 0.8 sice A[0,6] = 5 ad Up = 0.8. Similarly, the hole capacity at t = is E 1 = ( 7.5) = 1.2. The hole capacity E 0 is assiged a deadlie d 0 = 6, ad released at time 0, while E 1 is assiged a dead

6 lie d 1 = ad released at time 6. The hole capacities for the etire meta hyper-period are calculated offlie. They are released olie accordig to Algorithm 2. Algorithm 2 HOLE CAPACITY RELEASE k 0 loop t = curret time() if r k = t mod H the Isert (E k, ( t H H) + r k, ( t H H) + d k ) ito the global capacity queue. k (k+ 1) mod listsize ed if ed loop It is importat to ote that holes correspod to idle itervals i the task schedule with iflated executio times; however, idetifyig holes makes it extremely efficiet to exploit spare capacity i the system this approach is far better tha backgroud executio. The keystoe for this work o exploitig holes is to trasform backgroud time ito reserved badwidth by reclaimig resources. I fact, each CBS server is able to reclaim badwidth by cosumig spare capacity while preservig its ow budget. A formal discussio of this ituitio follows Theorems ad Proofs Theorem 2 Give a set Γ = {T i (,c i,s i )} of firm periodic tasks with a equivalet processor utilizatio factor U p 1, the iflated task set Γ = {T i (,c i,s i)} where c i = c i /U s schedulable. Proof. We eed to prove that L 0 : L where D(i,[0, L]) = ( L D(i,[0,L]) ) L c i. s i Sice c i = c i/u p, alteratively, we eed to prove that L 0 : L ( ( L ) ) L c i s i Up. By the defiitio of U p (Equatio (6)), we have U p L ( ( ( L ( L L s i L s i ) c i L ) ) c i U p ). Theorem 3 Algorithm 1 preserves the aperiodic badwidth, i.e., U sa = 1 U p over ay time iterval [t 1,t 2 ]. Proof. We will cosider three cases. Case 1: Processor is fully occupied durig iterval [t 1,t 2 ] Algorithm 1 assumes that the time betwee [t 1,t 2 ] is divided ito discrete time uits such that each uit resembles Figure 3. 1 Up U p Figure 3. A sigle uit of time. Hece for every = t 2 t 1, U sa is available as aperiodic badwidth. This however icreases the computatio for task T i to c i /U p; the schedulability for which is proved i Theorem 2. Case 2: Processor is idle durig iterval [t 1,t 2 ] I Algorithm 1, whe hole capacity is calculated at every skip deadlie, oly a fractio, U p, of it is idetified. The remaiig spare capacity is the aperiodic badwidth U sa = 1 U p. Therefore, for ay idle iterval, [t 1,t 2 ], the aperiodic badwidth is coserved. Case 3: Processor is partially busy durig iterval [t 1,t 2 ] This case is a combiatio of Case 1 ad Case 2. Sice the theorem holds for Case 1 ad Case 2, it holds for this case. Theorem 4 Additio of hole capacities does ot affect the schedulability of the origial task set Γ = {T i (,c i,s i )}. Proof. We eed to prove L : D(i,[0,L])+(1 Up)L+ E k (9) k,d k L where D(i,[0,L]) is the effective time demaded by T i (,c i,s i ), (1 U p)l is the total aperiodic badwidth i [0,L] ad E k is the hole capacity with deadlie d k. By takig the (1 U p)l term to the left-had side ad i (9) ad the dividig throughout by U p, we eed to show L D(i,[0,L]) U p + k,d k L E k Up. ()

7 Algorithm 1 uses the task set with iflated computatio times, Γ. However, sice Up 1, Γ is schedulable. From (8), we have: ( ) } L L A[0, L] c i s i Up { = D(i,[0,L]) Up. Normalized aperiodic respose time S U * p = 0.90, U sh = 0.12, U spare = 0.22 Usig the above iequality i (), to prove the theorem, we eed to show that (L A[0,L]) U p k,dk L E k. This, however, follows directly from Algorithm 1 whe L is a skip deadlie because E k,dk =L = (L A[0,L]) U p j,d j <L E j. If L is ot a skip deadlie, let L be the greatest skip deadlie such that L < L (L ca be 0.) The, we have (L A[0,L ]) U p = k,d k L E k. (11) Rewritig L A[0,L] as (L L ) A[L,L] + L A[0,L ], we eed to show that ((L L ) A[L,L]+ L A[0,L ]) Up k,dk <L E k. Usig (11), we simply eed to prove that (L L ) A[L,L] 0. This is trivial because the activity over a time iterval caot exceed the legth of the iterval. 4. Experimetal Results Spare has bee simulated usig RTSIM [1] to evaluate the performace of the proposed techique. I this sectio, we preset the results of our experimets. To evaluate the performace improvemet, the Spare algorithm is compared agaist the algorithm that utilizes oly uiformly distributed portio of space capacity. Periodic tasks are hadled usig the RTO schedulig policy. The six experimets described i this sectio ca be grouped ito two sets. The first set shows the performace of the algorithms as a fuctio of the aperiodic load, for three differet values of hole capacity (U sh ). The secod set of experimets tests the sesitivity of the algorithms to the average computatio time of aperiodic requests. The performace of the algorithms were measured by computig the average aperiodic respose time as a fuctio of the mea aperiodic load. Each aperiodic respose time has bee ormalized with respect to the average aperiodic computatio time. Thus, a value of 5 o the y-axis actually meas a average respose time five times loger tha the task computatio time; a value of 1 correspods to the miimum achievable respose time. The results have bee averaged over 20 rus, each of duratio 1,000,000 time uits. The 98% cofidece iterval is tight (but ot plotted) ad demostrates the accuracy of the simulatios. Executio Figure 4. Performace results of simulatio 1. times of aperiodic requests were chose from a uiform distributio over a predefied iterval, whereas their iter-arrival times were geerated accordig to a expoetial distributio, with the mea computed to impose a specific aperiodic load ρ a. The periodic task set cosists of five periodic tasks with U p = 0.90 ad differet hole capacities, U sh. The objective of the experimets is to measure the improvemet i the respose time of aperiodic tasks whe usig Spare. We use the mechaism to queue holes that result from skips ad our itetio is ot to model early completios that motivated the iitial developmet of [8] Varyig Aperiodic Load The first set of experimets icludes three simulatios which show the performace of the algorithms as a fuctio of the aperiodic load for low, medium ad high values of U sh. Executio times of aperiodic requests were chose to be uiformly distributed i the iterval [2, ]. Periods, computatio times, ad skip parameters of the tasks for every simulatio are show i Table 3. Notice that the value of U sh is icreased from the first to the third simulatio which meas that more istaces are skipped i the secod ad third experimet. The equivalet processor utilizatio, U p, is kept costat at 0.90 for all three experimets ad thus the aperiodic server has a fixed badwidth U sa = 1 U p = 0.. Figure 4 shows the results of the first experimet, with U sh = 0.12, i which very few periodic istaces are skipped ad icludes a periodic hard task. U spare represets the total spare capacity, which is U sa +U sh. As the reader ca see, the Spare algorithm outperforms for values of ρ a i the rage [0.08,0.22]. This rage is approximately equal to U sh. For values of ρ a outside this rage the aperiodic respose behavior for both the algorithms are similar. The aperiodic respose time uder the algorithm grows at a moderate pace after a iitial spurt sice aperiodic requests cotiue to be serviced durig the holes with deadlies periodically postpoed

8 Simulatio # Task Task1 Task2 Task3 Task4 Task5 Computatio I Period Skip Parameter Computatio II Period Skip Parameter Computatio III Period Skip Parameter Table 3. Parameters: first set of simulatios. U * p = 0.90, U sh = 0.20, U spare = 0.30 U * p = 0.90, U sh = 0.27, U spare = 0.38, ACT = [5-] 35 S 16 S Normalized aperiodic respose time Normalized aperiodic respose time Figure 5. Performace results of simulatio 2. Figure 7. Performace results of simulatio 4. U * p = 0.90, U sh = 0.27, U spare = 0.38 U * p = 0.90, U sh = 0.27, U spare = 0.38, ACT = [15-20] 25 S 16 S Normalized aperiodic respose time 15 Normalized aperiodic respose time Figure 6. Performace results of simulatio 3. Figure 8. Performace results of simulatio 5.

9 16 14 S U * p = 0.90, U h = 0.27, U s max = 0.38, ACT = [25-30] for both ad Spare. The aperiodic tasks saturate all capacity ad this leads to the covergece i performace. Normalized aperiodic respose time Figure 9. Performace results of simulatio 6. accordig to CBS rules. Figure 5 refers to the secod experimet, i which U sh = More periodic istaces are skipped which results i a lower aperiodic respose whe compared to the first experimet, as aperiodic load remais idetical. Spare improves aperiodic respose time for values of ρ a i the rage [0.08,0.28]. This rage is higher tha the first experimet sice U sh = 0.20 > Agai, the performace of both algorithms is see to be similar for values outside this rage. The results of the third experimet is show i Figure 6. I this case, U sh = 0.27, the highest value i all the experimets. The improvemet i aperiodic respose time occurs over a larger rage [ ]; thus the Spare algorithm performs best for higher values of hole capacities. The Spare algorithm works by locatig hole capacities ad placig them i the global capacity queue. Thus the aperiodic server ca prevet uecessary deadlie postpoemets while executig i the hole regio, eablig a better aperiodic respose time. Accordig to the first set of experimets, three distict zoes ca be idetified i terms of achieved performace: 1. ρ a U sa : I this zoe, aperiodic respose of ad Spare are idetical. If aperiodic load is less tha U sa, ca be as competitive as Spare is schedulig aperiodic tasks. The hole capacity, U sh, is ot utilized much i this traffic zoe. 2. U sa < ρ a U spare : Here Spare outperforms. The workload is cosistetly greater tha U sa ad therefore the holes are ecessary ad Spare is able to serve aperiodic tasks better. 3. ρ a > U spare : Aperiodic respose is idetical agai. Whe the aperiodic workload exceeds U sa + U sh, the respose times icrease rapidly Moreover, we observe that whe the umber of skips icreases, the gap betwee ad Spare decreases. This might seem to be couterituitive but the reaso is straightforward: whe more jobs are skipped, U sa also icreases ad reclamatio of holes is overshadowed by the icrease i U sa. Thus the gap betwee ad Spare reduces whe we icrease the skips. I our experimet, whe all tasks have a skip factor of 2, the effect is almost the same as doublig the period of the tasks. A lot of spare capacity is uiformly distributed ad ca be reclaimed quite easily by a simple server Aalysis of Sesitivity to Aperiodic Computatio Times To test the sesitivity of the algorithms with respect to the legth of aperiodic tasks, three simulatios were carried out usig task sets with short, medium, ad log aperiodic computatio times (ACT). I particular, executio times of aperiodic requests were chose from the uiform distributio over the iterval [5,] for Simulatio 4, [15,20] for Simulatio 5, ad [25,30] for Simulatio 6. To limit the total umber of graphs, the periodic tasks used were oly those used i Simulatio 3. The results of these experimets are show i Figures 7, 8, ad 9 respectively. The improvemet i performace achieved by Spare over is more sigificat whe aperiodic requests have short computatio times. As the ACTs become loger, the performace of Spare teds to be similar to the oe achieved by. This is because, for log aperiodic tasks, advacig the positio of small slack itervals i the schedule does ot create a great impact o the respose times. 5. Coclusio I this paper, we preseted a algorithm for reclaimig holes that are created whe schedulig tasks that allow skips. The holes are idetified offlie, ad are itroduced olie as capacities i the [8] queue. These holes are the utilized for miimizig the respose times of aperiodic tasks. To the best of our kowledge, this is the first work that describes a techique for reclaimig holes i a firm periodic real time eviromet. Idetifyig ad reclaimig holes trasforms backgroud capacity ito reserved capacity; this trasformatio results i improved behavior of Costat Badwidth Servers. I this work, we push the evelope o the

10 applicatios of the techique by utilizig it i a firm real time eviromet. We have cosidered the RTO (Red Tasks Oly) strategy for schedulig periodic tasks. Extesive experimetatio reveals that our approach ca sigificatly improve respose times for aperiodic tasks. I our future ivestigatios, we will seek for algorithms that are more efficiet, especially extedig our approach to the BWP strategy. We would also like to improve the idle time hadlig of the algorithm to obtai eve better respose times. So far, we have dealt with oly the deeply-red task model ad we eed to geeralize our approach to systems with arbitrary offsets because our approach requires us to kow how holes are distributed i a schedule ad this distributio will chage whe tasks have differet offsets. Fially, we ote that we ca improve respose times by advacig hole deadlies (usig techiques similar to TB* [7]) ad pla to study this extesio ad preset our results i a subsequet publicatio. Refereces [1] Real-time system simulator (RTSIM). [2] L. Abei ad G. Buttazzo. Itegratig multimedia applicatios i hard real-time systems. I Proceedigs of the IEEE Real-Time Systems Symposium, December [3] N.C. Audsley, A. Burs, M. Richardso, K. Tidell, ad A. Welligs. Applyig ew schedulig theory to static priority preemptive schedulig. Software Egieerig Joural, 5(8): , September [4] T.P. Baker. Stack-based schedulig of real-time processes. The Joural of Real-Time Systems, 1(3):67 0, [5] G. Berat ad A. Burs. Combiig (, m)-hard deadlies ad dual priority schedulig. I Proceedigs of the IEEE Real-Time Systems Symposium, pages 46 57, December [6] G. Buttazzo ad M. Caccamo. Miimizig aperiodic respose times i a firm real-time eviromet. IEEE Trasactios o Software Egieerig, 25(1):22 32, Jauary/February [7] G.C. Buttazzo ad F. Sesii. Optimal deadlie assigmet for schedulig soft aperiodic tasks i hard real-time eviromets. IEEE Trasactios o Computers, 48():35 52, October [8] M. Caccamo, G. Buttazzo, ad L. Sha. Capacity sharig for overru cotrol. I Proceedigs of the IEEE Real-Time Systems Symposium, December [9] T.M. Ghazalie ad T.P. Baker. Aperiodic servers i a deadlie schedulig eviromet. Real-Time Systems, pages 21 36, [] K. Jeffay ad D. Stoe. Accoutig for iterrupt hadlig costs i dyamic priority task systems. I Proceedigs of the 14th IEEE Real-Time Systems Symposium, pages , December [11] G. Kore ad D. Sasha. Skip-over: Algorithms ad complexity for overloaded systems that allow skips. I Proceedigs of the IEEE Real-Time Systems Symposium, pages 1 117, December [12] J.P. Lehoczky ad S.R. Ramos-Thuel. A optimal algorithm for schedulig soft-aperiodic tasks i fixed-priority preemptive systems. I Proceedigs of the 13th IEEE Real-Time Systems Symposium, pages 1 123, December [13] J.P. Lehoczky, L. Sha, ad J.K. Strosider. Ehaced aperiodic resposiveess i hard real-time eviromets. I Proceedigs of the IEEE Real-Time Systems Symposium, pages , December [14] C.L. Liu ad J.W. Laylad. Schedulig algorithms for multiprogrammig i a hard real-time eviromet. Joural of the ACM, 1(20):40 61, [15] D. Liu, X.S. Hu, M.D. Lemmo, ad Q. Lig. Firm real-time system schedulig based o a ovel QoS costrait. I Proceedigs of the IEEE Real-Time Systems Symposium, December [16] A. Marchad ad M. Silly-Chetto. QoS ad aperiodic tasks schedulig for real-time liux applicatios. I Proceedigs of the 6th RTL Workshop, November [17] M.Hamdaoui ad P. Ramaatha. A dyamic priority assigmet techique for streams with (m, k)- firm deadlies. IEEE Trasactios o Computers, 12(44): , December [18] L. Sha, R. Rajkumar, ad J.P. Lehoczky. Priority iheritace protocols: A approach to real-time sychroizatio. IEEE Trasactios o Computers, 9(39): , September [19] B. Sprut, L. Sha, ad J. P. Lehoczky. Aperiodic schedulig for hard real-time system. Joural of Real-Time Systems, (1):27 60, [20] M. Spuri ad G.C. Buttazzo. Efficiet aperiodic service uder earliest deadlie schedulig. I Proceedigs of the IEEE Real-Time System Symposium, pages 2 11, December [21] M. Spuri ad G.C. Buttazzo. Schedulig aperiodic tasks i dyamic priority systems. Real-Time Systems, 2():179 2, 1996.