Period and Deadline Selection for Schedulability in Real-Time Systems

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1 Perod and Deadlne Selecton for Schedulablty n Real-Tme Systems Thdapat Chantem, Xaofeng Wang, M.D. Lemmon, and X. Sharon Hu Department of Computer Scence and Engneerng, Department of Electrcal Engneerng Unversty of Notre Dame Notre Dame, IN {tchantem, xwang13, lemmon, Abstract Task perod adaptatons are often used to allevate temporal overload condtons n real-tme systems. Exstng frameworks assume that only task perods are adjustable and that task deadlnes reman unchanged at all tmes. Ths paper formally ntroduces a more general real-tme task model where task deadlnes, whch are less than or equal to task perods, are functons of task perods. Ths tght couplng between task deadlnes and task perods has been dscussed n a recent work n control systems and presents a novel real-tme schedulng challenge. To solve the perod and deadlne selecton problem, ths artcle dentfes a feasble perod-deadlne combnaton and proposes a heurstc, whch teratvely adjusts task perods and deadlnes n such a way that the task set becomes schedulable. Expermental results show that the heurstc fnds a soluton to the perod and deadlne selecton problem over 73% of the tme, usng less than three search teratons. When t s unable to fnd a soluton to the problem, the heurstc requres less than 0.02s to run n the worstcase (wth at most 100 search teratons). 1 Introducton Task schedulng has long been an mportant research topc n real-tme systems, where the man requrement conssts of performng some functons correctly and on tme (.e., by some specfc deadlne). Mssng a deadlne n a hard real-tme system may lead to catastrophc consequences, such as falure to stop an automatcally controlled tran on tme [24]. Despte havng been tradtonally treated as hard realtme systems, many control systems are qute robust n the presence of certan tmng perturbatons. Generally speakng, dependng on the system state, the samplng rate of a control system can vary wthn some nterval wthout causng sgnfcant performance degradaton. Ths observaton s very useful when temporal overload stuatons occur. A real-tme system s sad to experence an overload when t cannot fnsh executng one or more tasks on tme due to resource constrants. Although robust, f too many deadlnes have been mssed or f such msses occur n a hghly unpredctable manner, a control system may no longer stablze, even f all system resources are now dedcated to t. There are two man approaches to dealng wth overloads n real-tme systems: () droppng some nstances of tasks (.e., jobs) n a controlled manner, and () ncreasng task perods, equvalently decreasng the samplng rates, n such a way that no deadlnes are mssed and the performance of the system remans acceptable. Many algorthms have been proposed to control job droppng patterns. Some examples are the (m, k) schedulng algorthms [19], the Dynamc Wndow-Constraned Schedulng (DWCS) algorthm [30], the skp-over algorthms [20], and the algorthms for the weakly hard realtme systems [6]. In other works such as the mprecse computaton model [15] and reward-based model [2], the am s to maxmze system workload, whch s assumed to be proportonal to the qualty of servce (QoS). Snce t s sometmes more sutable to execute jobs less often nstead of droppng them or allocatng fewer cycles [3], we focus on such an approach n ths paper. Many prevous works can be found on the management of overloads n real-tme systems based on task perod adjustments (e.g., [22]). The works n [26], [27] and [7] solve the perod selecton problem for the earlest-deadlne frst (EDF), ratemonotone (RM), and fxed-prorty schedulng algorthms, respectvely. Cervn et al. propose an onlne perod adjustment mechansm, whle varyng task computaton tmes s handled n [21]. In [13], Caccamo et al. consder scenaros where the worst-case task executon tmes can be large but the normal task executon tmes tend to be very small. To effcently use system resources whle avodng overruns, the method of task rate adaptaton s combned wth the use of a constant bandwdth server to guarantee hard real-tme deadlnes. Buttazzo et al. propose an opt-

2 mal perod selecton algorthm n [10] based on the elastc task model. Many extensons to the elastc task model can be found n [9, 11, 8, 12]. In terms of schedulablty tests for task sets wth deadlnes less than perods, Baruah et al. proposed an exact test wth pseudo-polynomal runnng tme [5]. For effcency, we wll use the suffcent test provded n [14]. However, there exsts other suffcent condtons for schedulablty when task deadlnes are less than task perods. For nstance, Dev proposed a set of suffcent schedulablty tests n [16]. The man dfference between ths set of tests and the one n [14] s that the former requres N checks whle the latter requres only one check. Some extensons to Dev s work nclude, but are not lmted to, an approxmate schedulablty test [1], an adaptaton to fxed-prorty systems [17], and novel feasblty tests that are shown to outperform Dev s schedulablty condtons [25]. Most prevous works on overload management assume that only task perods can change. In [28], task deadlnes vary wth tme, but the tasks do not have perods (.e., tasks are non-perodc). There has also been work on determnng the lower bound on task deadlnes usng senstvty analyss n a perodc task model [4]. However, to the best of our knowledge, there has been no work that allows task perods and deadlnes to change smultaneously. Our frst man contrbuton s the ntroducton of a more general and realstc task model where both task deadlnes and task perods can vary wthn some ntervals. The deadlne n the real-tme system sense really denotes the maxmum allowable delay that a gven task (a control task, for nstance) can tolerate. As shown by the authors n [29], dfferent samplng rates for a control system lead to dfferent acceptable maxmum delays (deadlnes). Specfcally, a hgher samplng rate means that the correspondng control task executes more often, whch, n turns, allows the system to be more tolerant to a larger delay. Conversely, a larger samplng perod could make the system more susceptble to delays and thus a smaller deadlne may be requred. In other words, the deadlne of a task s a functon of ts perod. The relatonshp between task perods and task deadlnes poses an nterestng schedulng problem, as one can no longer assume that ncreasng task perods wll always mprove schedulablty. Although t s possble to set task deadlnes to be the smallest deadlnes (specfed by the applcatons) and only vary task perods, dong so may sgnfcantly worsen schedulablty. As our second man contrbuton, we study some nterestng relatonshps between task perods and task deadlnes that wll help to solve the perod and deadlne selecton problem. We then propose an effcent heurstc that can be used to fnd a set of feasble task perods and deadlnes and allevate an overload stuaton. Our heurstc can be appled to any real-tme task set where task deadlnes are less than or equal to task perods and where task deadlnes are pecewse frst-order dfferentable functons of ther respectve perods. Based on some expermental results, our heurstc fnds a soluton to the perod and deadlne selecton problem over 73% of the tme usng less than 0.02s n the worst-case. We ntroduce the system model and some mportant assumptons n Secton 2. Secton 3 provdes a motvatng example to hghlght the mportance and usefulness of our work. We present our formal analyss and heurstc n Sectons 4. Secton 5 summarzes some expermental results and the paper concludes wth Secton 6. 2 Prelmnares In ths secton, we descrbe the system model and mportant assumptons, as well as revew some relevant schedulablty tests. We also gve a formal defnton of the perod and deadlne selecton problem. 2.1 System Model and Assumptons Our system conssts of a set of N perodc, synchronous tasks specfed by the followng 5-tuple: (C,, mn, max, D ), = 1,..., N, where C s the worst-case executon tme of task τ, and s τ s actual perod, whch must le somewhere between mn and max. The parameter mn denotes the most desrable perod of τ, as specfed by the applcaton, whereas max represents the maxmum perod beyond whch the system performance s no longer acceptable. The parameter D s the deadlne of τ, and s dependent on the actual task perod. That s, the deadlne of a task s a functon of ts perod. Specfcally, D, [mn, max ] and D s some functon that s pecewse frst-order dfferentable. The utlzaton of each task τ s defned as U = C / and denotes system resources dedcated to τ. Snce the perod of τ, = 1,..., N, can vary between mn and max, the mnmum utlzaton of τ, U mn = C /max, and ts maxmum (desred) utlzaton, U max = C /mn, are also defned, for = 1,..., N. 2.2 Schedulablty Test Throughout ths paper, we wll assume that the Earlest Deadlne Frst (EDF) schedulng algorthm [23] s used. When one or more tasks need to decrease ther perod and/or deadlne n response to ether nternal (e.g., change n samplng rate of one or more tasks n the system) or external (e.g., network traffc) factors, a schedulablty test must be used to assess whether the task set s stll schedulable. A schedulablty test may also provde some gudance on how to adjust task parameters n such a way that a feasble task set can be obtaned. Based on the assumpton that the

3 EDF schedulng algorthm s used, there exst some useful schedulablty condtons that are brefly revewed here. A necessary condton for schedulablty of any gven task set s stated n the followng lemma. Lemma 1 [14] Gven a task set Γ, let C and D be the executon tme and the deadlne of task τ, = 1,..., N, respectvely. In addton, let all tasks start at tme 0 and let the tasks n Γ be ordered n a non-decreasng order of deadlnes. Regardless of the choces of perods, any task set that s schedulable must satsfy the followng property j C D j, j = 1,..., N. (1) Snce task deadlnes can be less than or equal to perods, there exst an exact, albet complex, schedulablty test for EDF as specfed by Baruah et al [5]. Sad test s restated n the followng theorem. Theorem 1 [5] Gven a perodc task set wth C, D, and as the executon tme, deadlne, and perod of task τ, = 1,..., N, respectvely. Let D, = 1,..., N, the task set s schedulable f and only f the followng constrant s satsfed L {k + D mn(b p, H)} and k N (the set of natural numbers ncludng 0), where B p and H denote the busy perod and hyperperod, respectvely, L ( ) + 1 C. (2) Verfyng that (2) s satsfed for all L s the man source of complexty n the above schedulablty test. To reduce the complexty of the test n Theorem 1, the authors n [14] proposed the followng suffcent condton for schedulablty. Theorem 2 [14] Gven a set Γ of N tasks that satsfy Lemma 1. Let C, D, and be the executon tme, deadlne, and perod of task τ, = 1,..., N, respectvely. In addton, let the tasks n Γ be sorted n a non-decreasng order of deadlnes. The task set Γ s schedulable f where L = L ( L ) D + 1 C (3) { D2 : D 1 + T 1 D 2 mn N ( + D ) : otherwse. For completeness, we nclude another exstng suffcent condton for EDF schedulablty. Theorem 3 [24] Consder a set Γ of N tasks where C and D are the executon tme and deadlne of task τ, = 1,..., N, respectvely. The task set Γ s schedulable by the EDF polcy f C D 1. (4) We wll use some of these schedulablty condtons n Secton Problem Defnton We consder the followng problem: Gven an ntally nfeasble set Γ of N real-tme tasks where the perod of task τ must le somewhere between [mn, max ], and the deadlne D of τ s some functon of ts perod, determne a perod-deadlne combnaton (, D ), = 1,..., N, such that the task set Γ becomes schedulable. In other words, we wsh to fnd (, D ), = 1,..., N, such that ( ) + 1 C L (5) mn for = 1, 2,, N (6) max for = 1, 2,, N (7) where L s defned as n Theorem 1, C s the worst-case executon tme of τ, and both mn and max are specfed by the applcatons under consderaton. The constrant n (5) ensures the schedulablty of the task set. The constrants n (6) and (7) bound the perod of τ, = 1,..., N, to ensure performance. 3 Motvatons In control systems, an advantage n usng the tradtonal perodc task model where task deadlnes are fxed s that these systems can be treated as dscrete-tme systems for whch there exsts a varety of mature controller synthess methods. However, when the perodc task model s used, task perods and deadlnes are often chosen conservatvely to guarantee stablty. Ths leads to wasted resources and perhaps even system over-provsonng. For these reasons, there has been a recent movement n the control system communty to nvestgate alternatve approaches to the perodc task model. The work n [29] s such an example. Each task determnes ts next release tme based on the current system state as sampled by the current job. Ths type of control systems s known as state-based self-trggerng systems. Selftrggerng can be vewed as a closed-loop form of releasng tasks for executon, whereas the tradtonal perodc task

4 Table 1. Task set for motvatng example Task C mn max D τ T 1e T 1, T 1 [0.5, 3.5] τ T 2e T 2, T 2 [0.5, 3.5] model s consdered open-loop. Snce each control task s aware of ts system state, t can adjust ts perod and deadlne n such a way that only the requred system resource s requested. More precsely, wth a small perod, a task s executed relatvely often and the system s thus more tolerant to delays, permttng the task deadlne to be larger (e.g., perhaps almost as large as the task perod tself). On the other hand, when the task perod s large, the system s more susceptble to dsturbances, requrng that the task deadlne be smaller (compared to the task perod) to reduce jtters. To understand how the deadlne as a functon of the perod affects schedulablty, let us consder a smple task set, whch conssts of two dentcal tasks whose attrbutes are shown n Table 1. The deadlne of each task can be computed as shown n the last column of Table 1 (all unts are n mllseconds). Fgure 1 plots the task deadlnes as a functon of task perods where the vertcal dotted lnes lmt the acceptable perod range for the example tasks. Intally, the task set s not schedulable wth T 1 = T 2 = 0.5ms, snce the ntal deadlnes D 1 = D 2 = 0.303ms and the aggregate executon tme requred s 0.36ms. If we smply set T 1 = T 2 = 3.5ms, whch s the maxmum allowable perods, then the correspondng deadlnes wll be D 1 = D 2 = 0.106ms. The task set s, agan, not schedulable and one may wrongly conclude that the task set cannot be made feasble. However, there exsts many feasble perod-deadlne combnatons. For example, when T 1 = T 2 = 1ms and D 1 = D 2 = 0.368ms, the task set s schedulable. In the tradtonal perodc task model, snce task deadlnes are consdered fxed, system desgners must use the smallest possble deadlnes to ensure that, gven a specfc range of task perods, the system wll always meet the mnmum performance requrements. For the above example, the smallest deadlne for both tasks s 0.106ms, whch means that the task set can never be made schedulable usng exstng technques. It s not dffcult to see n ths example that the task deadlnes can be set to 0.36ms for the task set to be feasble, regardless of the resultant perods. In general, however, both task perods and task deadlnes must be consdered smultaneously, snce dfferent tasks may have dfferent tmng requrements. 4 Perod and Deadlne Selecton As shown n the prevous secton, snce a task s deadlne s a functon of ts perod, adjustng the perod affects both the correspondng deadlne and the schedulablty of the en- D1 = D2 (ms) T 1mn = T 2mn T 1 = T 2 (ms) T 1max = T 2max Fgure 1. Deadlne as a functon of perod tre task set. Due to the condton n (5), the problem defned n Secton 2.3 s nonlnear, non-convex, and non-contnuous snce L,, and D, = 1,..., N, are varables, and because of the floor functon. Solvng the above problem drectly usng a nonlnear solver s neffcent and t cannot be guaranteed that a soluton wll be found, even f one exsts. For these reasons, we propose usng a heurstc whch uses a number of fast suffcent condtons to fnd a soluton. In a nutshell, the heurstc starts by performng some smple schedulablty tests to determne a feasble perod-deadlne combnaton. Such tests also serve to elmnate some nfeasble perod and deadlne values should they fal to dentfy a feasble task set. The heurstc then uses ths knowledge to conduct an effcent search process. 4.1 Identfyng Infeasble Regons Usng Smple Tests We now descrbe our dea of usng the smple tests n more detal. We frst determne the mnmum and maxmum deadlnes, D mn and D max, respectvely, for each task τ, = 1,..., N. The maxmum deadlne of τ, D max can drectly be solved by fndng the maxmum of D. (Recall that the maxmum of a functon can be obtaned by takng ts dervatve and subsequently fndng the root(s) of sad dervatve.) The correspondng perod value s denoted T Dmax, = 1,..., N. To determne the lower bound on the deadlne of a task τ, = 1,..., N, we would deally use Lemma 1. However, Lemma 1 requres that tasks be sorted n a non-decreasng order of deadlnes. Snce a task deadlne s a varable to be determned, we cannot drectly use Lemma 1 to compute the mnmum deadlne. Instead, let D be the smallest deadlne of task τ,.e., D D( ), [mn, max ], = 1,..., N. We say that task τ domnates task τ j (denoted by τ τ j ) f D > D jmax. Otherwse, we say

5 that τ and τ j are non-comparable. Usng the above domnance defnton, a partal order can be bult for a gven set of tasks. It s easy to see that Lemma 1 holds true for tasks wth deadlnes as varables f we sort the tasks usng the partal order establshed above. For example, consder a smple task set consstng of task τ j and τ k. If τ j τ k then D kmn = C k and D jmn = C k + C j. In general, for a task τ, D mn = max{ds} + C k, where DS s the set of deadlnes of tasks that are domnated by τ. Snce D mn set n ths way s a lower bound on the mnmum task deadlne for task τ, = 1,..., N, we can elmnate some nfeasble perod-deadlne combnatons (shown by the rght-slanted pattern n Fgure 2). The task perod that corresponds to when the task deadlne s D mn = 1,..., N. Once we have found the mnmum and maxmum deadlnes for each task n the task set, we can apply a seres of effcent schedulablty tests to avod searchng for a soluton, f possble. We start wth the suffcent condton from Theorem 3 usng D max, = 1,..., N, as the task deadlnes. The followng lemma helps to explan why only D max, = 1,..., N, need to be consdered when applyng Theorem 3 on the current task set. s referred to as T Dmn, Lemma 2 Gven a set Γ of N tasks. Let C and D be the executon tme and deadlne of task τ, = 1,..., N, respectvely. If the schedulablty condton from Theorem 3 s not satsfed for D max, = 1,..., N, then t s not satsfed for any D < D max, = 1,..., N. Proof: If the task set Γ fals the schedulablty test n Theorem 3, then C > 1. (8) D max Usng any D < D max, = 1,..., N, would yeld C D > C D max > 1. (9) Therefore, the lemma holds. Note that f the condton from Theorem 3 s satsfed for D max, = 1,..., N, then we have dentfed a feasble soluton. Otherwse, we apply the schedulablty test from Theorem 2 for a specal pont (T Dmax, D max ), = 1,..., N (see Fgure 2). (To use Theorem 2, we order the tasks n a non-decreasng order of deadlnes usng D = D max and = T Dmax, = 1,..., N, whenever L needs to be determned). We choose to test the pont (T Dmax, D max ), = 1,..., N, because f the task set s not schedulable at ths pont accordng to Theorem 2, then t s not schedulable for any (, D ), T Dmax and D D max, = 1,..., N. The followng theorem proves ths clam and explans why the left-slanted regon n Fgure 2 can be elmnated from further consderaton f the Deadlne (ms) (T Dmax, D max ) Perod (ms) (T Dmn, D mn ) Fgure 2. Infeasble schedulablty regons pont (T Dmax, D max ), = 1,..., N, are found not to be schedulable accordng to Theorem 2. Lemma 3 Gven a set Γ of N tasks. Let C and D be the executon tme and deadlne of task τ, = 1,..., N, respectvely. Let be the perod obtaned when D = D max, = 1,..., N. If the condton n Theorem 2 s not satsfed for (, D ), = 1,..., N, then t s not satsfed for any (, D ), D D,, = 1,..., N. Proof: Snce the task set s not schedulable at (, D ), = 1,..., N, we have L < ( ) + 1 C. (10) In addton, snce D D and, = 1,..., N, L < < ( ) + 1 C ( ) + 1 C, (11) snce L D > L D and <, = 1,..., N. Fnally, as L > L, where L = D 2 f D 1 + T 1 D 2 and L = mn N ( + D ) otherwse, L < ( ) + 1 C. (12) Observe that we use the schedulablty condtons n Theorems 2 and 3 n conjuncton to one another. Ths s because a task set that s feasble accordng to one of the aforementoned schedulablty condtons s not necessarly feasble accordng to the other (and vce versa).

6 The left-slanted regon n Fgure 2 s a result of Lemma 3 and can be elmnated from further consderaton. (Note that snce we test the pont (T Dmax, D max ) usng the schedulablty test from Theorem 2, such a pont may n fact be feasble. However, to exactly determne schedulablty, the condton n Theorem 1 needs to be satsfed and s often too tme consumng to be used durng an overload stuaton.) The area wth no pattern, however, ndcates the remanng search regon. Note that for a specfc perod, = 1,..., N, any deadlne 0 < D < D s also acceptable from the system performance pont of vew. However, snce usng D wll worsen schedulablty, we only consder D. All the smple tests descrbed thus far appear as part of our heurstc and are shown n Algorthm 1. Lnes 1 4 show the frst smple test dscussed n Lemma 2. The second smple test from Lemma 3 s shown n Lnes 5 9. Fnally, Lnes show that we perform an addtonal schedulablty test for (T Dmn, D mn ), = 1,..., N, snce ths pont has already been computed and such an addtonal test does not ncur a sgnfcant amount of addtonal overhead. 4.2 Effcently Conductng the Search Process If all the aforementoned smple tests fal, we wll have to search along the unpatterned regon of Fgure 2 to fnd a feasble perod-deadlne combnaton, (, D ), = 1,..., N (Algorthm 2). Snce the man source of complexty of the problem defned n Secton 2.3 s that (5) must be satsfed for all possble values of L, the search process wll nstead use the schedulablty test from Theorem 2. In other words, the problem n Secton 2.3 s modfed to ( ) C L where T Dmn C (13) { D2 : D L = 1 + T 1 D 2 (14) mn ( + D ) : otherwse mn { max, T Dmn }, = 1, 2,, N (15) max { mn, T Dmax }, = 1, 2,, N (16) and T Dmax, = 1,..., N are as defned prevously. Gven an ntally nfeasble task set, one can compute the correspondng value of L as n (14). Let us frst assume that the value of L s fxed once t has been computed. To satsfy the condton n (13), observe that snce the rghthand sde of (13) can be treated as a constant, one way to solve the above problem s to adjust task perods and deadlnes such that the left-hand sde becomes as small as possble. We can express ths dea mathematcally as the follow- Algorthm 1 SmpleTests(Γ) 1: result N C D max 2: f result 1 then 3: return [ ] D max, T Dmax, for = 1,..., N 4: end f 5: compute L as n (14) usng D max and T Dmax, 1 N 6: result ( ) N L Dmax + 1 C T Dmax 7: f result L then 8: return [ D max, T Dmax ], for = 1,..., N 9: end f 10: compute L as n (14) usng D mn and T Dmn 1 N 11: result ( ) N L Dmn + 1 C T Dmn 12: f result L then, 13: return [ ] D mn, T Dmn, for = 1,..., N 14: end f 15: return ng constraned optmzaton problem. mn : s.t. : (L D ) U (17) U U max for = 1, 2,, N (18) U U mn for = 1, 2,, N (19) where D s the task deadlne functon that depends on U,.e., D D (U ) = D (C /U ). (For notatonal smplcty, D always refers to D ( ) and D to D (U ).) U = C /, U max = C max{mn,t Dmax }, and U mn = C mn{max,t Dmn }. Solvng the above constraned optmzaton problem s attractve because f a soluton to the problem n (13) (16) exsts for a fxed value of L, then we wll fnd t by solvng the above constraned optmzaton problem. Ths clam s formally stated n the followng lemma. Lemma 4 Gven an ntally nfeasble task set Γ where C, U, and D denote the executon tme, utlzaton, and deadlne (as a functon of the utlzaton) of task τ, = 1,..., N, respectvely. For a fxed value of L, f there exsts a soluton to the problem n (13) (16), t wll be found by solvng the problem defned n (17) (19). Proof: The lemma can be trvally proved by observng that (13) can be rewrtten as (L D ) U L C. (20) The constraned optmzaton problem n (13) (16) mnmzes the left-hand sde of the above equaton. Thus, f we

7 can adjust task perods and deadlnes such that (20) s true, then the soluton to the optmzaton problem n (17) (19) wll also be a soluton to the problem n (13) (16). The followng theorem presents a globally optmal soluton to the problem n (17) (19) and hence a soluton to the problem n (13) (16), for a fxed value of L. Theorem 4 Gven the constraned optmzaton problem as specfed n (17) (19). Let U be the utlzaton of task τ, = 1,..., N. Let D be the deadlne of τ where D s a functon of the U,.e., D D (U ) = D (C /U ), and let G (U ) = (L D ) U. (21) For a fxed value of L, the soluton, U, s optmal f and only f U = argmn j ff {G (U )} (22) U U U mn U max where U s a set of U such that L D D U = 0. Proof: We prove that f U, = 1,..., N, s an optmal soluton to the constraned optmzaton problem n (17) (19), then (22) must be true by utlzng the Kuhn-Tucker (KKT) necessary condtons for optmalty for constraned optmzaton problem, whch can be wrtten n terms of the Lagrangan functon for the problem as J a (U, µ) = + (L D ) U + µ (U mn U ) λ (U U max ) (23) where µ s and λ s are Lagrange multplers, µ 0 and λ 0, = 1,..., N. The necessary condtons for the exstence of a relatve mnmum at U are, for = 1,..., N, From (24) 0 = L D U D µ + λ (24) 0 = µ (U mn U ) (25) 0 = λ (U U max ) (26) L D U D = µ λ (27) If L D U D < 0 for U [ ] U mn, U max, then µ must be 0 and λ > 0. Hence, U = U max. If L D U D > 0 for U [ ] U mn, U max then λ = 0 and µ > 0. Therefore, U = U mn. Otherwse, L D U D = 0 at least once when U [ ] U mn, U max. In such a case, we can fnd the value(s) of U by fndng all the extreme ponts n the nterval [ ] U mn, U max, whch s equvalent to solvng the equaton L D D U = 0 for U. Note that snce the KKT condtons are necessary for optmalty, we have completed the proof for ths part. Now, we prove that f U, = 1,..., N, s determned as n (22), then t s an optmal soluton to the constraned optmzaton problem n (17) (19). We start by observng that, gven a pecewse dfferentable functon G (U ), the global ], U max must e- ], U max or at the boundares,.e., at U mn or U max. Ths s ndeed captured by the expresson n (22). Fnally, snce the objectve functon n (17) can be rewrtten as mn N G (U ), mnmzng each ndvdual G (U ), mnmum of G (U ) n the nterval [ U mn ther be at one of the extreme ponts nsde [ U mn = 1,..., N, s equvalent to mnmzng (17). We use the result from the above theorem drectly n the man part of our heurstc (Lne 26 n Algorthm 2). Although the heurstc can optmally solve the problem n (17) (19) for a fxed value of L, t needs to teratvely search for a feasble task set. Ths s because the value of L may ether ncrease or decrease as D and, = 1,..., N, change. Consder two consecutve teratons h and h + 1. If the task set wth perods T (h) and deadlnes D (h), = 1,..., N, satsfes the constrants n (13) (16) gven some fxed value of L (h) and L (h+1) L (h), then the task set s guaranteed to be schedulable (as shown n the followng lemma) and the search process ends. Remark: If the left-hand sde of (17) s a convex functon, then the KKT necessary condtons for optmalty also become suffcent condtons. In such a case, a global optmal soluton to the optmzaton problem n (17) (19) for a nonfxed L can be found usng Theorem 4. Lemma 5 Gven a set Γ of N tasks, and let C,, and D be the executon tme, perod, and deadlne of task τ, = 1,..., N, respectvely. If the task set satsfes the condton n Theorem 2 for some L, then t also satsfes the condton n Theorem 2 for any L L. Proof: We have L ( L 1 L L C C ) ( ) + 1 C (28) C D C (29) C D C (30) whch clearly holds for any L L. Now, f L (h+1) < L (h), the schedulablty condton n Theorem 2 must explctly be checked (Lnes n Algorthm 2). In ths way, the heurstc wll ether return a feasble task set or contnue searchng untl the number of

8 Algorthm 2 FndFeasblePerodsDeadlnes(Γ, maxiter) 1: for each τ Γ do 2: D max max T [mn,mn ] D 3: T Dmax perod when deadlne s D max 4: D mn C 5: T Dmn perod when deadlne s D mn 6: end for 7: result SmpleTests(Γ) 8: f result then 9: return [D, ], = 1,..., N 10: end f 11: D D max, = 1,..., N 12: T Dmax, = 1,..., N 13: done false 14: ternum 0 15: whle not done do 16: ternum ternum : compute L as n (14) usng D and, 1 N 18: result ( ) N L D + 1 C 19: f result L then 20: return [D, ], = 1,..., N 21: end f 22: f ternum > maxiter then 23: done true 24: end f 25: for each τ Γ do 26: compute U as n Theorem 4 27: C U 28: determne D accordngly 29: end for 30: end whle 31: return maxmum teratons, maxiter, has been reached (Lne 22). The value maxiter s a user-defned constant and, from our experments n the next secton, can be set to some small number such as 100. The tme complexty of our heurstc s domnated by the whle loop on Lne 15 of Algorthm 2. Insde the whle loop, the most tme consumng operatons appear nsde the for-loop on Lne 25. Let U be sze of U as defned n Theorem 4 over all teratons and let O(G ) be the worstcase tme complexty requred to fnd all solutons to the equaton L D D U = 0, also from Theorem 4. The runnng tme of our heurstc s then O(maxIter N ( U + O(G ))), where N s the number of tasks n the task set. Remark: In our approach, we assume that when a task set s nfeasble, each task s equally responsble for reducng ts processor demand (f possble) to allevate the overload stuaton. In practce, however, some tasks may be more mportant than the others. As a result, a weght may be Table 2. Man results Method Number of solutons found % solutons found Fxed deadlne technque 0/80 0% Our heurstc 59/ % Table 3. Heurstc total runnng tme and number of teratons Number of task sets Runnng Tme (s) Number of teratons needed 37 (soluton found) < 0.01 < 3 13 (soluton not found) < 0.02 > 100 assocated wth each task to denote ts mportance. In such a case, our approach can be extended by factorng n the weght of each task when decdng the amount of processor demand reducton that each task should be responsble for. Specfcally, the problem formulaton n Secton 2.3 can be modfed to a constraned optmzaton problem of the form mn : s.t. : w (mn ) 2 (31) ( ) + 1 C L (32) mn for = 1, 2,, N (33) max for = 1, 2,, N (34) where w s the weght of the task τ, = 1,..., N, and all other parameters retan ther meanng as prevously defned. Clearly, the modfed problem can be too tme consumng (and perhaps too dffcult) to solve usng a nonlnear solver and thus the use of a heurstc smlar to the one presented earler s recommended. 5 Expermental Results Snce drectly solvng the perod and deadlne selecton problem n Secton 2.3 usng a commercal non-lnear solver can be very tme consumng and t cannot be guaranteed that a soluton wll be found, even f one exsts, we proposed an effcent heurstc n Secton 4. In ths secton, we evaluate the performance of our approach. Due to the lack of realstc benchmarks sutable for the ntended experment, we randomly generated 80 task sets consstng of 5 tasks each. In order to scrutnze the search aspect of the heurstc, each task set s chosen such that t s ntally nfeasble wth the guarantee that all the three smple tests from Algorthm 1 wll fal. In addton, gven a task set, there exsts at least one perod-deadlne combnaton, (T, D ), for each task τ, = 1,..., N, such that the task set can be made schedulable usng the schedulablty test from Theorem 2 (and hence satsfes the necessary and suffcent condton from Theorem 1).

9 In our experment, we use the followng deadlne functon, whose curve s representatve of the relatonshp between task perods and task deadlnes of the type of control systems under consderaton. (It s worth notng, however, that any deadlne functon can be used, as long as t s pecewse frst-order dfferentable.) D = k1 k2, (35) = 1,..., N, where k1 and k2, = 1,..., N, are some constants that depend on the specfc task under consderaton. To fnd k1 and k2, = 1,..., N, we start by randomly generatng the pont (T Dmax, D max ), = 1,..., N, whch denotes the pont where the deadlne for the task τ s maxmum. In addton, we ensure that the pont (T Dmax, D max ), = 1,..., N, s not schedulable accordng to Theorems 2 and 3. (Recall that the purpose of the experment s to test the search aspect of the heurstc and therefore we have to ensure that the smple tests fal.) Note that the deadlne functon n (35) s defned only for T Dmax, = 1,..., N, snce accordng to Lemma 3, any task set that s not schedulable for (T Dmax, D max ), = 1,..., N, wll not be schedulable for any (, D ), T Dmax, D D max, = 1,..., N. In other words, any perod < T Dmax, = 1,..., N, can be gnored by the search process. The followng steps were taken to generate a task set. Frst of all, the followng parameters were specfed: utlzaton level, maxmum hyperperod, mnmum perod, maxmum perod, precson, and maxmum number of tres. Based on these parameters, task perods are generated n such a way that the hyperperod s no larger than the maxmum hyperperod. (Ths could take a number of tres.) In our experment, we set the maxmum hyperperod, mnmum perod, and maxmum perod to 500,000, 10,000, and 40,000, respectvely. The precson was specfed to be 100, whereas the maxmum number of tres was set to 10,000. The precson denotes the mnmum ncrement n any task perod. For example, f the precson s set to 100, a task perod could be 5200, but not Fnally, for the task sets used n our experment, the range for the utlzaton level was between 0.5 and 0.7. Each task s randomly assgned an executon tme such that the total utlzaton equals that specfed by the user. No task wll have an utlzaton that s greater than half of the specfed total utlzaton. Then, each task s randomly assgned a deadlne D that ensures that N C D > 1. As a fnal step, the random task generator tests the schedulablty of the task set usng the necessary and suffcent condton from Theorem 1. If the task set s unschedulable, task deadlnes are randomly ncreased such that the new deadlne s greater than the prevous deadlne but N C D s stll greater than 1. Ths fnal step s repeated untl ether a feasble task set has been found or the maxmum number of tres has been reached. After the generaton of the aforementoned random ponts, each task set wll be assocated wth two ponts:, D max ) and (T, D ), = 1,..., N, where the former pont s not schedulable accordng to Theorem 2, but the latter pont s. Usng these two ponts, the constants k1 and k2, = 1,..., N, can be found. Fnally, the pont (T Dmax, D mn ), = 1,..., N can be determned as descrbed n Secton 4. We mplemented the heurstc proposed n the last secton n C++. The user-defned parameters maxiter was set to 100, whch means that at most 100 search teratons were (T Dmn conducted for each task set (benchmark). The proposed heurstc found a feasble perod-deadlne combnaton for 59 out of the 80 task sets. For these benchmarks, f we were to use exstng technques where task perods are fxed (whch do not drectly apply to the system model under consderaton), then no soluton wll be found for any of these task sets because these technques assume that f the task set s not schedulable for (T Dmn, D mn ), = 1,..., N, then t cannot be made feasble. (In other words, the schedulablty test from Theorem 2 s performed for (T Dmn, D mn ), = 1,..., N. Ths test s referred to as the fxed deadlne technque n Table 2.) Clearly, due to the dependency between task perods and task deadlnes, the fxed deadlne technque s shown to be too pessmstc. Table 2 summarzes the results whch show that our heurstc has an overall success rate of over 73% whle the fxed deadlne technque has a success rate of 0%. Further, snce the left-hand sde of (17) s a convex functon (due to the deadlne functon used), the solutons found by the heurstc are also optmal solutons to the optmzaton problem n (17) (19). Table 3 shows the runnng tme as well as the number of teratons needed by the proposed heurstc to fnd a soluton for each task set. For reference, the case of the fxed deadlne technque requred the runnng tme of less than 0.01s n all cases. As can be seen from the table, the task sets that the heurstc could make feasble took less than 0.01s to run wth no more than 3 search teratons. On the other hand, 100 search teratons were not enough to fnd a feasble perod-deadlne combnaton for 13 task sets. 6 Conclusons In ths paper, we proposed a more general and realstc real-tme task model where each task deadlne s a functon of the correspondng perod. Ths task model facltates the feasblty analyss of the real-tme control systems where task deadlnes reflect the maxmum allowable delays as tolerated by any gven system and vary accordng to the samplng perods. Snce exstng technques cannot adequately be used to determne schedulablty for ths

10 novel task model, we also proposed a heurstc to dentfy a schedulable perod-deadlne combnaton. Our heurstc mnmzes the search regon and teratvely fnds a feasble perod-deadlne combnaton. Expermental results show that our method of solvng the perod and deadlne selecton problem s much less pessmstc than exstng technques that consder task deadlnes to be fxed parameters; our heurstc found a soluton to the problem over 73% of the tme usng less than 3 search teratons and requrng less than 0.02s to run n the worst-case. As future work, we ntend on () obtanng more expermental results, partcularly usng benchmarks derved from real-world applcatons, and () mplementng the proposed heurstc on a real-tme operatng system such as the S.Ha.R.K. kernel [18]. References [1] K. Albers and F. Slomka. Effcent feasblty analyss for real-tme systems wth edf schedulng. In Proc. Desgn, Automaton and Test n Europe, pages , [2] H. Aydn, R. Melhem, D. Mosse, and P. Alvarez. Optmal reward-based schedulng for perodc real-tme tasks. In Proc. Real-Tme Systems Symposum, pages 79 89, [3] J. Balleul and P. A. (edtors). Specal ssue on networked control systems. Transactons on Automatc Control, 49(9): , Sept [4] P. Balbastre, I. Rpoll, and A. Crespo. Mnmum deadlne calculaton for perodc real-tme tasks n dynamc prorty systems. Transactons on Computers, 57(1):96 109, Jan [5] S. Baruah, L. Roser, and R. Howell. Algorthms and complexty concernng the preemptve schedulng of perodc, real-tme tasks on one processor. Real-Tme Systems, 2(4): , Nov [6] G. Bernat, A. Burns, and A. Llamosí. Weakly hard real-tme systems. Transactons on Computers, 50(4): , Apr [7] E. Bn and M. D. Natale. Optmal task rate selecton n fxed prorty systems. In Proc. Real-Tme Systems Symposum, pages , [8] G. Buttazzo and L. Aben. Adaptve workload management through elastc schedulng. Real-Tme Systems, 23(1 2):7 24, July [9] G. Buttazzo and L. Aben. Smooth rate adaptaton through mpedance control. In Proc. Euromcro Conf. on Real-Tme Systems, pages 3 10, [10] G. Buttazzo, G. Lpar, and L. Aben. Elastc task model for adaptve rate control. In Proc. Real-Tme Systems Symposum, pages , [11] G. Buttazzo, G. Lpar, M. Caccamo, and L. Aben. Elastc schedulng for flexble workload management. Transactons on Computers, 51(3): , Mar [12] M. Caccamo, G. Buttazzo, and L. Sha. Elastc feedback control. In Proc. Euromcro Conf. on Real-Tme Systems, pages , [13] M. Caccamo, G. Buttazzo, and L. Sha. Handlng executon overruns n hard real-tme control systems. Transactons on Computers, 51(7): , July [14] T. Chantem, X. Hu, and M. Lemmon. Generalzed elastc schedulng. In Proc. Real-Tme Systems Symposum, pages , [15] J.-Y. Chung, J. W. Lu, and K.-J. Ln. Schedulng perodc jobs that allow mprecse results. Transactons on Computers, 39(9): , Sept [16] U. Dev. An mproved schedulablty test for unprocessor perodc task systems. In Proc. Euromcro Conf. on Real- Tme Systems, pages 23 30, [17] N. Fsher and S. Baruah. A polynomal-tme approxmaton scheme for feasblty analyss n statc-prorty systems wth bounded relatve deadlnes. In Proc. Internatonal Conference on Real-Tme Systems, pages , [18] P. Ga, L. Aben, M. Gorg, and G. Buttazzo. A new kernel approach for modular real-tme systems development. In Proc. Euromcro Conf. on Real-Tme Systems, pages , [19] M. Hamdaou and P. Ramanathan. A dynamc prorty assgnment technque for streams wth (m,k)-frm deadlnes. Transactons on Computers, 44(12): , [20] G. Koren and D. Shasha. Skp-over: Algorthms and complexty for overloaded systems that allow skps. In Proc. Real-Tme Systems Symposum, pages , [21] X. Koutsoukos, R. Tekumalla, B. Natarajan, and C. Lu. Hybrd supervsory utlzaton control of real-tme systems. In Proc. Real-Tme & Embedded Technology and Applcatons Symposum, pages 12 21, [22] T.-W. Kuo and A. Mok. Load adjustment n adaptve realtme systems. In Proc. Real-Tme Systems Sympsum, pages , [23] C. Lu and J. Layland. Schedulng algorthms for multprogrammng n a hard real-tme envronment. Journal of the ACM, 20(1):46 61, Jan [24] J. W. S. Lu. Real-Tme Systems. Prentce-Hall, NJ, [25] A. Masrur, S. Drossler, and G. Farber. Improvements n polynomal-tme feasblty testng for edf. In Proc. Desgn, Automaton and Test n Europe, pages , [26] D. Seto, J. Lehoczky, and L. Sha. Task perod selecton and schedulablty n real-tme systems. In Proc. Real-Tme Systems Symposum, pages , [27] D. Seto, J. Lehoczky, L. Sha, and K. Shn. On task schedulablty n real-tme control systems. In Proc. Real-Tme Systems Symposum, pages 13 21, [28] C.-S. Shh and J. W. Lu. State-dependent deadlne schedulng. In Proc. Real-Tme Systems Symposum, pages 3 14, [29] X. Wang and M. Lemmon. Self-trggered feedback control systems wth fnte-gan l2 stablty. Accepted to Transactons on Automatc Control, [30] R. West and C. Poellabauer. Analyss of a wndowconstraned scheduler for real-tme and best-effort packet streams. In Proc. Real-Tme Systems Symposum, pages , 2000.

Rate Monotonic (RM) Disadvantages of cyclic. TDDB47 Real Time Systems. Lecture 2: RM & EDF. Priority-based scheduling. States of a process

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