Period and Deadline Selection for Schedulability in RealTime Systems


 Florence Holt
 1 years ago
 Views:
Transcription
1 Perod and Deadlne Selecton for Schedulablty n RealTme 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 realtme 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 realtme 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 realtme schedulng challenge. To solve the perod and deadlne selecton problem, ths artcle dentfes a feasble peroddeadlne 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 realtme 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 realtme 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 realtme 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 realtme 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 WndowConstraned Schedulng (DWCS) algorthm [30], the skpover algorthms [20], and the algorthms for the weakly hard realtme systems [6]. In other works such as the mprecse computaton model [15] and rewardbased 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 realtme systems based on task perod adjustments (e.g., [22]). The works n [26], [27] and [7] solve the perod selecton problem for the earlestdeadlne frst (EDF), ratemonotone (RM), and fxedprorty 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 worstcase 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 realtme 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 pseudopolynomal 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 fxedprorty 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 nonperodc). 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 realtme 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 realtme task set where task deadlnes are less than or equal to task perods and where task deadlnes are pecewse frstorder 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 worstcase. 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 5tuple: (C,, mn, max, D ), = 1,..., N, where C s the worstcase 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 frstorder 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 nondecreasng 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 nondecreasng 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 realtme 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 peroddeadlne 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 worstcase 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 dscretetme 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 overprovsonng. 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 statebased selftrggerng systems. Selftrggerng can be vewed as a closedloop 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 openloop. 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 peroddeadlne 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, nonconvex, and noncontnuous 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 peroddeadlne 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 nondecreasng 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 noncomparable. 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 peroddeadlne combnatons (shown by the rghtslanted 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 nondecreasng 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 leftslanted 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 leftslanted 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 peroddeadlne 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 lefthand 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 lefthand 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 KuhnTucker (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 lefthand 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 userdefned 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 forloop 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 nonlnear 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 peroddeadlne 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 frstorder 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 userdefned 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 peroddeadlne 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 lefthand 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 peroddeadlne combnaton for 13 task sets. 6 Conclusons In ths paper, we proposed a more general and realstc realtme task model where each task deadlne s a functon of the correspondng perod. Ths task model facltates the feasblty analyss of the realtme 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 peroddeadlne combnaton. Our heurstc mnmzes the search regon and teratvely fnds a feasble peroddeadlne 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 worstcase. As future work, we ntend on () obtanng more expermental results, partcularly usng benchmarks derved from realworld applcatons, and () mplementng the proposed heurstc on a realtme operatng system such as the S.Ha.R.K. kernel [18]. References [1] K. Albers and F. Slomka. Effcent feasblty analyss for realtme systems wth edf schedulng. In Proc. Desgn, Automaton and Test n Europe, pages , [2] H. Aydn, R. Melhem, D. Mosse, and P. Alvarez. Optmal rewardbased schedulng for perodc realtme tasks. In Proc. RealTme 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 realtme 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, realtme tasks on one processor. RealTme Systems, 2(4): , Nov [6] G. Bernat, A. Burns, and A. Llamosí. Weakly hard realtme systems. Transactons on Computers, 50(4): , Apr [7] E. Bn and M. D. Natale. Optmal task rate selecton n fxed prorty systems. In Proc. RealTme Systems Symposum, pages , [8] G. Buttazzo and L. Aben. Adaptve workload management through elastc schedulng. RealTme Systems, 23(1 2):7 24, July [9] G. Buttazzo and L. Aben. Smooth rate adaptaton through mpedance control. In Proc. Euromcro Conf. on RealTme Systems, pages 3 10, [10] G. Buttazzo, G. Lpar, and L. Aben. Elastc task model for adaptve rate control. In Proc. RealTme 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 RealTme Systems, pages , [13] M. Caccamo, G. Buttazzo, and L. Sha. Handlng executon overruns n hard realtme control systems. Transactons on Computers, 51(7): , July [14] T. Chantem, X. Hu, and M. Lemmon. Generalzed elastc schedulng. In Proc. RealTme 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 polynomaltme approxmaton scheme for feasblty analyss n statcprorty systems wth bounded relatve deadlnes. In Proc. Internatonal Conference on RealTme Systems, pages , [18] P. Ga, L. Aben, M. Gorg, and G. Buttazzo. A new kernel approach for modular realtme systems development. In Proc. Euromcro Conf. on RealTme 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. Skpover: Algorthms and complexty for overloaded systems that allow skps. In Proc. RealTme Systems Symposum, pages , [21] X. Koutsoukos, R. Tekumalla, B. Natarajan, and C. Lu. Hybrd supervsory utlzaton control of realtme systems. In Proc. RealTme & Embedded Technology and Applcatons Symposum, pages 12 21, [22] T.W. Kuo and A. Mok. Load adjustment n adaptve realtme systems. In Proc. RealTme Systems Sympsum, pages , [23] C. Lu and J. Layland. Schedulng algorthms for multprogrammng n a hard realtme envronment. Journal of the ACM, 20(1):46 61, Jan [24] J. W. S. Lu. RealTme Systems. PrentceHall, NJ, [25] A. Masrur, S. Drossler, and G. Farber. Improvements n polynomaltme 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 realtme systems. In Proc. RealTme Systems Symposum, pages , [27] D. Seto, J. Lehoczky, L. Sha, and K. Shn. On task schedulablty n realtme control systems. In Proc. RealTme Systems Symposum, pages 13 21, [28] C.S. Shh and J. W. Lu. Statedependent deadlne schedulng. In Proc. RealTme Systems Symposum, pages 3 14, [29] X. Wang and M. Lemmon. Selftrggered feedback control systems wth fntegan l2 stablty. Accepted to Transactons on Automatc Control, [30] R. West and C. Poellabauer. Analyss of a wndowconstraned scheduler for realtme and besteffort packet streams. In Proc. RealTme Systems Symposum, pages , 2000.
Rate Monotonic (RM) Disadvantages of cyclic. TDDB47 Real Time Systems. Lecture 2: RM & EDF. Prioritybased scheduling. States of a process
Dsadvantages of cyclc TDDB47 Real Tme Systems Manual scheduler constructon Cannot deal wth any runtme changes What happens f we add a task to the set? RealTme Systems Laboratory Department of Computer
More information1 Approximation Algorithms
CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons
More informationbenefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).
REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or
More informationRealTime Process Scheduling
RealTme Process Schedulng ktw@cse.ntu.edu.tw (RealTme and Embedded Systems Laboratory) Independent Process Schedulng Processes share nothng but CPU Papers for dscussons: C.L. Lu and James. W. Layland,
More informationProject Networks With MixedTime Constraints
Project Networs Wth MxedTme Constrants L Caccetta and B Wattananon Western Australan Centre of Excellence n Industral Optmsaton (WACEIO) Curtn Unversty of Technology GPO Box U1987 Perth Western Australa
More informationSolution of Algebraic and Transcendental Equations
CHAPTER Soluton of Algerac and Transcendental Equatons. INTRODUCTION One of the most common prolem encountered n engneerng analyss s that gven a functon f (, fnd the values of for whch f ( = 0. The soluton
More informationA Computer Technique for Solving LP Problems with Bounded Variables
Dhaka Unv. J. Sc. 60(2): 163168, 2012 (July) A Computer Technque for Solvng LP Problems wth Bounded Varables S. M. Atqur Rahman Chowdhury * and Sanwar Uddn Ahmad Department of Mathematcs; Unversty of
More informationSupport Vector Machines
Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.
More informationThe Development of Web Log Mining Based on ImproveKMeans Clustering Analysis
The Development of Web Log Mnng Based on ImproveKMeans Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.
More informationInstitute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic
Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange
More informationChapter 4 ECONOMIC DISPATCH AND UNIT COMMITMENT
Chapter 4 ECOOMIC DISATCH AD UIT COMMITMET ITRODUCTIO A power system has several power plants. Each power plant has several generatng unts. At any pont of tme, the total load n the system s met by the
More informationThe OC Curve of Attribute Acceptance Plans
The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4
More informationLuby s Alg. for Maximal Independent Sets using Pairwise Independence
Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent
More informationModule 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..
More informationRecurrence. 1 Definitions and main statements
Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.
More informationFORMAL ANALYSIS FOR REALTIME SCHEDULING
FORMAL ANALYSIS FOR REALTIME SCHEDULING Bruno Dutertre and Vctora Stavrdou, SRI Internatonal, Menlo Park, CA Introducton In modern avoncs archtectures, applcaton software ncreasngly reles on servces provded
More informationPowerofTwo Policies for Single Warehouse MultiRetailer Inventory Systems with Order Frequency Discounts
Powerofwo Polces for Sngle Warehouse MultRetaler Inventory Systems wth Order Frequency Dscounts José A. Ventura Pennsylvana State Unversty (USA) Yale. Herer echnon Israel Insttute of echnology (Israel)
More informationEnabling P2P Oneview Multiparty Video Conferencing
Enablng P2P Onevew Multparty Vdeo Conferencng Yongxang Zhao, Yong Lu, Changja Chen, and JanYn Zhang Abstract MultParty Vdeo Conferencng (MPVC) facltates realtme group nteracton between users. Whle P2P
More informationCommunication Networks II Contents
8 / 1  Communcaton Networs II (Görg)  www.comnets.unbremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP
More informationAn Alternative Way to Measure Private Equity Performance
An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate
More informationSchedulability Bound of Weighted Round Robin Schedulers for Hard RealTime Systems
Schedulablty Bound of Weghted Round Robn Schedulers for Hard RealTme Systems Janja Wu, JyhCharn Lu, and We Zhao Department of Computer Scence, Texas A&M Unversty {janjaw, lu, zhao}@cs.tamu.edu Abstract
More informationAvailabilityBased Path Selection and Network Vulnerability Assessment
AvalabltyBased Path Selecton and Network Vulnerablty Assessment Song Yang, Stojan Trajanovsk and Fernando A. Kupers Delft Unversty of Technology, The Netherlands {S.Yang, S.Trajanovsk, F.A.Kupers}@tudelft.nl
More informationFisher Markets and Convex Programs
Fsher Markets and Convex Programs Nkhl R. Devanur 1 Introducton Convex programmng dualty s usually stated n ts most general form, wth convex objectve functons and convex constrants. (The book by Boyd and
More informationOn the Optimal Control of a Cascade of HydroElectric Power Stations
On the Optmal Control of a Cascade of HydroElectrc Power Statons M.C.M. Guedes a, A.F. Rbero a, G.V. Smrnov b and S. Vlela c a Department of Mathematcs, School of Scences, Unversty of Porto, Portugal;
More informationThe Greedy Method. Introduction. 0/1 Knapsack Problem
The Greedy Method Introducton We have completed data structures. We now are gong to look at algorthm desgn methods. Often we are lookng at optmzaton problems whose performance s exponental. For an optmzaton
More informationPSYCHOLOGICAL RESEARCH (PYC 304C) Lecture 12
14 The Chsquared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed
More informationAn MILP model for planning of batch plants operating in a campaignmode
An MILP model for plannng of batch plants operatng n a campagnmode Yanna Fumero Insttuto de Desarrollo y Dseño CONICET UTN yfumero@santafeconcet.gov.ar Gabrela Corsano Insttuto de Desarrollo y Dseño
More informationParticle Swarm Optimization for Scheduling to Minimize Tardiness Penalty and Power Cost
Partcle Swarm Optmzaton for Schedulng to Mnmze Tardness Penalty and Power Cost KueTang Fang and Bertrand M.T. Ln Department of Informaton and Fnance Management Insttute of Informaton Management Natonal
More informationA hybrid global optimization algorithm based on parallel chaos optimization and outlook algorithm
Avalable onlne www.ocpr.com Journal of Chemcal and Pharmaceutcal Research, 2014, 6(7):18841889 Research Artcle ISSN : 09757384 CODEN(USA) : JCPRC5 A hybrd global optmzaton algorthm based on parallel
More information8 Algorithm for Binary Searching in Trees
8 Algorthm for Bnary Searchng n Trees In ths secton we present our algorthm for bnary searchng n trees. A crucal observaton employed by the algorthm s that ths problem can be effcently solved when the
More informationLogical Development Of Vogel s Approximation Method (LDVAM): An Approach To Find Basic Feasible Solution Of Transportation Problem
INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME, ISSUE, FEBRUARY ISSN 77866 Logcal Development Of Vogel s Approxmaton Method (LD An Approach To Fnd Basc Feasble Soluton Of Transportaton
More informationResearch Article Enhanced TwoStep Method via Relaxed Order of αsatisfactory Degrees for Fuzzy Multiobjective Optimization
Hndaw Publshng Corporaton Mathematcal Problems n Engneerng Artcle ID 867836 pages http://dxdoorg/055/204/867836 Research Artcle Enhanced TwoStep Method va Relaxed Order of αsatsfactory Degrees for Fuzzy
More informationU.C. Berkeley CS270: Algorithms Lecture 4 Professor Vazirani and Professor Rao Jan 27,2011 Lecturer: Umesh Vazirani Last revised February 10, 2012
U.C. Berkeley CS270: Algorthms Lecture 4 Professor Vazran and Professor Rao Jan 27,2011 Lecturer: Umesh Vazran Last revsed February 10, 2012 Lecture 4 1 The multplcatve weghts update method The multplcatve
More information9.1 The Cumulative Sum Control Chart
Learnng Objectves 9.1 The Cumulatve Sum Control Chart 9.1.1 Basc Prncples: Cusum Control Chart for Montorng the Process Mean If s the target for the process mean, then the cumulatve sum control chart s
More information"Research Note" APPLICATION OF CHARGE SIMULATION METHOD TO ELECTRIC FIELD CALCULATION IN THE POWER CABLES *
Iranan Journal of Scence & Technology, Transacton B, Engneerng, ol. 30, No. B6, 789794 rnted n The Islamc Republc of Iran, 006 Shraz Unversty "Research Note" ALICATION OF CHARGE SIMULATION METHOD TO ELECTRIC
More information2008/8. An integrated model for warehouse and inventory planning. Géraldine Strack and Yves Pochet
2008/8 An ntegrated model for warehouse and nventory plannng Géraldne Strack and Yves Pochet CORE Voe du Roman Pays 34 B1348 LouvanlaNeuve, Belgum. Tel (32 10) 47 43 04 Fax (32 10) 47 43 01 Emal: corestatlbrary@uclouvan.be
More informationMultivariate EWMA Control Chart
Multvarate EWMA Control Chart Summary The Multvarate EWMA Control Chart procedure creates control charts for two or more numerc varables. Examnng the varables n a multvarate sense s extremely mportant
More informationII. PROBABILITY OF AN EVENT
II. PROBABILITY OF AN EVENT As ndcated above, probablty s a quantfcaton, or a mathematcal model, of a random experment. Ths quantfcaton s a measure of the lkelhood that a gven event wll occur when the
More informationLecture 3. 1 Largest singular value The Behavior of Algorithms in Practice 2/14/2
18.409 The Behavor of Algorthms n Practce 2/14/2 Lecturer: Dan Spelman Lecture 3 Scrbe: Arvnd Sankar 1 Largest sngular value In order to bound the condton number, we need an upper bound on the largest
More informationData Broadcast on a MultiSystem Heterogeneous Overlayed Wireless Network *
JOURNAL OF INFORMATION SCIENCE AND ENGINEERING 24, 819840 (2008) Data Broadcast on a MultSystem Heterogeneous Overlayed Wreless Network * Department of Computer Scence Natonal Chao Tung Unversty Hsnchu,
More informationNasdaq Iceland Bond Indices 01 April 2015
Nasdaq Iceland Bond Indces 01 Aprl 2015 Fxed duraton Indces Introducton Nasdaq Iceland (the Exchange) began calculatng ts current bond ndces n the begnnng of 2005. They were a response to recent changes
More informationDEFINING %COMPLETE IN MICROSOFT PROJECT
CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMISP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,
More informationCalculation of Sampling Weights
Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a twostage stratfed cluster desgn. 1 The frst stage conssted of a sample
More informationJ. Parallel Distrib. Comput.
J. Parallel Dstrb. Comput. 71 (2011) 62 76 Contents lsts avalable at ScenceDrect J. Parallel Dstrb. Comput. journal homepage: www.elsever.com/locate/jpdc Optmzng server placement n dstrbuted systems n
More informationMoment of a force about a point and about an axis
3. STATICS O RIGID BODIES In the precedng chapter t was assumed that each of the bodes consdered could be treated as a sngle partcle. Such a vew, however, s not always possble, and a body, n general, should
More informationThis article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and
Ths artcle appeared n a journal publshed by Elsever. The attached copy s furnshed to the author for nternal noncommercal research and educaton use, ncludng for nstructon at the authors nsttuton and sharng
More informationSoftware project management with GAs
Informaton Scences 177 (27) 238 241 www.elsever.com/locate/ns Software project management wth GAs Enrque Alba *, J. Francsco Chcano Unversty of Málaga, Grupo GISUM, Departamento de Lenguajes y Cencas de
More informationEDFVD Scheduling of MixedCriticality Systems with Degraded Quality Guarantees
EDFVD Schedulng of MxedCrtcalty Systems wth Degraded Qualty Guarantees D Lu 1, Jelena Spasc 1 Gang Chen 2, Nan Guan 3, Songran Lu 2, Todor Stefanov 1, Wang Y 2, 4 1 Leden Unversty, The Netherlands 2
More informationPOLYSA: A Polynomial Algorithm for Nonbinary Constraint Satisfaction Problems with and
POLYSA: A Polynomal Algorthm for Nonbnary Constrant Satsfacton Problems wth and Mguel A. Saldo, Federco Barber Dpto. Sstemas Informátcos y Computacón Unversdad Poltécnca de Valenca, Camno de Vera s/n
More informationLecture 7 March 20, 2002
MIT 8.996: Topc n TCS: Internet Research Problems Sprng 2002 Lecture 7 March 20, 2002 Lecturer: Bran Dean Global Load Balancng Scrbe: John Kogel, Ben Leong In today s lecture, we dscuss global load balancng
More informationJoint Scheduling of Processing and Shuffle Phases in MapReduce Systems
Jont Schedulng of Processng and Shuffle Phases n MapReduce Systems Fangfe Chen, Mural Kodalam, T. V. Lakshman Department of Computer Scence and Engneerng, The Penn State Unversty Bell Laboratores, AlcatelLucent
More information8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by
6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng
More informationNonlinear data mapping by neural networks
Nonlnear data mappng by neural networks R.P.W. Dun Delft Unversty of Technology, Netherlands Abstract A revew s gven of the use of neural networks for nonlnear mappng of hgh dmensonal data on lower dmensonal
More informationOptimal portfolios using Linear Programming models
Optmal portfolos usng Lnear Programmng models Chrstos Papahrstodoulou Mälardalen Unversty, Västerås, Sweden Abstract The classcal Quadratc Programmng formulaton of the well known portfolo selecton problem,
More informationHow Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence
1 st Internatonal Symposum on Imprecse Probabltes and Ther Applcatons, Ghent, Belgum, 29 June 2 July 1999 How Sets of Coherent Probabltes May Serve as Models for Degrees of Incoherence Mar J. Schervsh
More informationIdentifying Workloads in Mixed Applications
, pp.395400 http://dx.do.org/0.4257/astl.203.29.8 Identfyng Workloads n Mxed Applcatons Jeong Seok Oh, Hyo Jung Bang, Yong Do Cho, Insttute of Gas Safety R&D, Korea Gas Safety Corporaton, ShghungSh,
More informationRESEARCH ON DUALSHAKER SINE VIBRATION CONTROL. Yaoqi FENG 1, Hanping QIU 1. China Academy of Space Technology (CAST) yaoqi.feng@yahoo.
ICSV4 Carns Australa 9 July, 007 RESEARCH ON DUALSHAKER SINE VIBRATION CONTROL Yaoq FENG, Hanpng QIU Dynamc Test Laboratory, BISEE Chna Academy of Space Technology (CAST) yaoq.feng@yahoo.com Abstract
More informationDynamic Constrained Economic/Emission Dispatch Scheduling Using Neural Network
Dynamc Constraned Economc/Emsson Dspatch Schedulng Usng Neural Network Fard BENHAMIDA 1, Rachd BELHACHEM 1 1 Department of Electrcal Engneerng, IRECOM Laboratory, Unversty of Djllal Labes, 220 00, Sd Bel
More information1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP)
6.3 /  Communcaton Networks II (Görg) SS20  www.comnets.unbremen.de Communcaton Networks II Contents. Fundamentals of probablty theory 2. Emergence of communcaton traffc 3. Stochastc & Markovan Processes
More informationBERNSTEIN POLYNOMIALS
OnLne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful
More informationSurvey on Virtual Machine Placement Techniques in Cloud Computing Environment
Survey on Vrtual Machne Placement Technques n Cloud Computng Envronment Rajeev Kumar Gupta and R. K. Paterya Department of Computer Scence & Engneerng, MANIT, Bhopal, Inda ABSTRACT In tradtonal data center
More informationSection B9: Zener Diodes
Secton B9: Zener Dodes When we frst talked about practcal dodes, t was mentoned that a parameter assocated wth the dode n the reverse bas regon was the breakdown voltage, BR, also known as the peaknverse
More informationA generalized hierarchical fair service curve algorithm for high network utilization and linksharing
Computer Networks 43 (2003) 669 694 www.elsever.com/locate/comnet A generalzed herarchcal far servce curve algorthm for hgh network utlzaton and lnksharng Khyun Pyun *, Junehwa Song, HeungKyu Lee Department
More informationAN APPOINTMENT ORDER OUTPATIENT SCHEDULING SYSTEM THAT IMPROVES OUTPATIENT EXPERIENCE
AN APPOINTMENT ORDER OUTPATIENT SCHEDULING SYSTEM THAT IMPROVES OUTPATIENT EXPERIENCE YuL Huang Industral Engneerng Department New Mexco State Unversty Las Cruces, New Mexco 88003, U.S.A. Abstract Patent
More informationA GENERAL APPROACH FOR SECURITY MONITORING AND PREVENTIVE CONTROL OF NETWORKS WITH LARGE WIND POWER PRODUCTION
A GENERAL APPROACH FOR SECURITY MONITORING AND PREVENTIVE CONTROL OF NETWORKS WITH LARGE WIND POWER PRODUCTION Helena Vasconcelos INESC Porto hvasconcelos@nescportopt J N Fdalgo INESC Porto and FEUP jfdalgo@nescportopt
More informationSensitivity Analysis in a Generic MultiAttribute Decision Support System
Senstvty Analyss n a Generc MultAttrbute Decson Support System Sxto RíosInsua, Antono Jménez and Alfonso Mateos Department of Artfcal Intellgence, Madrd Techncal Unversty Campus de Montegancedo s/n,
More informationFeature selection for intrusion detection. Slobodan Petrović NISlab, Gjøvik University College
Feature selecton for ntruson detecton Slobodan Petrovć NISlab, Gjøvk Unversty College Contents The feature selecton problem Intruson detecton Traffc features relevant for IDS The CFS measure The mrmr measure
More informationA FASTER EXTERNAL SORTING ALGORITHM USING NO ADDITIONAL DISK SPACE
47 A FASTER EXTERAL SORTIG ALGORITHM USIG O ADDITIOAL DISK SPACE Md. Rafqul Islam +, Mohd. oor Md. Sap ++, Md. Sumon Sarker +, Sk. Razbul Islam + + Computer Scence and Engneerng Dscplne, Khulna Unversty,
More informationActivity Scheduling for CostTime Investment Optimization in Project Management
PROJECT MANAGEMENT 4 th Internatonal Conference on Industral Engneerng and Industral Management XIV Congreso de Ingenería de Organzacón Donosta San Sebastán, September 8 th 10 th 010 Actvty Schedulng
More informationGenerator WarmUp Characteristics
NO. REV. NO. : ; ~ Generator WarmUp Characterstcs PAGE OF Ths document descrbes the warmup process of the SNAP27 Generator Assembly after the sotope capsule s nserted. Several nqures have recently been
More informationDynamic Resource Allocation and Power Management in Virtualized Data Centers
Dynamc Resource Allocaton and Power Management n Vrtualzed Data Centers Rahul Urgaonkar, Ulas C. Kozat, Ken Igarash, Mchael J. Neely urgaonka@usc.edu, {kozat, garash}@docomolabsusa.com, mjneely@usc.edu
More informationGraph Theory and Cayley s Formula
Graph Theory and Cayley s Formula Chad Casarotto August 10, 2006 Contents 1 Introducton 1 2 Bascs and Defntons 1 Cayley s Formula 4 4 Prüfer Encodng A Forest of Trees 7 1 Introducton In ths paper, I wll
More informationThe eigenvalue derivatives of linear damped systems
Control and Cybernetcs vol. 32 (2003) No. 4 The egenvalue dervatves of lnear damped systems by YeongJeu Sun Department of Electrcal Engneerng IShou Unversty Kaohsung, Tawan 840, R.O.C emal: yjsun@su.edu.tw
More informationAnts Can Schedule Software Projects
Ants Can Schedule Software Proects Broderck Crawford 1,2, Rcardo Soto 1,3, Frankln Johnson 4, and Erc Monfroy 5 1 Pontfca Unversdad Católca de Valparaíso, Chle FrstName.Name@ucv.cl 2 Unversdad Fns Terrae,
More informationEE201 Circuit Theory I 2015 Spring. Dr. Yılmaz KALKAN
EE201 Crcut Theory I 2015 Sprng Dr. Yılmaz KALKAN 1. Basc Concepts (Chapter 1 of Nlsson  3 Hrs.) Introducton, Current and Voltage, Power and Energy 2. Basc Laws (Chapter 2&3 of Nlsson  6 Hrs.) Voltage
More informationTo manage leave, meeting institutional requirements and treating individual staff members fairly and consistently.
Corporate Polces & Procedures Human Resources  Document CPP216 Leave Management Frst Produced: Current Verson: Past Revsons: Revew Cycle: Apples From: 09/09/09 26/10/12 09/09/09 3 years Immedately Authorsaton:
More informationIMPACT ANALYSIS OF A CELLULAR PHONE
4 th ASA & μeta Internatonal Conference IMPACT AALYSIS OF A CELLULAR PHOE We Lu, 2 Hongy L Bejng FEAonlne Engneerng Co.,Ltd. Bejng, Chna ABSTRACT Drop test smulaton plays an mportant role n nvestgatng
More informationJoint Resource Allocation and BaseStation. Assignment for the Downlink in CDMA Networks
Jont Resource Allocaton and BaseStaton 1 Assgnment for the Downlnk n CDMA Networks Jang Won Lee, Rav R. Mazumdar, and Ness B. Shroff School of Electrcal and Computer Engneerng Purdue Unversty West Lafayette,
More informationA linear recurrence sequence of composite numbers
LMS J Comput Math 15 (2012) 360 373 C 2012 Author do:101112/s1461157012001143 A lnear recurrence sequence of composte numbers Jonas Šurys Abstract We prove that for each postve nteger k n the range 2 k
More informationA New Task Scheduling Algorithm Based on Improved Genetic Algorithm
A New Task Schedulng Algorthm Based on Improved Genetc Algorthm n Cloud Computng Envronment Congcong Xong, Long Feng, Lxan Chen A New Task Schedulng Algorthm Based on Improved Genetc Algorthm n Cloud Computng
More informationPower Low Modified Dual Priority in Hard Real Time Systems with Resource Requirements
Power Low Modfed Dual Prorty n Hard Real Tme Systems wth Resource Requrements M.Angels Moncusí, Alex Arenas {amoncus,aarenas}@etse.urv.es Dpt d'engnyera Informàtca Matemàtques Unverstat Rovra Vrgl Campus
More informationCalculating the high frequency transmission line parameters of power cables
< ' Calculatng the hgh frequency transmsson lne parameters of power cables Authors: Dr. John Dcknson, Laboratory Servces Manager, N 0 RW E B Communcatons Mr. Peter J. Ncholson, Project Assgnment Manager,
More informationA ReplicationBased and Fault Tolerant Allocation Algorithm for Cloud Computing
A ReplcatonBased and Fault Tolerant Allocaton Algorthm for Cloud Computng Tork Altameem Dept of Computer Scence, RCC, Kng Saud Unversty, PO Box: 28095 11437 RyadhSaud Araba Abstract The very large nfrastructure
More informationGeneral Auction Mechanism for Search Advertising
General Aucton Mechansm for Search Advertsng Gagan Aggarwal S. Muthukrshnan Dávd Pál Martn Pál Keywords game theory, onlne auctons, stable matchngs ABSTRACT Internet search advertsng s often sold by an
More informationSolutions to the exam in SF2862, June 2009
Solutons to the exam n SF86, June 009 Exercse 1. Ths s a determnstc perodcrevew nventory model. Let n = the number of consdered wees,.e. n = 4 n ths exercse, and r = the demand at wee,.e. r 1 = r = r
More informationA Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy Scurve Regression
Novel Methodology of Workng Captal Management for Large Publc Constructons by Usng Fuzzy Scurve Regresson ChengWu Chen, Morrs H. L. Wang and TngYa Hseh Department of Cvl Engneerng, Natonal Central Unversty,
More informationPAS: A Packet Accounting System to Limit the Effects of DoS & DDoS. Debish Fesehaye & Klara Naherstedt University of IllinoisUrbana Champaign
PAS: A Packet Accountng System to Lmt the Effects of DoS & DDoS Debsh Fesehaye & Klara Naherstedt Unversty of IllnosUrbana Champagn DoS and DDoS DDoS attacks are ncreasng threats to our dgtal world. Exstng
More informationb) The mean of the fitted (predicted) values of Y is equal to the mean of the Y values: c) The residuals of the regression line sum up to zero: = ei
Mathematcal Propertes of the Least Squares Regresson The least squares regresson lne obeys certan mathematcal propertes whch are useful to know n practce. The followng propertes can be establshed algebracally:
More informationAnswer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy
4.02 Quz Solutons Fall 2004 MultpleChoce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multplechoce questons. For each queston, only one of the answers s correct.
More informationForecasting the Demand of Emergency Supplies: Based on the CBR Theory and BP Neural Network
700 Proceedngs of the 8th Internatonal Conference on Innovaton & Management Forecastng the Demand of Emergency Supples: Based on the CBR Theory and BP Neural Network Fu Deqang, Lu Yun, L Changbng School
More information1 Example 1: Axisaligned rectangles
COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton
More informationFormula of Total Probability, Bayes Rule, and Applications
1 Formula of Total Probablty, Bayes Rule, and Applcatons Recall that for any event A, the par of events A and A has an ntersecton that s empty, whereas the unon A A represents the total populaton of nterest.
More informationConferencing protocols and Petri net analysis
Conferencng protocols and Petr net analyss E. ANTONIDAKIS Department of Electroncs, Technologcal Educatonal Insttute of Crete, GREECE ena@chana.tecrete.gr Abstract: Durng a computer conference, users desre
More informationWhat is Candidate Sampling
What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble
More informationTime Series Analysis in Studies of AGN Variability. Bradley M. Peterson The Ohio State University
Tme Seres Analyss n Studes of AGN Varablty Bradley M. Peterson The Oho State Unversty 1 Lnear Correlaton Degree to whch two parameters are lnearly correlated can be expressed n terms of the lnear correlaton
More informationFeasibility of Using Discriminate Pricing Schemes for Energy Trading in Smart Grid
Feasblty of Usng Dscrmnate Prcng Schemes for Energy Tradng n Smart Grd Wayes Tushar, Chau Yuen, Bo Cha, Davd B. Smth, and H. Vncent Poor Sngapore Unversty of Technology and Desgn, Sngapore 138682. Emal:
More informationANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING
ANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING Matthew J. Lberatore, Department of Management and Operatons, Vllanova Unversty, Vllanova, PA 19085, 6105194390,
More informationTHE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek
HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo
More informationDynamic Pricing for Smart Grid with Reinforcement Learning
Dynamc Prcng for Smart Grd wth Renforcement Learnng ByungGook Km, Yu Zhang, Mhaela van der Schaar, and JangWon Lee Samsung Electroncs, Suwon, Korea Department of Electrcal Engneerng, UCLA, Los Angeles,
More informationAn Analysis of Central Processor Scheduling in Multiprogrammed Computer Systems
STANCS73355 I SUSE73013 An Analyss of Central Processor Schedulng n Multprogrammed Computer Systems (Dgest Edton) by Thomas G. Prce October 1972 Techncal Report No. 57 Reproducton n whole or n part
More information