Improved multiprocessor global schedulability analysis

Transcription

1 Iproved ultiprocessor global schedulability analysis Sanjoy Baruah The University of North Carolina at Chapel Hill Vincenzo Bonifaci Max-Planck Institut für Inforatik Sebastian Stiller Technische Universität Berlin Alberto Marchetti-Spaccaela Sapienza Università di Roa Abstract A new technique was recently introduced by Bonifaci et al. for the analysis of real-tie systes scheduled on ultiprocessor platfors by the global Earliest Deadline First (EDF) scheduling algorith. In this paper, this technique is generalized so that it is applicable to the schedulability analysis of realtie systes scheduled on ultiprocessor platfors by any work-conserving algorith. The resulting analysis technique is applied to obtain a new sufficient global Deadline Monotonic (DM) schedulability test. It is shown that this new test is quantitatively superior to pre-existing DM schedulability analysis tests; in addition, the degree of its deviation fro any hypothetical optial scheduler (that ay be clairvoyant) is quantitatively bounded. A new global EDF schedulability test is also proposed here that builds on the results of Bonifaci et al. This new test is shown to be less pessiistic and ore widely applicable than the earlier result was, while retaining the strong theoretical properties of the earlier result. Keywords. Global ultiprocessor scheduling; sporadic tasks; Deadline Monotonic; EDF; processor speedup factor Supported in part by NSF Grant Nos. CNS , CNS , and CNS , ARO Grant No. W911NF , and funding fro IBM, Sun Microsystes, and the Intel Corporation. Supported in part by grants EU FET project FRONTS n and by MIUR FIRB project Italy-Israel RBIN047MH9. Supported by the DFG-Research Center MATHEON.

2 1 Introduction Real-tie systes coprised of recurrent processes or tasks are often odeled using the sporadic tasks odel. In this odel, each sporadic task τ i is characterized by three paraeters a worst-case execution tie C i, a relative deadline D i, and a period T i. Such a task generates a potentially infinite sequence of jobs. Successive jobs of τ i arrive at least T i tie units apart, with each job having an execution requireent C i and a deadline D i tie units after its arrival tie. We consider here real-tie systes that can be odeled in this anner as collections of independent sporadic tasks, and that are ipleented upon a platfor coprised of several identical processors. We assue that the platfor is fully preeptive, and that it allows for global inter-processor igration. (However, each job ay execute on at ost one processor at each instant in tie.) We study the behavior of two well-known algoriths when scheduling systes of sporadic tasks upon such preeptive platfors: Earliest Deadline First [16, 20] and Deadline-Monotonic (DM) [19]. Schedulability tests for sporadic task systes. It is evident fro the definition that a sporadic task syste ay generate infinitely any different collections of jobs during different executions. In hardreal-tie systes, it ust be guaranteed prior to syste run tie that all deadlines will be et. Such guarantees are ade by schedulability tests. Let A denote a scheduling algorith. A sporadic task syste is said to be A-schedulable upon a specified platfor if A eets all deadlines when scheduling each of the potentially infinite different collections of jobs that could be generated by the sporadic task syste, upon the specified platfor. An A-schedulability test accepts as input the specifications of a sporadic task syste and a ultiprocessor platfor, and deterines whether the task syste is A-schedulable. An A-schedulability test is said to be exact if it identifies all A-schedulable systes, and sufficient if it identifies only soe A-schedulable systes. (Of course, an A-schedulability test ay never incorrectly is-identify soe syste that is not A-schedulable, as being A-schedulable.) A sufficient schedulability test that is not exact is pessiistic, but for any situations an exact schedulability test is unknown or is coputationally intractable. Fro an engineering point of view, a tractable schedulability test that is exact is ideal, while a tractable sufficient test with low pessiis is acceptable. No exact schedulability tests are known, of any coputational coplexity, for the global EDF and DM scheduling of sporadic task systes (although an exact test of unacceptably high coputational coplexity, based on brute force state-space search, has been proposed [3] for EDF, for the special case when all task paraeters are restricted to be integers). A nuber of sufficient schedulability tests have been proposed, including [1, 4, 8 10, 14, 18] see [12] for a fairly coprehensive survey. It has been observed that none of these sufficient tests doinates all the others there are schedulable systes deeed to be so by each test, which the other tests fail to identify as being schedulable. In the absence of such doination, extensive siulation experients have been perfored [5, 6, 12, 13] coparing the efficacy of the different tests in scheduling large collections of randoly-generated task systes. Although such siulations are very useful particularly in that they provide insight into the kinds of scenarios under which the different schedulability tests are superior to each other, the applicability of such siulation-based analysis is liited by the doubts raised regarding how realistic the rando tasksyste generators are. In essence, it is difficult to generalize the validity of these results beyond the kinds of workloads that are odeled by the rando task-syste generator.

3 Processor speedup factor. Processor speedup factors represent a quantitative approach towards coparing different sufficient schedulability tests. A schedulability test is defined to have a processor speedup factor f, f 1, if any task syste not deeed to be schedulable by this test upon a particular platfor is guaranteed to not be feasible schedulable by an optial clairvoyant scheduler upon a platfor in which each processor is at ost 1/f ties as fast. More forally, Definition 1 (Processor speedup factor) A schedulability test has a processor speedup factor f, f 1, if it is guaranteed that any task syste that is feasible upon a specified platfor is deeed to be schedulable by the test upon a platfor in which each processor is at least f ties as fast. Intuitively, the processor speedup factor of a schedulability test quantifies both the pessiis of the test and the non-optiality of the scheduling algorith. According to this etric, schedulability tests with saller processor speedup factors are superior to ones with larger processor speedup factors, with a processor speedup factor equal to one iplying both that the scheduling algorith is optial and that the test is exact. It is reasonable to ask whether it akes sense to bundle both the non-optiality of the scheduling algorith and the pessiis of the test into a single etric. Fro a pragatic perspective, we believe that the answer is yes for our doain of interest, which is hard-real-tie systes. Since it ust be a priori guaranteed in such hard-real-tie systes that all deadlines will be et during run-tie, a scheduling algorith is only as good as its associated schedulability test. In other words, a scheduling algorith, no atter how close to optial, cannot be used in the absence of an associated schedulability test able to guarantee that all deadlines will be et; what atters is that the cobination of scheduling algorith and schedulability test together have desirable properties. This research. Soe novel techniques were introduced in [15], for global EDF schedulability analysis of sporadic task systes. These techniques were applied to design an EDF schedulability test that is optial fro the perspective of processor speedup factor (this will be explained in greater detail in Section 2). In this paper, we generalize the analysis technique of [15] to render it applicable to algoriths other than EDF; in particular, so that it is applicable to DM 1. Also, we present soe pragatic iproveents that reduce soe the pessiis of the [15] test for EDF, and indicate how such pragatic iproveents can be done for DM scheduling as well. Organization. The reainder of this paper is organized as follows. In Section 2, we briefly review soe related work that is ost relevant to the research presented here, and that helps place our results in context. In Section 3, we present the task and syste odels used in this work, and describe techniques deand bound function and its refineent the axin or the forced-forward deand bound function that are used for quantifying the cuulative execution requireent of a task syste upon a platfor. In Section 4 we generalize the novel analysis technique fro [15] to extend its applicability to both EDF and DM scheduling (in fact, to any work-conserving scheduling discipline), and thereby obtain necessary un-schedulability conditions for both EDF and DM. In Section 5 we use these unschedulability conditions to obtain processor speedup bounds for both EDF and DM (while the derivation for 1 However unlike the results in [15] which are applicable for arbitrary sporadic task systes, we restrict our attention in this paper to systes in which each task s deadline is no larger than its period: D i T i for all tasks τ i. Such systes are called constrained sporadic task systes.

4 EDF repeats the one fro [15], the bound for DM is new). In Section 6 we present an algorith ipleenting an EDF schedulability test that strictly doinates the one fro [15], and indicate how this ipleentation can be odified to coe up with a DM schedulability test as well. 2 Prior results It has been shown [21] that any collection of independent jobs that are feasible upon a platfor coprised of speed-( ) processors is EDF-schedulable upon a platfor coprised of speed processors. It was also shown that this bound is tight: there are task systes feasible on speed- ( +ɛ) processors that are not EDF-schedulable on speed-1 processors, for arbitrary sall positive 2 1 ɛ. Although the result in [21] is very interesting and iportant fro a theoretical perspective, it does not yield an EDF-schedulability test for sporadic task systes, since it does not tell us how to deterine whether a given sporadic task syste is feasible (upon the slower platfor). Recently [15], a new sufficient schedulability test for sporadic task systes was derived, which was shown to possess a processor speedup factor of (2 1 ) on an -processor platfor2. It follows fro the result fro [21] cited above that this EDF-schedulability test is in fact speedup-optial. (By speedupoptial, we ean that no EDF-schedulability test for sporadic task systes can possibly have a processor speedup factor less than this test s.) With regards to sufficient schedulability tests for global DM scheduling 3, a test was presented in [17] for deterining whether a given sporadic task syste is global-dm schedulable upon a preeptive ultiprocessor platfor coprised coprised of unit-capacity processors. It was shown that the processor speedup factor for this global-dm schedulability test is at ost (4 1 ), when ipleented upon -processor platfors. This result was subsequently iproved [7], and extended to sporadic task systes [11] which are not constrained (i.e., in which individual tasks relative deadlines ay exceed their periods). These iproved results yield a processor speedup factor equal to 2( 1) (4 1) This approaches 2 3 (which is 3.73) as. The global EDF schedulability test of [15] is, despite its theoretical significance as a speedup-optial test, of liited applicability in the analysis of actual real-tie systes. ) First, it fails to identify as being EDF-schedulable any syste τ for which ax τi τ (C i /D i, C i /T i is larger than /(2 1) (where denotes the nuber of processors), and hence, for exaple, cannot be used to deterine the EDFschedulability of any task syste containing even ) a single task with C i /D i or C i /T i > 0.5. Even for task systes τ satisfying ax τi τ (C i /D i, C i /T i /(2 1), we will see in Section 6 that the test of [15] is overly pessiistic. One of the contributions of this paper is to flesh out the details of the test 2 [15] also presents a fully polynoial-tie approxiation schee that trades off accuracy for coputational efficiency: given any task syste and an ɛ > 0, it correctly decides, in polynoial tie, that either the syste is EDF-schedulable on speed-(2 1/ + ɛ) processors, or it is infeasible on speed-1 achines. 3 We will focus on those tests for which quantitative perforance bounds, in the for of processor speedup factors, have been deterined.

5 in [15] and thereby derive a sufficient schedulability test of wider applicability and lower pessiis, while retaining the strong theoretical properties in particular, the optial processor speedup factor. This new schedulability test can be ipleented to have a run-tie that is pseudo-polynoial in the representation of the task syste being analyzed, and is thus efficient enough to be used in practice. 3 Model and Notation In the reainder of this paper, we will let τ denote a syste of n sporadic tasks: τ = {τ 1, τ 2,... τ n }, with τ i = (C i, D i, T i ) for all i, 1 i n. Task syste τ is said to be a constrained sporadic task syste if it is guaranteed that each task τ i τ has its relative deadline paraeter no larger than its period: D i T i for all τ i τ. We restrict our attention here to constrained task systes. In order to devise schedulability tests, it is necessary to quantify the cuulative execution requireent that sequences of jobs ay place on a coputing platfor. Given a sequence of jobs J and a specified tie-interval [t a, t f ), the deand of J over [t a, t f ) is defined to be the su of the execution requireents of all jobs in J with arrival ties t a and deadlines t f. The deand bound function DBF(τ, t) of a sporadic task syste τ for an interval length t is then defined to be the largest deand by any legal sequence of jobs that ay be generated by τ over any interval of length t. The load LOAD(τ) of a sporadic task syste τ is defined as ax t>o DBF(τ, t)/t. Baker and Cirinei [2] observed that soe jobs arriving and/or having deadlines outside an interval could also contribute to the cuulative execution requireent placed on the coputing platfor within the interval. They introduced a notion that they call iniu deand. The iniu deand of a given collection of jobs in any specific tie interval is the iniu aount of execution that the sequence of jobs could require within that interval in order to eet all its deadlines 4. We illustrate the difference between deand and iniu deand by a siple exaple. Exaple 1 Consider a sequence of jobs coprised of a single job that arrives at tie-instant zero, has an execution requireent equal to 5, and a deadline at tie-instant 10. The deand of this sequence of jobs over the tie-interval [0, t) is 0 for all t < 10, and 5 for all t 10. The iniu deand of this sequence of jobs over the tie-interval [0, t) is equal to zero, for values of t 5; t 5, for t (5, 10], since the sole job ust execute for at least t 5 units over the interval if it is to eet its deadline; and five, for t > 10. The iniu deand concept is extended to sporadic tasks (and task systes) as follows. By definition, a sporadic task τ i ay generate infinitely any different collections of jobs. For a given intervallength t, τ i s axin deand is defined to be the largest iniu deand over an interval of length t, by any collection of jobs that could legally be generated by τ i. The axin deand of a sporadic task syste τ for interval-length t is defined as the su of the axin deands of the tasks in τ, each for an interval-length t. The axin load of τ is defined as the axiu value of the axin deand of τ for t, noralized by the interval length. 4 An essentially identical concept was independently introduced in [15], and called the necessary deand; a related

6 3C i 2C i C i C i σ C i σ D i T i T i + D i 2T i 2T i + D i 3T i Figure 1. Illustrating FF-DBF(τ i, t, σ). C i σ t Forced-forward deand bound function. In [15], these concepts of axin deand and axin load were generalized to be applicable to execution on speed-σ processors, for arbitrary σ > 0. We now discuss a odified for of these ideas fro [15]. Let τ i denote a sporadic task, t any positive real nuber, and σ any positive real nuber 1. The forced-forward deand bound function FF-DBF(τ i, t, σ) is defined as follows: where FF-DBF(τ i, t, σ) = def q i C i + def t q i = T i C i if r i D i C i (D i r i )σ if D i > r i D i C i σ 0 otherwise and def r i = t od T i. Inforally speaking, FF-DBF(τ i, t, σ) can be thought of as denoting the axin deand of τ i for interval-length t, when execution outside the interval occurs on a speed-σ processor see Figure 1. Siilarly, it is easy to see that the traditional deand bound function is a special case of FF-DBF, in which it is assued that execution outside the interval occurs on an infinite-speed processor: Soe additional notation: for a task syste τ DBF(τ i, t) = FF-DBF(τ i, t, ). Utilization U(τ) = def C l /T l. τ l τ (1) concept called forced-forward deand was also introduced. Maxiu density δ ax (τ) = def ax C l/d l. τ l τ

7 j i j i 1 j 3 j 2 j 1 t t i t i 1 t 2 t 1 t 0 Figure 2. Notation used in Section 4. Hyperperiod P (τ) = def lc τl τ{t l }. FF-DBF(τ, t, σ) = def FF-DBF(τ l, t, σ). (2) τ l τ ( ) FF-DBF(τ, t, σ) FF-LOAD(τ, σ) = def ax t>0 t. (3) 4 Sufficient schedulability conditions for EDF and DM Both EDF and DM are work-conserving algoriths: they never idle any processor while there are jobs awaiting execution. As we will see below, this work-conserving property iplies that a deadline iss can only follow an interval during which a considerable aount of execution ust have occurred. This property is foralized in the following discussion. Let A denote soe work-conserving algorith that isses a deadline while scheduling soe legal collection of jobs generated by τ. Let us now exaine A s behavior on soe inial 5 legal collection of jobs of τ on which it isses a deadline. Let t 0 denote the instant at which the deadline iss occurs. Let j 1 denote a job that isses its deadline at t 0, and let t 1 denote j 1 s arrival-tie. Let s denote any constant satisfying δ ax (τ) s 1. (Observe that, since s δ ax (τ) and j 1 has not copleted execution by t 0, it has executed for strictly less than (t 0 t 1 ) s units over the interval [t 1, t 0 ).) We define a sequence of jobs j i, tie-instants t i, and an index k, according to the following pseudo-code (also see Figure 2): for i 2, 3,... do let j i denote a job that arrives at soe tie-instant t i < t i 1 ; has a deadline after t i 1 ; has not copleted execution by t i 1 ; and has executed for strictly less than (t i 1 t i ) s units over the interval [t i, t i 1 ). if there is no such job then k (i 1) break (out of the for loop) end if end for 5 By inial we ean that A is able to successfully schedule every proper subset of this collection of jobs.

8 Let W denote the aount of execution that occurs in this schedule over the interval [t k, t 0 ). Observation 1 derives a lower bound on W for any work-conserving algorith A. Observation 1 W > ( ( 1)s ) (t 0 t k ). Proof: For each i, 1 i k, let W i denote the total aount of execution that occurs over the interval [t i, t i 1 ). Hence, W = k i=1 W i. Let x i denote the total length of the tie-intervals over [t i, t i 1 ) during which job j i executes. By choice of job j i, it is the case that x i < (t i 1 t i ) s. By choice of job j i, it has not copleted execution by tie-instant t i 1. Hence over [t i, t i 1 ), all processors ust be executing whenever j i is not; it follows that W i (t i 1 t i x i ) + x i = (t i 1 t i ) ( 1)x i > (t i 1 t i ) ( 1)(t i 1 t i )s = ( ( 1)s) (t i 1 t i ). The observation, follows, by suing k i=1 W i. Observation 1 above derived a lower bound on the aount of work W that is executed over [t k, t 0 ). In the following two observations, we will derive upper bounds on W when the scheduling algorith is EDF and DM respectively; necessary conditions for non-schedulability under EDF and DM will follow, by requiring that the respective upper bounds be at least as large as the lower bound. Observation 2 (EDF) If the work-conserving algorith A is EDF, then W FF-DBF(τ, (t 0 t k ), s). Proof: Recall our assuption that we are analyzing a inial unschedulable collection of jobs. If EDF is the scheduling algorith, then such a inial unschedulable collection will not contain any job that has its deadline > t 0 (since the presence of such jobs cannot effect the scheduling of jobs with deadline t 0, the collection of jobs obtained by reoving all such jobs is also unschedulable by EDF.) Thus all jobs that execute in [t k, t 0 ) (and thereby contribute to W ) have their deadlines within the interval [t k, t 0 ). Soe of the will also have arrived within this interval, while others ay not. Now it ay be verified that the aount of execution that jobs of any task τ l contribute to W is bounded fro above by the scenario in which a job of τ l has its deadline coincident with the end of the interval t 0, and prior jobs have arrived exactly T l tie-units apart. Under this scenario, the jobs of τ l that ay contribute to W include def at least q l = (t 0 t k )/T l jobs of τ l that lie entirely within the interval [t k, t 0 ); and def (perhaps) an additional job that has its deadline at tie-instant t k + r l, where r l = (t 0 t k ) od T l. We now consider two separate cases:

9 1. r l D l ; i.e., the additional job with deadline at t k + r l arrives at or after t k. In this case, its contribution is C l. 2. r l < D l ; i.e., the additional job with deadline at t k + r l arrives prior to t k. Fro the exit condition of the for-loop, it ust be the case that this job has copleted at least (D l r l ) s units of execution prior to tie-instant t k ; hence, its reaining execution is at ost ax(0, C l (D l r l ) s). In either case, it ay be seen that the upper bound on the total contribution of τ l to W is equal to FF-DBF(τ l, t 0 t k, s) (see Equation 1). Suing over all l, we conclude that the total contribution of all the tasks to W is bounded fro above by n l=1 (FF-DBF(τ l, t 0 t k, s)), which is, by definition, FF-DBF(τ, L, s). Observation 3 (DM) If the work-conserving algorith A is DM, then W FF-DBF(τ, 2(t 0 t k ), s). Proof: Recall once again our assuption that we are analyzing a inial unschedulable collection of jobs. If DM is the scheduling algorith, then such a inial unschedulable collection will not contain any job with relative deadline greater than the relative deadline of j 1, the job that isses its deadline at tie-instant t 0 (since such jobs are assigned lower priority than j 1 under DM, and hence cannot effect the ability or otherwise of j 1 to eet its deadline.) By definition of t 1, the relative deadline of j 1 is (t 0 t 1 ). For any job with relative deadline < (t 0 t 1 ) to contribute to W (and hence execute prior to t 0 ), it ust have a deadline (t 0 +(t 0 t 1 )), i.e., 2t 0 t 1. As in the proof of Observation 2 above, it ay be verified that the aount of execution that jobs of any task τ l contribute to W is bounded fro above by the scenario in which a job of τ l has its deadline coincident with the end of the interval (i.e., at 2t 0 t 1 ), and prior jobs have arrived exactly T l tie-units apart. Under this scenario, it follows by an arguent essentially identical to the one used in the proof of Observation 2 that the total contribution of all the tasks to W is bounded fro above by FF-DBF(τ, 2t 0 t 1 t k, s). But since t k t 1, (2t 0 t 1 t k ) 2(t 0 t k ), and the observation is proved. Lea 1 (EDF) Suppose that constrained-deadline sporadic task syste τ is not schedulable by global EDF upon unit-speed processors. For each s, s δ ax (τ), there is an interval-length L such that FF-DBF(τ, L, s) > ( ( 1)s)L. Proof: Follows by chaining the lower bound on W of Observation 1 with the upper bound of Observation 2, with L (t 0 t k ). Lea 2 (DM) Suppose that constrained-deadline sporadic task syste τ is not schedulable by global DM upon unit-speed processors. For each s, s δ ax (τ), there is an interval-length L such that FF-DBF(τ, L, s) > ( ( 1)s) L 2. Proof: Follows by chaining the lower bound on W of Observation 1 with the upper bound of Observation 3, with L 2(t 0 t k ).

10 5 Deterining processor speedup factor The contrapositive of Lea 1 above represents a global EDF schedulability condition: any task syste τ satisfying ( ( ( ))) σ : σ δ ax (τ) : : 0 : FF-DBF(τ,, σ) ( ( 1)σ) (4) is EDF-schedulable upon unit-speed processors. Siilarly, the contrapositive of Lea 2 represents a global DM schedulability condition: any task syste τ satisfying ( ( ( σ : σ δ ax (τ) : : 0 : FF-DBF(τ,, σ) ( ( 1)σ) 2 is DM-schedulable upon unit-speed processors. We now deterine, in Theores 1 and 2 below, the processor speedup factors of schedulability test for EDF and DM respectively, based on checking these conditions. While the result in Theore 1 is a rederivation of the one in [15], the one in Theore 2 is new and represents a significant iproveent over the previously best-known result in [11]. In [2, Theore 2], necessary conditions for global ultiprocessor schedulability were identified; these necessary conditions generalize to our context and notation in the following anner: Lea 3 If FF-LOAD(τ, σ) > σ then τ is not feasible on speed-σ processors. Proof: Suppose that FF-LOAD(τ, σ) > σ. Let t 0 denote a value for t that axiizes the RHS of Equation 3, and hence deterines the value of FF-LOAD(τ, σ): ( ) FF-LOAD(τ, σ) = FF-DBF(τ l, t 0, σ) /t 0 τ l τ By definition of FF-DBF, each task τ l can generate a sequence of jobs that together require FF-DBF(τ l, t 0, σ) units of execution over soe interval of length t 0, when executing upon speed-σ processors. Since the different tasks of a sporadic task syste are assued to be independent of each other, such intervals for the different tasks can be aligned; the total execution requireent by all the tasks over the aligned interval is τ l τ FF-DBF(τ l, t 0, σ) = FF-LOAD(τ, σ) t 0 (By definition of t 0 ) > σt 0 (Since FF-LOAD(τ, σ) is assued to be > σ) But σt 0 denotes the total coputing capacity over an interval of size t 0 upon speed-σ processors. We therefore conclude that the total execution requireent by all the tasks in τ over the interval cannot be et, and soe deadline ust necessarily be issed. Lea 4 If task syste τ does not satisfy Condition 4, then it is not feasible upon a platfor coprised of speed 2 1 processors. ))) (5)

11 Proof: First, any τ not satisfying Condition 4 that has δ ax (τ) > /(2 1) is trivially not feasible on speed- processors. 2 1 Next, consider τ with δ ax (τ) /(2 1). Suppose that τ does not satisfy Condition 4: for all values of σ > δ ax (τ), there is an interval-length 0 such that FF-DBF(τ,, σ) > ( ( 1)σ). Let us instantiate this inequality for σ /(2 1): 0 : FF-DBF(τ,, 2 1 ) ( ) > ( 1) 2 1 FF-LOAD(τ, ) > ( 1) FF-LOAD(τ, 2 1 ) > FF-LOAD(τ, 2 1 ) > 2 1 It therefore follows, fro Lea 3, that τ is not feasible upon a platfor coprised of speed- 2 1 processors. By taking the contrapositive of Lea 4 above and observing that 1/( ) is equal to (2 1 ), we 2 1 have Theore 1 The processor speedup factor of an EDF schedulability test based on checking Condition 4 has a processor speedup factor of (2 1 ) on an -processor platfor. Reasoning very siilar to that used in Lea 4 and Theore 1 above are now applied to DM scheduling: Lea 5 If task syste τ fails the DM schedulability test of Condition 5, then it is not feasible upon a platfor coprised of speed 3 1 processors. Proof: Suppose that τ fails the schedulability test of Condition 5. If δ ax (τ) > (/(3 1) then τ is trivially not feasible on a platfor coprised of (any nuber of) speed-(/(3 1) processors, and we are done. Assue now that δ ax (τ) (/(3 1). Suppose that τ does not satisfy Condition 5: for all values of σ > δ ax (τ), there is an interval-length 0 such that FF-DBF(τ,, σ) > ( ( 1)σ) 2. Let us instantiate this inequality for σ /(3 1): 0 : FF-DBF(τ,, 3 1 ) ( ) > ( 1) FF-LOAD(τ, ) > ( ( 1) ) 1 2 FF-LOAD(τ, 3 1 ) > (3 1) FF-LOAD(τ, 3 1 ) > 3 1

12 It therefore follows fro Lea 3 above that τ is not feasible upon a platfor coprised of speed- processors. 3 1 By taking the contrapositive of Lea 4 above and observing that 1/( ) is equal to (3 1 ), we 3 1 have Theore 2 The processor speedup factor of a DM schedulability test based on checking Condition 5 has a processor speedup factor of (3 1 ) on an -processor platfor. 6 Iproved schedulability tests Condition 4, the contrapositive of the EDF unschedulability condition of Lea 1, asserts that in order to show a given task syste τ EDF-schedulable, it suffices to deonstrate the existence of a σ δ ax (τ) such that FF-DBF(τ, t, σ) ( ( 1)σ) t (6) for all values of t 0. Let us refer to such a σ as a witness to the EDF-schedulability of τ. In order to obtain a schedulability test with the optial processor speedup factor of (2 1 ), we have seen (Theore 1 above) that it suffices to only consider σ /(2 1) as a potential witness, declaring the task set not schedulable if this value of σ fails to satisfy Inequality 6 for all t 0. Indeed, this is in essence the schedulability test presented in [15]: deterine whether σ /(2 1) satisfies Inequality 6 for all t 0. However, by testing only one out of all the values of σ that could bear witness to a task syste s schedulability, this test clearly fails to ake full use of the insight into EDF-schedulability that Lea 1 affords us. In the reainder of this section, we derive an algorith that fully exploits the insight of Lea 1, by correctly identifying all task systes for which any σ would cause Inequality 6 to evaluate to true for all t 0. In other words, the algorith we derive in this section identifies all systes satisfying Condition 4, whereas the test in [15] only identifies those systes that satisfy this condition instantiated with σ /(2 1). (A siilar exercise can easily be conducted for the DM schedulability test derived fro Condition 5, the contrapositive of the DM unschedulability condition of Lea 2. The steps are essentially identical to the ones for EDF as described below; hence, we oit the details for DM schedulability analysis.) Now there are infinitely any different values of σ that could potentially be witnesses to the EDFschedulability of a task syste; for each such potential witness, there are infinitely any values of t for which it ust be validated that Inequality 6 is satisfied. Two questions ust therefore now be answered: Q1: What values of σ would we need to consider as potential witnesses to the EDF-schedulability of τ? and Q2: In order to deterine whether a particular σ is indeed a witness or not, for which values of t do we need to evaluate Condition 6? We address the second of these questions first, in Section 6.1 below; the second one is addressed in Section 6.2.

13 6.1 Bounding the range of tie-values that ust be tested We now address Q2, the second of the questions listed above: for a given value of σ, for which values of t ust we validate Condition 6 in order to be able to conclude that it holds for all t? Clai 1 For a given σ and τ, if Condition 6 is violated for any t then it is violated for soe t in { } kt i + D i, kt i + D i in(c i /σ, D i ) k N τ i τ (7) Proof Sketch: This follows fro the observation (also see Figure 1) that FF-DBF(τ i, t, σ) increases with a constant slope between kt i + D i in(c i /σ, D i ) and kt i + D i, and reains unchanged elsewhere, for all k N. Hence the LHS of Condition 6 increases with constant slope with increasing t between two consecutive t s in this set; since the RHS also increases with constant slope with increasing t, it is guaranteed that if this condition is violated at soe t it will be violated at one of the two t s in this set that neighbor t. Clai 1 above tells us that we need evaluate Condition 6 for only countably any t s; Clais 2 and 3 below allow us to bound the actual nuber. Clai 2 If Condition 6 is violated at soe t for a given τ and, and a particular σ ( U(τ))/( 1), then it is violated at soe t no larger than P (τ). Proof: Recall that P (τ) denotes the hyperperiod the least coon ultiple of the task period paraeters of τ. Since for all τ i the period T i divides the hyperperiod P (τ), it follows fro Equation 1 (also see Figure 1) that FF-DBF(τ, t + P (τ), σ) = P (τ) U(τ) + FF-DBF(τ i, t, σ) P (τ) ( ( 1)σ) + FF-DBF(τ i, t, σ) (since we are assuing that σ ( (U(τ))/( 1)). Hence if Condition 6 is to be violated for soe t v > P (τ), it will also be violated for t v od P (τ). Clai 2 tells us that P (τ) is an upper bound on the values of t for which Condition 6 needs to be evaluated. Clai 3 below provides another upper bound. Clai 3 If Condition 6 is violated at soe t for a given τ and, and a particular σ ( U(τ))/( 1), then t is no larger than τ i τ C i (8) ( 1)σ U(τ) Proof: We first observe that it directly follows fro the definition of FF-DBF (Equation 1 also see Figure 1) that for all t D i and for all σ, ( ) t FF-DBF(τ i, t, σ) + 1 C i T i

14 Suppose that FF-DBF(τ, t, σ) > ( ( 1)σ ) t for soe t > ax τi τ{ Di }. We then have and the lea is proved. FF-DBF(τ, t, σ) > ( ( 1)σ ) t ( t C ) i + C i > ( ( 1)σ ) t T i τ i τ tu(τ) + τ i τ C i > ( ( 1)σ ) t t < τ i τ C i ( 1)σ U(τ) Testing set. For a given σ and τ, let T S(τ, σ) denote the testing set of values of t that lie in the set defined in Equation 7 and are no larger than both P (τ) and the bound defined by Equation 8. How large can this testing set be? As shown in Clai 2, we need not consider any t exceeding the hyperperiod P (τ). It is easily seen that there are at ost exponentially any points in the set defined in Equation 7 not exceeding P (τ); hence, the testing set contains at ost exponentially any points. Suppose, however, that we were to enforce an additional restriction that we would not consider any σ greater than ( ) U(τ) ɛ /( 1) (9) where ɛ is an arbitrarily sall positive constant. It would then follow fro Inequality 8 that the upper bound on the values in T S(τ, σ) is guaranteed to be ( τ i τ C i) /ɛ, which is pseudo-polynoial in the representation of the task syste τ. Thus, this restriction iediately yields a pseudo-polynoial upper bound on the nuber of eleents in T S(τ, σ). The consequence of enforcing the restriction of Equation 9 above is that the test we develop is no longer able to identify all task systes satisfying Condition 4: task systes that only satisfy Condition 4 for values of σ in (( U(τ))/( 1) ɛ, ( U(τ))/( 1)] would not be identified by our test. In exchange for this slight loss of optiality (the degree of which can be controlled by choosing ɛ to be appropriately sall), we would restrict the size of the testing set to be pseudo-polynoial. 6.2 Choosing potential witnesses to test In this section, we address Q1, the first of the two questions listed earlier in this section. That is, we set about restricting the candidate field of σ s that need be tested as potential witnesses to the EDFschedulability of τ. Clai 4 No value of σ that is greater than ( U(τ))/( 1) can possibly result in Condition 6 evaluating to true for all values of t (and hence, no such value of σ can attest to the EDF-schedulability of τ). Proof: Observe that FF-DBF(τ i, t, σ) asyptotically approaches t (C i /T i ) as t. Hence FF-DBF(τ, t, σ) asyptotically approaches t U(τ) with increasing t. In order to have FF-DBF(τ, t, σ)

15 ( ( 1)σ)t for all t, therefore, we need U(τ) ( 1)σ σ U(τ). 1 As a consequence of Clai 4 above, we can restrict the range of values for σ that are potential witnesses to the EDF-schedulability of τ. However, there are still infinitely any distinct values in this range, and we clearly cannot exhaustively check all these infinitely any values. Fortunately it so happens that we can restrict the actual nuber of values of σ within this range that need be considered as potential witnesses to the EDF-schedulability of τ, as we will now show. Let us suppose that we have identified a particular σ cur, such that we know that no σ < σ cur can possibility bear witness to the EDF-schedulability of τ. Suppose that we then test σ cur, and deterine that it is not a witness to the EDF-schedulability of τ, either t cur is a value of t that causes Condition 6 to evaluate to false when σ σ cur. Let σ new denote the sallest value of σ > σ cur such that Condition 6 evaluates to true with (σ σ new ; t t cur ). (We describe below, in Section 6.4, how the value of σ new is coputed.) It is clear that t cur rules out the possibility of any σ [σ cur, σ new ) bearing witness to the EDF-schedulability of τ; hence, the next value of σ that we will need to consider is σ new. So we ve seen how, if we know a constant σ cur such that no σ σ cur can be a witness to the EDFschedulability of τ, we can deterine the next potential witness σ new that we ust consider. Clai 5 below tells us that in considering σ new, we need not revisit values of T S(τ, σ new ) that are t cur : Clai 5 Suppose that τ is not EDF-schedulable. Consider soe s 1 and t 1 such that For any s 2 > s 1, there is a t 2 t 1 such that FF-DBF(τ, t 1, s 1 ) > ( ( 1)s 1 ) t 1. FF-DBF(τ, t 2, s 2 ) > ( ( 1)s 2 ) t 2. Proof: This follows fro the observation that the jobs j 1, j 2,... that are defined according to the pseudo-code given in Section 4 for a given value of s (say, s s 1 ), are also valid for larger values of s (say, s s 2 ). This is easily shown by induction. Assue that j 1,..., j i 1 as defined for s s 1 are valid for s s 2 : this iplies that t i 1 is the sae when s s 1 and s s 2. The job j i as defined for s s 1 has executed for less than (t i 1 t i )s 1 prior to t i 1. But since s 2 > s 1, it has also executed for less than (t i 1 t i )s 2 prior to t i 1, and hence satisfies the condition to be considered as job j i for s s 2 as well. 6.3 Putting the pieces together: the EDF schedulability test We are now ready to put the pieces together, and specify our schedulability test. This schedulability test is a ethodical quest for a value of σ for which there is no t causing Condition 6 to evaluate to false (and which is thus a witness to the EDF-schedulability of τ). Based on Clai 5 above, we will start out testing a sall value for σ; if this fails, we can try a larger value for σ and use the result of Clai 5 to tri the set of potential values of t that need to be tested for this larger value of σ. In greater detail, our algorith is the following.

16 S1 Let σ cur denote the value of σ currently being evaluated (i.e., the potential witness currently under consideration). This is initialized as follows: σ cur δ ax (τ). We will also use an additional variable t cur, initialized to zero: t cur 0. ) S2 If σ cur is larger than ɛ where ɛ is an arbitrarily sall positive constant that has been a ( U(τ) 1 priori deterined, then we exit the test, having failed to show τ is EDF-schedulable. (Here, we are using the result shown in Clai 4, odified as discussed in Equation 9 to yield a testing set of pseudo-polynoial size, to restrict the range of values of σ that we need test as potential witnesses to the EDF-schedulability of τ.) Otherwise by the results of Section 6.1, we need only evaluate Condition 6 for values of t T S(τ, σ cur ) to deterine whether it is satisfiable or not. We begin at the sallest value in T S(τ, σ cur ) that is greater than t cur, and consider the values in T S(τ, σ cur ) in increasing order. If no value of t in T S(τ, σ cur ) causes Condition 6 to evaluate to false for this current value of σ cur, then we exit the test, having succeeded in showing that τ is EDF-schedulable. S3 Suppose, however, that there is soe value of t that causes Condition 6 to evaluate to false for this value of σ cur. Assign t cur this value of t. By Clai 5, if τ is not EDF-schedulable then for all values of σ > σ cur there is soe t t cur which causes Condition 6 to evaluate to false. Let σ new denote the sallest value of σ > σ cur, such that FF-DBF(τ, t cur, σ ) ( ( 1)σ ) t cur. We copute σ new using the technique described in Section 6.4 below, assign σ cur this value σ new, and goto Step S2. Coputational coplexity. Observe that the values assigned to the variable t cur during the above algorith are onotonically increasing once we assign t cur a particular value, we never assign it a saller value even after we have changed the value assigned to σ cur. This observation can be used to show that the total nuber of values assigned to t cur is no ore than the cardinality of T S(τ, σ), for the largest value of σ that is tested. And we have seen in Section 6.1 that this nuber is pseudo-polynoially bounded in the representation of the task syste τ. We will see in Section 6.4 below that σ new can be coputed in polynoial tie, while the rest of the processing above for a given value of t cur is easily seen to also take polynoial tie. This yields the following result: Theore 3 This EDF-schedulability test has pseudo-polynoial tie coplexity. 6.4 Coputing σ new Given fixed values for t cur and σ cur such that FF-DBF(τ, t cur, σ cur ) > ( ( 1)σ cur ) t cur, our objective is to copute σ new, the sallest σ > σ cur such that FF-DBF(τ, t cur, σ ) ( ( 1)σ ) t cur.

17 Let us exaine how FF-DBF(τ i, t cur, σ) changes as σ is increased in the neighborhood of σ cur, while the task τ i and the interval-length t cur are kept unchanged. Fro Equation 1, we know that FF-DBF(τ i, t cur, σ) depends on q i and r i, where q i = t cur /T i and r i = t cur od T i. Notice that the values of q i and r i do not depend on σ. Hence, (a) FF-DBF(τ i, t cur, σ) does not vary with σ if r i D i (the first case in Equation 1). (b) It varies linearly with σ while D i > r i D i C i σ. That is, if r i < D i then FF-DBF(τ i, t cur, σ) decreases linearly with increasing σ while σ C i /(D i r i ). This is the second case in Equation 1. (c) Once σ increases such that it is > C i /(D i r i ), FF-DBF(τ i, t cur, σ) reains unchanged with further increase in the value of σ. This is the third case in Equation 1. In order to copute σ new given values for t cur and σ cur, we would therefore L1 Classify each task τ i as being either in class (a), (b), or (c) according to the above classification. That is, a task τ i for which r i D i would be classified as being in class (a); one with r i < D i and r i D i C i σ cur would be classified as being in class (b); while one with r i < D i C i σ cur would be classified as being in class (c). def For each task τ i in class (b), let ˆσ i = C i /(D i r i ). Increasing σ to becoe greater than ˆσ i would cause τ i to no longer be a class (b) task. L2 Observing that only tasks in class (b) have their FF-DBF s change linearly with changing σ, we can set up and solve a linear equation to deterine σ o, the sallest σ > σ cur for which FF-DBF(τ, t cur, σ o ) = ( ( 1)σ o ) t cur. L3 If this coputed value of σ o is ˆσ i for all tasks τ i in class (b), then we have coputed the desired value for σ new. Else, (a) assign σ cur the value of the sallest ˆσ i fro aong all those coputed for tasks in class (b), and (b) repeat fro step L1 above. Coputational coplexity. It is not difficult to see that σ new can be coputed in tie polynoial in the representation of τ. This follows fro the observations that Steps L1, L2, and L3 above each take polynoial tie. During each iteration of the 3-step process L1-L3 either (i) we deterine the value of σ new and exit; or (ii) at least one task that was in class (b) will henceforth be placed in class (c) in the subsequent iteration, whereas no additional tasks becoe class (b) tasks. Thus, the nuber of iterations of L1-L3 is bounded fro above by the nuber of tasks initially in class (b), which is, of course, itself bounded by the nuber of tasks in τ.

18 7 Suary and conclusions Recently, Bonifaci et al. have obtained [15] a speedup-optial global EDF schedulability test for sporadic task systes ipleented upon identical ultiprocessor platfors. In this paper, we have generalized the technique so that it is applicable to the analysis of any work-conserving global scheduling algorith. We have used this generalization to coe up with a schedulability test for global DM which has a speedup factor superior to any previously known. Although the speedup optiality property akes the EDF schedulability condition in [15] very significant fro a conceptual perspective, its applicability in the analysis of actual real-tie systes is soewhat liited. We have built upon the theoretical foundations provided by the [15] test, to obtain a sufficient EDF schedulability test of wider applicability and lower pessiis that retains the optial processor speedup factor. We have shown that this schedulability test can be ipleented to have a run-tie that is pseudo-polynoial in the representation of the task syste being analyzed. Since any algoriths that are currently used in the analysis of real-tie systes are of siilar coputational coplexity, this suggests that our schedulability test is efficient enough to be of use in the design and analysis of actual ultiprocessor real-tie application systes. References [1] BAKER, T. Multiprocessor EDF and deadline onotonic schedulability analysis. In Proceedings of the IEEE Real-Tie Systes Syposiu (Deceber 2003), IEEE Coputer Society Press, pp [2] BAKER, T., AND CIRINEI, M. A necessary and soeties sufficient condition for the feasibility of sets of sporadic hard-deadline tasks. In Proceedings of the IEEE Real-tie Systes Syposiu (Rio de Janeiro, Deceber 2006), IEEE Coputer Society Press, pp [3] BAKER, T., AND CIRINEI, M. Brute-force deterination of ultiprocessor schedulability for sets of sporadic hard-deadline tasks. In Proceedings of the 10th International Conference on Principles of Distributed Systes (Guadeloupe, Deceber 2007), pp [4] BAKER, T. P. An analysis of EDF schedulability on a ultiprocessor. IEEE Transactions on Parallel and Distributed Systes 16, 8 (2005), [5] BAKER, T. P. Coparison of epirical success rates of global vs. partitioned fixed-priority and EDF scheduling for hard real tie. Tech. Rep. TR , Departent of Coputer Science, Florida State University, [6] BAKER, T. P. A coparison of global and partitioned EDF schedulability tests for ultiprocessors. In Proceeding of the International Conference on Real-Tie and Network Systes (Poitiers, France, 2006). [7] BARUAH, S. Schedulability analysis of global deadline-onotonic scheduling. Available at baruah, 2007.

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