Multiprocessor Extensions to Real-Time Calculus

Transcription

1 Noname manuscript No. (wi be inserted by the editor) Mutiprocessor Extensions to Rea-Time Cacuus Hennadiy Leontyev Samarjit Chakraborty James H. Anderson the date of receipt and acceptance shoud be inserted ater Abstract Many embedded patforms consist of a heterogeneous coection of processing eements, memory modues, and communication subsystems. These components often impement different scheduing/arbitration poicies, have different interfaces, and are suppied by different vendors. Hence, compositiona techniques for modeing and anayzing such patforms are of interest. In prior work, the rea-time cacuus framework has proven to be very effective in this regard. However, rea-time cacuus has heretofore been imited to systems with uniprocessor processing eements, which is a serious impediment given the advent of muticore technoogies. In this paper, a two-step approach is proposed that aows the power of rea-time cacuus to be appied in gobay-schedued mutiprocessor systems: first, assuming that job response-time bounds are given, determine whether these bounds are met; second, using these bounds, determine the resuting residua processor suppy and streams of job competion events using formaisms from rea-time cacuus. For this methodoogy to be appied in settings where response-time bounds are not specified, such bounds must be determined. Cosed-form expressions for cacuating such responsetime bounds are presented for a arge famiy of fixed-job-priority scheduers. The utiity of the proposed anaysis framework is demonstrated using a case study. Keywords Mutiprocessor Scheduing Streaming Task Mode Response-Time Anaysis Rea-Time Cacuus H. Leontyev Department of Computer Science, University of North Caroina at Chape Hi Te.: E-mai: eontyev@cs.unc.edu S. Chakraborty Institute for Rea-Time Computer Systems (RCS), Technica University of Munich E-mai: samarjit@tum.de J. H. Anderson Department of Computer Science, University of North Caroina at Chape Hi Te.: Fax.: E-mai: anderson@cs.unc.edu

2 2 1 Introduction The increasing compexity and heterogeneity of modern embedded patforms have ed to growing interest in compositiona modeing and anaysis techniques (Richter et a 2003). In devising such techniques, the goa is not ony to anayze the individua components of a patform in isoation, but aso to compose different anaysis resuts to estimate the timing and performance characteristics of the entire patform. Such anaysis shoud be appicabe even if individua processing and communication eements impement different scheduing/arbitration poicies, have different interfaces, and are suppied by different vendors. These compicating factors often cause standard event modes (e.g., periodic, sporadic, etc.) and scheduabiity-anaysis techniques to ead to overy pessimistic resuts or to be atogether inappicabe. To overcome this difficuty, a compositiona framework often referred to as rea-time cacuus was proposed by Chakraborty et a (2003) and then subsequenty extended in a number of papers (e.g., see (Chakraborty et a 2006)). Reatime cacuus is a speciaization of network cacuus, which was proposed by Cruz (1991a,b) and has been widey used to anayze communication networks. Rea-time cacuus speciaizes network cacuus to the domain of rea-time and embedded systems by, for exampe, adding techniques to mode different scheduers and mode/statebased information (e.g., see (Phan et a 2008)). A number of scheduabiity tests have aso been derived based upon network cacuus. An overview of these tests can be found in (Wu et a 2005). In rea-time cacuus, timing properties of event streams are represented using upper and ower bounds on the number of events that can arrive over any time interva of a specified ength. These bounds are given by functions α u (Δ) and α (Δ), which specify the maximum and minimum number of events, respectivey, that can arrive at a processing/communication resource within any time interva of ength Δ (or the maximum/minimum number of possibe task activations within any Δ). The service offered by a resource is simiary specified using functions β u (Δ) and β (Δ), which specify the maximum and minimum number of serviced events, respectivey, within any interva of ength Δ. Given the functions α u (Δ) and α (Δ) corresponding to an event stream arriving at a resource, and the service β u (Δ) and β (Δ) offered by it, it is possibe to compute the timing properties of the processed stream and remaining processing capacity, i.e., functions α u (Δ), α (Δ), β u (Δ), and β (Δ), as iustrated in Figure 1(a), as we as the maximum backog and deay experienced by the stream. As shown in the same figure, the computed functions α u (Δ) and α (Δ) can then serve as inputs to the next resource on which this stream is further processed. By repeating this procedure unti a resources in the system have been considered, timing properties of the fuy-processed stream can be determined, as we as the endto-end event deay and tota backog. This forms the basis for composing the anaysis for individua resources, to derive timing/performance resuts for the fu system. Simiary, for any resource with tasks being schedued according to some scheduing poicy, it is aso possibe to compute service bounds (β u (Δ) and β (Δ)) avaiabe to its individua tasks. Figure 1(b) shows how this is done for the fixed-priority (FP) and time-division-mutipe-access (TDMA) poicies. As shown in this figure, for the FP poicy, the remaining service after processing Stream A serves as the input (or,

3 3 (a) (b) Fig. 1: (a) Computing the timing properties of the processed stream using rea-time cacuus. (b) Scheduing networks for fixed priority and TDMA scheduers. is avaiabe) to Stream B. On the other hand, for the TDMA poicy, the tota service β(δ) is spit between the services avaiabe to the two streams. Simiar so caed scheduing networks (Chakraborty et a 2006) can be constructed for other scheduing poicies as we. Various operations on the arriva and service curves α(δ) and β(δ), as we as procedures for the anaysis of scheduing networks on uniprocessors (and partitioned systems) have been impemented in the RTC (rea-time cacuus) toobox (Wandeer and Thiee 2006), which is a MATLAB-based ibrary that can be used for modeing and anayzing distributed rea-time systems. Unfortunatey, the compositiona techniques described above can be used for the anaysis of mutiprocessor systems ony if the mutiprocessor under consideration is viewed as a coection of independent uniprocessors and the workoad is spit using some partitioning technique.

4 4 This precudes supporting workoads that fundamentay require goba scheduing approaches. Under goba scheduing, tasks are independenty seected by processors from a singe ready queue. In particuar, when such agorithms are used, processors may be ide even though tasks are avaiabe for execution, as tasks must execute sequentiay; this situation does not arise on uniprocessors and thus is not addressed in uniprocessor compositiona techniques. There are two reasons why existing compositiona techniques need to be extended to incorporate such mutiprocessors. First, muticore chips are becoming increasingy common. Second, goba scheduers are often ess resource-demanding than partitioned ones. Exampe 1. Consider a MPEG-2 video decoder appication that has been studied previousy in (Chakraborty et a 2006; Phan et a 2008). The originay-studied appication, shown in Figure 2(a), is partitioned and mapped onto two PEs, PE1 and PE2. PE1 runs the VLD (variabe-ength decoding) and IQ (inverse quantization) tasks, whie PE2 runs the IDCT (inverse discrete cosine transform) and MC (motion compensation) tasks. The (coded) input bit stream enters this system and is stored in the input buffer B. The macrobocks (frame pieces of size pixes) in B are first processed by PE1 and the corresponding partiay decoded macrobocks are stored in the buffer B before being processed by PE2. The resuting stream of fuy decoded macrobocks is written into a payout buffer B prior to transmission by the output video device. In the above system, the coded input event stream arrives at a constant bit-rate. It has been found from measurements that, in the above appication, each task consumes 70% of the time on a dedicated 500 MHz processor. Thus, a singe instance of the appication requires two processors. Suppose that we want to support the simutaneous decoding of four video streams S 1,...,S 4 such that stream S 1 has the owest possibe atency. Such a task system coud be used in a virtua reaity appication, where mutipe video streams need to be processed. If a partitioned approach is used, then eight processors are needed as shown in Figure 2(b). (For conciseness, the intermediate buffers are not shown.) This is because no two tasks can be assigned to a singe processor without overoading it. However, if goba scheduing is used, then six processors are sufficient. In the system shown in Figure 2(c), the six processors are divided into two groups of three. One processor in each group is dedicated to running tasks processing stream S 1. The remaining time on this processor and the time avaiabe on the other two processors in each group is used for processing streams S 2, S 3, and S 4 as shown. The tota ong-term execution demand of a tasks running on each processor group is =2.8, whie there are three processors avaiabe to each such group. In Figure 2(c), abeed down-arrows indicate the ong-term fraction of time avaiabe on each processor (i.e., the ong-term avaiabe per-processor utiization). The timing properties of stream S 1 can be anayzed using existing uniprocessor rea-time cacuus methods. However, these methods cannot be appied for the anaysis of streams S 2, S 3, and S 4. In the next section, we briefy overview our proposed extensions to the rea-time cacuus framework (Chakraborty et a 2003, 2006) that aow systems such as that

5 5 B PE1 PE2 VLD+IQ B IDCT+MC B T 1 T 2 (a) S 1 Proc 1 T 1 Proc 2 T 2 Three processors u=1.0 u=1.0 Three processors S 2 Proc 3 T 3 Proc 4 T 4 S 1 T 1 T 2 u=1.0 u=0.3 Proc 5 Proc 6 S 3 T 5 T 6 S 3 Proc 7 Proc 8 S 4 S 4 T 7 T 8 S 2 T 3 T 4 T 5 T 6 T 7 T 8 (b) (c) Fig. 2: (a) Exampe MPEG-2 decoder appication. Four MPEG decoders running using (b) partitioned and (c) goba scheduing. presented in Exampe 1 above to be anayzed. The proposed extension incorporates gobay-schedued mutiprocessors and is compatibe with the RTC toobox. 2 Anaysis Overview and Prior Work In this paper, we consider a PE that processes mutipe input event streams α 1,...,α n using the avaiabe resource suppy B(Δ). The appication of our resuts invoves severa theorems stated ater and is iustrated in Figure 3. The core of our framework is a pseudo-poynomia-time procedure that, given a coection of arriva curves for input streams α u (Δ) and α (Δ), their execution requirements, and B(Δ), checks that event deays on such a mutiprocessor reside within specified bounds. The set of deay bounds {Θ 1,...,Θ n }, where n is the number of streams, can be: specified (e.g., as reative deadines) for individua tasks; cacuated using Theorem 7 from the input if task deadines are not specified or not feasibe; determined by other means. We shoud note that Theorem 7, which can be used to derive event-deay bounds from inputs, is ony appicabe in settings in which fixed-job-priority scheduers such as eariest-deadine-first (EDF) or first-in-first-out (FIFO) are used. When other scheduers are used, maximum deay bounds shoud be specified or found using aternative anaysis techniques.

6 Event-deay cacuation 1 n Stream output cacuation n 1,..., n } Suppy output cacuation n Fig. 3: A mutiprocessor PE anayzed using mutiprocessor rea-time cacuus. In terms of computationa compexity, cacuating deay bounds using Theorem 7 (where appicabe) and comparing those to specified deadines is ess costy than checking whether specified deadine constraints are met using the pseudo-poynomia sheduabiity test. As Theorem 7 gives somewhat conservative estimations of deay bounds, they might be further tightened by iterativey decreasing them and appying the scheduabiity test. However, the detais of this tightening procedure are specific to a task set and running a scheduabiity test mutipe times might be time-consuming. In our case study (described ater), tightening the bounds obtained using Theorem 7 did not give much improvement. Once the event deays are identified, we can compute arriva curves for the processed streams α u (Δ) and α (Δ) using Theorem 1. This is done by agebraic manipuations invoving the specified input curves. As a resut, because the maximum event-deay is bounded, the ong-term departure rate of events is the same as the ong-term arriva rate, i.e., the ong-term growth rate of α u i (Δ) is the same as that of α u i (Δ) for each task T i. Aso, we can compute the remaining-tota-service curve B (Δ) using Theorem 2. In this case, we subtract the tota execution demand of tasks within an interva of ength Δ from the tota avaiabe suppy B(Δ). The obtained output curves as in the uniprocessor case can in turn be used as input for other resources, thereby resuting in a compositiona framework (as shown in Figure 1(a)). Prior work. Our work is based upon mutiprocessor scheduabiity tests by Baruah (2007) and Leontyev and Anderson (2008). In some aspects, the presented anaysis is aso simiar to resuts by Bertogna et a (2009), Shin et a (2008), and Zhang and Burns (2008). The main difference between our work and these prior efforts is that we consider more genera task arriva and execution modes, viz. those supported by the rea-time cacuus framework. Aso, we consider the case when one or more processors can be partiay avaiabe, which is simiar to anaysis in (Shin et a 2008), where partia avaiabiity is considered in the context of hierarchica scheduing.

7 7 Apart from the deveopment of rea-time cacuus, a number of scheduabiity tests have been independenty proposed for uniprocessor and partitioned settings that consider workoad modes incorporating ong-term arriva patterns and execution requirements. These resuts pertain to workoad modes that are simiar to that presented in our paper. An exampe of such anaysis aong with an overview of reated work is presented in (Wu et a 2005). In their paper, Wu et a consider uniprocessor systems and static-priority scheduing. In contrast, we are primariy concerned with mutiprocessor scheduing and EDF-ike agorithms. The rest of the paper is organized as foows. Section 3 presents our task mode. In Sections 4 and 5, timing characteristics of processed streams and the remaining suppy are computed. In Section 6, we present a basic response-time bound test. In Section 7, its time compexity is discussed. In Section 8, we improve the basic test for the case when an EDF-ike scheduer is used. In Section 9, cosed-form expressions for response-time bounds are derived. Section 10 presents a case study for our anaysis. Section 11 summarizes our contributions and discusses some directions for future work. 3 Task Mode In this paper, we consider a task set τ = {T 1,...,T n }. Each task has incoming jobs that are processed by a mutiprocessor consisting of m 2 unit-speed processors. We assume that n m. We aso assume that a time quantities except the interva ength Δ are integra. The j th job of T i, where j 1, is denoted T i,j. The arriva (or reease) time of T i,j is denoted r i,j. The competion time of T i,j is denoted f i,j and the deay between its start time and competion, f i,j r i,j, is caed its response time. Asin prior work on rea-time cacuus, we wish to be abe to accommodate very genera assumptions concerning job executions and arrivas and the avaiabe service. Most of the remaining definitions in this section are devoted to formaizing the assumptions we require. Tabe 1 summarizes the notation introduced in this section. Definition 1. γi u(k) (γ i (k)) denotes an upper (ower) bound on the tota execution time of any k consecutive jobs of T i. (We assume γi u(k) =γ i (k) =0for a k 0 and γi u(k) γu i (k +1)and γ i (k) γ i (k +1).) These definitions are equivaent to the workoad demand curves in (Maxiaguine 2005). Exampe 2. Suppose that task T i s job execution times foow a pattern 1, 5, 2, 1, 5, 2,... Then, γi u(1) = 5, γu i (2) = 7, γu i (3) = 8, γu i (4) = 13, etc. Aso, γ i (1) = 1, γi (2) = 3, γ i (3) = 8, γ i (4) = 9, etc. Definition 2. The arriva function αi u(δ) (α i (Δ)) provides an upper (ower) bound on the number of jobs of T i that can arrive within any time interva (x, x + Δ], where x 0 and Δ>0 (Chakraborty et a 2006). (We assume αi u (Δ) =0for a Δ 0.) α i (Δ) denotes the pair (αi u(δ),α i (Δ)). Exampe 3. The widey-studied periodic and sporadic task modes are subcases of this more genera task mode. In both modes, each job of T i requires at most e max i

8 8 Tabe 1: Mode notation. Input parameters αi u(δ) (α i (Δ)) Max. (min.) number of job arrivas of T i over Δ γi u(k) (γ i (k)) Max. (min.) execution demand of any k consecutive jobs of T i B(Δ) Min. guaranteed cumuative processor suppy over Δ Params. beow can be found using the RTC Toobox Û Long-term aviabe processor utiization σ tot Maximum backout time F Number of processors that are aways avaiabe A 1 i (k) K i Pseudo-inverse of αi u Min. integer s.t. A 1 (K i ) γi u(k i) e i T i s average worst-case job execution time v i Burstiness of the execution demand R i Long-term arriva rate of T i s jobs B i Burstiness of the arriva curve u i T i s ong-term utiization U sum Tota utiization Θ i beow can be checked using the test in Section 6 Θ i T i s response-time bound Output cacuated using the input and {Θ i } α u i(δ) (α i(δ)) Max. (min.) number of job competions of T i over Δ B (Δ) Min. guaranteed unused processor suppy over Δ execution units and consecutive job arrivas are separated by at east p i time, where p i is the period of T i. Therefore, for both modes, αi u(δ) = Δ p i and γi u (k) =k emax i. Definition 3. Let A 1 i (k) =inf{δ αi u (Δ) >k}, where Δ>0. This function characterizes the minimum ength of the time interva (x, x + Δ] during which jobs T i,j+1,...,t i,j+k can be reeased for some j, assuming T i,j is reeased at time x. We define A 1 i (0) = 0 and require that there exists K i 1 such that A 1 i (K i ) γi u (K i ). (1) α We further require that there exists R i > 0 and B i 0, where R i = im u i (Δ) Δ + Δ, such that αi u (Δ) R i Δ + B i for a Δ 0. (2) γ Aso, we assume that there exists e i > 0 and v i 0, where e i = im u i (k) k + k, such that γi u (k) e i k + v i for a k 1. (3) (1) is needed in order to prevent task T i from overoading the system. In (2), R i characterizes the ong-term arriva rate of task T i s jobs and B i characterizes the degree of burstiness of the arriva sequence. In (3), the parameter e i denotes the average worst-case job execution time of T i.

9 9 Definition 4. Let u i = R i e i. This quantity denotes the average ong-term utiization of task T i. We require that 0 <u i 1. Let U sum = T u i τ i. ( ) Δ Exampe 4. Under the sporadic task mode, R i = im Δ + p i +1 /Δ = 1 p i and e i = e max i,sou i = R i e i = emax i pi. Definition 5. Let suppy h (t, Δ) be the tota amount of processor time avaiabe to tasks in τ on processor h in the interva [t, t + Δ), where Δ 0. Let Suppy(t, Δ) = m h=1 suppy h(t, Δ) be the cumuative processor suppy in the interva [t, t + Δ). Though we desire to make our anaysis compatibe with the rea-time cacuus framework, which requires that individua processor suppies be known, there exist many settings in which individua processor suppy functions are not known and a ower bound on the cumuative avaiabe processor time is provided instead. (In uniprocessor rea-time cacuus, the avaiabe service is described as the number of incoming events processed by a PE during a time interva.) Note that if individua processor suppy guarantees are known, a ower bound on the cumuative guaranteed suppy can be computed easiy. Definition 6. Let B(Δ) Suppy(t, Δ) be the guaranteed tota time that a processors can provide to the tasks in τ during any time interva [t, t + Δ), where Δ 0. We assume that B(Δ) max(0, Û (Δ σ tot)), (4) where Û (0,m] and σ tot 0. WeetF be the number of processors that are aways avaiabe at any time. If a processors have unit speed (as we have assumed), then F = max{y Δ 0::B(Δ) y Δ}. In the above definition, the parameters Û, which is the tota ong-term fraction of processor time avaiabe to the tasks in τ on the entire patform, and σ tot, which is the maximum duration of time when a processors are unavaiabe, are simiar to those in the bounded deay mode (Mok et a 2001). Exampe 5. Consider a system with one processor (m =1) that is not fuy avaiabe. The avaiabiity pattern, which repeats every eight time units, is shown in Figure 4(a); intervas of unavaiabiity are shown as shaded regions. For processor 1, the minimum amount of time that is guaranteed to rea-time tasks over any interva of ength Δ is zero if Δ 2, Δ 2 if 2 Δ 4, and so on. Figure 4(b) shows the minimum amount of time B(Δ) that is avaiabe on processor 1 for tasks over any interva [t, t + Δ]. It aso shows a ower bound max(0, Û(Δ σ tot)), where Û = 5 8 and σ tot =2, which bounds B(Δ) from beow. We require that (5) beow hods for otherwise the system is overoaded and job response times coud be unbounded. U sum Û (5) We assume that reeased jobs are paced into a singe goba ready queue. When choosing a new job to schedue, the scheduer seects (and dequeues) the ready job of highest priority. An unfinished job is pending if it is reeased. A pending job is ready if its predecessor (if any) has competed execution. Note that, the jobs of each task execute sequentiay. Job priorities are determined as foows.

10 10 Processor (a) t max(0,5/8( -2)) (b) Fig. 4: (a) Unavaiabe time instants and (b) service function in Exampe 5. Definition 7. (prioritization rues) Associated with each job T i,j is a constant vaue χ i,j.ifχ i,j <χ k,h or χ i,j = χ k,h (i <k (i = k j<h)), then the priority of T i,j is higher than that of T k,h, denoted T i,j T k,h. Additionay, we assume j<h impies χ i,j χ i,h for each task T i. Exampe 6. Goba eariest-deadine-first (GEDF) priorities can be defined by setting χ i,j = r i,j + D i for each job T i,j, where D i is T i s reative deadine. Goba first-in-first-out (FIFO) priorities can be defined by setting χ i,j = r i,j, and static priorities can be defined by setting χ i,j to a constant (Leontyev and Anderson 2009a). The technica contributions of this paper are threefod. First, given a task set τ = {T 1,..., T n } and a mutiprocessor patform characterized by a cumuative guaranteed processor time B(Δ), we deveop a sufficient test that verifies whether the maximum job response time of a task T i τ, max j (f i,j r i,j ), is at most Θ i, where Θ i max j 1 (γu i (j) A 1 i (j 1)). (6) The right-hand side of (6) is the maximum job response time bound of T i when it is schedued on a dedicated processor. Consider a sequence of j consecutive jobs T i,a,...,t i,a+j 1 schedued on a dedicated processor such that T i,a starts its execution at r i,a and r i,k f i,k 1 for k [a +1,a+ j 1]. The response time of job T i,a+j 1 is f i,a+j 1 r i,a+j 1 =(f i,a+j 1 r i,a ) (r i,a+j 1 r i,a ). Because the processor is dedicated and jobs execute back-to-back, f i,a+j 1 r i,a γi u(j). Beow, in Section 6 in Lemma 2, we show that r i,a+j 1 r i,a A 1 i (j 1). Thus, f i,a+j 1 r i,a+j 1 γi u (j) A 1 i (j 1). If Θ i equas the reative deadine of a job, then the proposed test wi check whether the system is hard-rea-time scheduabe. Aternativey, if deadines are aowed to be missed and Θ i incudes the maximum aowed deadine tardiness, then the test wi check soft-rea-time scheduabiity. Such a test aows workoads to be considered that fundamentay require goba scheduing approaches. In settings where response-time bounds are not known, they must be determined. Our second contribution is a set of cosed-form expressions for cacuating responsetime bounds directy from task and suppy parameters for a arge cass of scheduing

11 11 agorithms. These response-time bounds can be directy used for cacuating stream and suppy outputs. It is aso possibe to refine the obtained response-time bounds by incrementay decreasing them and running the scheduabiity test to see if the smaer bounds are aso vaid. Finay, given per-task bounds on maximum job response times, we characterize the sequence of job competion events for each task T i by deriving the next-stage arriva functions α u i (Δ) and α i (Δ), and the remaining processor suppy B (Δ) (see Figure 3). These functions, in turn, can serve as inputs to subsequent PEs, thereby resuting in a compositiona technique. 4 Cacuating α u i and α i Let α u i (Δ) (α i (Δ)) be the maximum (respectivey, minimum) number of job competions of task T i over an interva (x, x + Δ], where x 0. Bounds on these functions can be computed using Theorem 1 beow. The foowing definition is used in the statement of the theorem. Definition 8. Let γ 1 i (Δ) =inf{k k is integra and γi (k) Δ}, where Δ>0,be the pseudoinverse function of the ower bound on the execution time function γi (k) (note that we use a non-strict inequaity because γi (k) is not defined for non-integra vaues of k). For Δ =0, we define γ 1 i (0) = 0. Exampe 7. The function γ 1 i (Δ) gives an upper bound on the number of jobs that can compete over any interva (x, x + Δ], where Δ>0. For exampe, if Δ = γi (1), then at most one job can compete over any interva (x, x + γi (1)]. Simiary, if Δ = γi (k) for some k, then at most k jobs can compete over any interva (x, x + γ i (k)]. Theorem 1. If the response time of any job of T i is at most Θ i, then αi u (Δ) ( ) min γ 1 i (Δ), αi u(δ + Θ i γi (1)) and αi (Δ) α i (Δ Θ i + γi (1)). Proof. We first prove the first inequaity. Consider an interva (t 1,t 2 ] such that at east one job of T i competes within it and et Δ = t 2 t 1. Let N 1,(N 2 ) be the index of the first (ast) job of T i competed within (t 1,t 2 ]. Then, f i,n1 >t 1 and f i,n2 t 2. (7) By the condition of the theorem, job T i,j s response time f i,j r i,j is at most Θ i. By the definition of response time and Definition 1, f i,j r i,j is at east γi (1). From (7), we thus have r i,n1 >t 1 Θ i and r i,n2 t 2 γi (1). Thus, the number of jobs competed within the interva (t 1,t 2 ], N 2 N 1 +1, is at most the number of jobs reeased within the interva (t 1 Θ i,t 2 γi (1)]. By Definition 2, we have N 2 N 1 +1 αi u(t 2 γi (1) t 1 + Θ i )=αi u(δ + Θ i γi (1)). Moreover, from Definition 8, it foows that at most γ 1 i (Δ) jobs can compete within an interva of ength Δ>0.

12 12 We now prove the second inequaity. Consider an interva (t 1,t 2 ] and et Δ = t 2 t 1. Let N 1,(N 2 ) be the index of the ast (respectivey, first) job of T i competed at or before time t 1 (respectivey, after time t 2 ). Then, f i,n1 t 1 and f i,n2 >t 2. (8) By the condition of the theorem, job T i,j s response time f i,j r i,j is at most Θ i. By the definition of response time and Definition 1, f i,j r i,j is at east γi (1). From (8), we thus have r i,n2 >t 2 Θ i and r i,n1 t 1 γi (1). Thus, the number of jobs competed within the interva (t 1,t 2 ], N 2 N 1 1, is at east the number of jobs reeased within the interva (t 1 γi (1),t 2 Θ i ]. By Definition 2, we have N 2 N 1 1 αi (t 2 Θ i t 1 + γi (1)) = α i (Δ Θ i + γi (1)). 5 Cacuating B (Δ) We now cacuate a ower bound B (Δ) on processor time that is avaiabe after scheduing tasks T 1,...,T n. We first upper-bound the tota aocation of jobs of T i over any interva of ength Δ. Definition 9. Let A(T i,y,i) (respectivey, A(T i,i)) be the amount of time for which job T i,y (respectivey, task T i ) executes within the set of intervas I. Lemma 1. If the response time of any job of T i is at most Θ i, then A(T i, [t, t+δ)) min(δ, γ u i (αu i (Δ + Θ i))). Proof. Consider an interva [t, t + Δ). The condition of the emma impies that a of T i s jobs reeased at or before time t Θ i compete by time t. Thus, the aocation of T i within [t, t + Δ), A(T i, [t, t + Δ)), is upper-bounded by the maximum execution demand of T i s jobs reeased within the interva (t Θ i,t+δ]. By Definition 2, there are at most α u i (Δ + Θ i) such jobs, and by Definition 1, their tota execution demand is at most γ u i (αu i (Δ+Θ i)). We thus have A(T i, [t, t+δ)) γ u i (αu i (Δ+Θ i)). Aso, A(T i, [t, t + Δ)) cannot exceed the ength of the interva [t, t + Δ). Theorem 2. If, for each task T i, the response time of any of its jobs is at most Θ i, then at east B (Δ) = sup (Z(y)) (9) 0 y Δ time units are avaiabe over any interva of ength Δ 0, where Z(y) = max ( 0, B(y) T i τ min(y, γu i (αu i (y +Θ i)) ). Additionay, (4) for B (Δ) hods with Û = Û U sum and σ tot =(Û σ tot + T i τ (u i Θ i + e i B i + v i ))/Û. Proof. Consider an interva [t, t + y), where y Δ. Let Suppy (t, y) be the amount of suppy that is avaiabe after scheduing the tasks in τ in this interva. By Defini-

13 13 tions 5 and 9, we have Suppy (t, y) =Suppy(t, y) T i τ A(T i, [t, t + y)) {by Definition 6} ( max 0, B(y) ) A(T i, [t, t + y)) T i τ {by Lemma 1} ( max 0, B(y) ) min(y, γi u (αi u (y + Θ i ))) T i τ {by the definition of Z(y) in the statement of the theorem} = Z(y). (10) Additionay, Suppy (t, Δ) sup 0 y Δ (Suppy (t, y)). From this inequaity and (10), we have Suppy (t, Δ) sup 0 y Δ (Z(y)) = B (Δ). We are eft with finding coefficients Û and σ tot such that (4) hods for B (Δ). Setting (4) (for B(Δ)) into the definition of Z(y), wehave ( Z(y) max 0, max(0, Û (y σ tot)) ) min(y, γi u (αi u (y+θ i ))) T i τ ( max 0, Û (y σ tot) ) min(y, γi u (αi u (y+θ i ))) T i τ {by (2) and (3)} ( max 0, Û (y σ tot) ) i (R i (y+θ i )+B i )+v i ) T i τ(e {by Definition 4} ( = max 0, Û (y σ tot) ) i y+u i Θ i +e i B i +v i ) T i τ(u ( = max 0, Û (y σ tot) U ) sum y+ i Θ i +e i B i +v i ) T i τ(u ( = max 0, (Û U sum) y Û σ tot ) i Θ i +e i B i +v i ) T i τ(u {by the definition of Û and σ tot in the statement of the theorem} ( ) = max 0, Û (y σ tot). (11) ( ( Finay, by (9) and (11), B (Δ) sup 0 y Δ max 0, Û (y σ tot) )) = max ( 0, Û (Δ σ tot) ). Thus, (4) hods with B (Δ), Û, and σ tot as defined.

14 14 6 Mutiprocessor Scheduabiity Test In this section, we present the core anaysis of our framework in the form of a scheduabiity test (given in Coroary 1 ater in this section) that checks whether a predefined response-time bound Θ i is not vioated for a task T i. As noted earier, the way jobs are prioritized according to Definition 7 is simiar to GEDF or static priority scheduing. A number of GEDF scheduabiity tests have been deveoped assuming that jobs arrive periodicay or sporadicay (e.g., (Baruah 2007; Bertogna et a 2009; Leontyev and Anderson 2008)). In this paper, we extend techniques from (Baruah 2007) and (Leontyev and Anderson 2008) in order to incorporate more genera job arrivas and execution modes. Simiary to (Devi 2006), we derive our test by ordering jobs by their priorities and assuming that T,q is the first job for which f,q >r,q + Θ. We further assume that, for each job T a,b such that T a,b T,q, f a,b r a,b + Θ a. (12) We first derive a necessary condition for T,q to vioate its response-time bound by considering an interva that incudes the time when T,q becomes ready and the atest time when T,q is aowed to compete, which is r,q + Θ. This interva is parametrized by a number k [1,K ] (see Definition 3) and δ (defined ater in this section), which determine its ength, δ + Θ. (The range of δ depends on k and.) In essence, the parameter k defines the number of T,q s predecessors (incuding T,q itsef) that compete too ate to warrant T,q s timey competion in the presence of other tasks. During this interva, we consider demand due to competing higherpriority jobs that can interfere with T,q or its predecessors. We then perform the foowing three steps: S1: Compute the minimum guaranteed suppy B(δ + Θ ) over the interva of interest. S2: Given a finite upper bound M (δ, τ, m) on the competing demand and a finite upper bound on the unfinished work due to job T,q and its predecessors, E (k), define a sufficient test for checking whether T s response-time bound is not vioated by checking that M (δ, τ, m)+(m 1) (E (k) 1) < B(δ + Θ ) hods for each k [1,K ] and δ defined with respect to k and. S3: Cacuate M (δ, τ, m) and E (k) as used in S Steps S1 and S2 To avoid distracting boundary cases, we henceforth assume that the schedue being anayzed is prepended with a schedue in which response-time bounds are not vioated that is ong enough to ensure that a predecessor jobs referenced in the proof exist. We begin with the foowing definition. Definition 10. Let α + i (Δ) = im ɛ +0 αi u (Δ + ɛ). This function provides an upper bound on the number of jobs reeased within any interva [x, x + Δ], where x 0 and Δ 0. (We assume α + i (Δ) =0for a Δ<0.)

15 15 The next exampe iustrates the difference between the functions αi u and α+ i. Exampe 8. Consider a task T i, whose jobs arrive periodicay with period p i. The maximum number of jobs that can arrive within an interva (x, x +2 p i ] is thus αi u(2 p 2 pi i)= p i =2. However, the maximum number of jobs that can arrive within the interva [x, x +2 p i ] is α + i (2 p i) = im ɛ +0 αi u(2 p i + ɛ) =3.In genera, under the sporadic task mode, α + i (Δ) = Δ p i +1. We start the derivation by proving a emma and severa caims. The foowing emma specifies the minimum time between the arrivas of jobs T h,g and T h,g i. Lemma 2. r h,g r h,g i A 1 h (i). Proof. Let Δ = r h,g r h,g i. Let Δ =inf{δ α + h (Δ) i +1}. (13) Because jobs T h,g i,...,t h,g are reeased within the interva [r h,g i,r h,g ], by Definition 10, α + h (Δ ) i +1. Therefore, by (13), We now consider two cases. Case 1: αh u {by Definition 10} (Δ ) >i. In this case, Δ (i). The emma foows from this and (14). A 1 h r h,g r h,g i = Δ Δ. (14) inf{δ α u h {by Definition 3} (Δ) >i} = Case 2: α u h (Δ ) i. Because α u h (Δ) is non-decreasing, αu h (Δ ) i impies α u (Δ) i for each Δ Δ. (15) Further, by (13), α + h (Δ) <i+1, for each Δ<Δ. (16) Suppose that for some Δ >Δ, αh u(δ ) i. Because αh u (Δ) is non-decreasing, this impies αh u(δ x) i for each Δ x [Δ,Δ ). The atter impies α + h (Δ x)= im ɛ +0 αh u(δ x + ɛ) i for each Δ x [Δ,Δ ). From this and (16), we have α + h (Δ) <i+1for each Δ<Δ. Since Δ >Δ, we have a contradiction to (13). Therefore, αh u(δ) >ifor each Δ>Δ. From this and (15), we have Δ = inf{δ αh u {by Definition 3} (Δ) >i} = A 1 h (i). The emma foows from this equaity and (14). The next two caims estabish a ower bound on the maximum job response time and an upper bound on the finish times of certain jobs that can be used in addition to (12). Caim 1: Θ i γi u(1). (1) A 1 i (0). By Defini- Proof. By (6), Θ i max j 1 (γi u tion 3, A 1 i (0) = 0. (j) A 1 i (j 1)) γi u

16 16 Caim 2: f,q K r,q + Θ γ u(k ). Proof. By (12), for i 1, f,q i r,q i + Θ = r,q i r,q + r,q + Θ {by Lemma 2} r,q + Θ A 1 (i). (17) By (1), A 1 (K ) γ u(k ). Setting this and i = K into (17), we get the required resut. Job T,q can vioate its response-time bound for the foowing reasons. If T,q 1 competes by time r,q +Θ γ u(1), then T,q may finish its execution after r,q +Θ if, after time max(f,q 1,r,q ), higher-priority jobs deprive it of processor time or one or more processors are unavaiabe. Aternativey, T,q 1 may compete after time r,q + Θ γ u (1), which can happen if the minimum job inter-arriva time for T is ess than γ u(1). In this situation, T,q coud vioate its response-time bound even if it executes uninterruptedy within [f,q 1,r,q + Θ ). In this case, T s responsetime bound is vioated because T,q 1 competes ate, namey after time r,q (reca that, by Caim 1, Θ γ u(1)). However, this impies that T is pending continuousy throughout the interva [r,q 1,r,q + Θ ), and hence, we can examine the execution of jobs T,q 1 and T,q together. In this case, we need to consider the competion time of job T,q 2.Iff,q 2 r,q + Θ γ u(2), then job T,q may exceed its response-time bound if this job and its predecessor, T,q 1, experience interference from higher-priority jobs or some processors are unavaiabe during the time interva [max(f,q 2,r,q 1 ),r,q + Θ ). On the other hand, if f,q 2 >r,q + Θ γ u(2), then T,q can compete after time r,q +Θ even if T executes uninterruptedy within [f,q 2,r,q +Θ ). Continuing by considering predecessor jobs T,q k in this manner, we wi exhaust a possibe reasons for the response-time bound vioation. Note that it is sufficient to consider ony jobs T,q 1,...,T,q K since, by Caim 2, f,q K r,q + Θ γ u(k ). Assuming that, for job T,q k, f,q k r,q + Θ γ u(k),we define the probem window for jobs T,q k+1,...,t,q as [r,q k+1,r,q +Θ ). (This probem window definition is a significant difference when comparing our anaysis to prior anaysis pertaining to periodic or sporadic systems.) Definition 11. Let λ [1,K ] be the smaest integer such that f,q λ r,q +Θ γ u (λ). By Caim 2, such a λ exists. Caim 3. T is ready (i.e., has a ready job) at each instant of the interva [r,q k+1, r,q + Θ ) for each k [1,λ]. Proof. To prove the caim, we first show that, for each k [1,λ], T is ready continuousy within [r,q k+1,f,q ). Because T is ready within the interva [r,q,f,q ), this is true for k =1.Ifk>1 (in which case λ>1), then f,q j >r,q + Θ γ u (j)

17 17 for each j [1,λ), by the seection of λ. From this, we have f,q j >r,q + Θ γ u (j) {because, by (6), Θ γ u (j) A 1 (j 1)} r,q A 1 (j 1) {by Lemma 2} r,q j+1. Thus, the intervas [r,q j,f,q j ) and [r,q j+1,f,q j+1 ), where consecutive jobs of T are ready, overap. Therefore, T is ready continuousy within [r,q j,f,q ) for each j [1,λ), and hence, T is ready continuousy within [r,q k+1,f,q ) for each k [2,λ]. The caim foows from [r,q k+1,r,q + Θ ) [r,q k+1,f,q );tosee this, note that f,q >r,q +Θ hods, since T,q vioates its response-time bound. Because T,q vioates its response-time bound, after time r,q k+1, there are other higher-priority jobs that deprive T of processor time or one or more processors are unavaiabe. Definition 12. Let τ p (t) ={T h for some y, T h,y is ready at time t and T h,y T,q }. (The subscript p denotes the fact that these jobs have higher or equa priority.) To indicate an excessive number of tasks with ready jobs of equa or higher priority at time t we wi use the foowing predicate. IS HP(t)=( τ p (t) m or fewer than τ p (t) tasks from τ p (t) execute at time t). (18) Definition 13. Let t 0 (k) r,q k+1 be the eariest instant such that t [t 0 (k), r,q k+1 ),IS HP(t) hods. If such an instant does not exist, then et t 0 (k) =r,q k+1. The definition beow defines the jobs that can compete with T,q or its predecessors. Definition 14. Let J be the set of jobs T i,y such that (i) T i,y T,q or (ii) T i,y T,q, i, and T i,y executes at some time t [t 0 (k),r,q + Θ ) and IS HP(t) hods. (More informay, J incudes higher-or-equa-priority jobs and ower-priority jobs that cause non-preemptive bocking.) Note that J does not contain T,q s successors. In this paper, we are mainy concerned with fuy preemptive scheduing (i.e., J contains ony higher-or-equa-priority jobs). However, the introduced definitions are constructed to support both fuy preemptive and non-preemptive execution. Uness stated otherwise, we do not distinguish between the preemptive and non-preemptive cases. However, if non-preemptive execution is assumed, then we assume that a processors are aways avaiabe. We discuss other differences in the anaysis of the non-preemptive case in Section 6.3 and eave the consideration of non-preemptivity in the face of partia avaiabiity to future work.

18 18 T,q competing jobs (in set J) unavaiabe time job reease m-1 T,q-1 T,q T,q T,q time t( 0 r,q r,q + Fig. 5: Conditions for a response-time bound vioation for λ =1. Definition 13 generaizes the we-known concept of an ide instant in uniprocessor scheduing with respect to jobs in J, as iustrated in Figure 5, which shows the response-time bound vioation for job T,q assuming λ =1. Our scheduabiity test for task T is based upon summing the demand of competing jobs as defined above in Definition 14 executing within the interva [t 0 (λ),r,q + Θ ), which has ength r,q t 0 (λ)+θ, and the unavaiabe time within this interva (see Figure 5). Definition 15. Let A J (T i,i)= T A(T i,y J i,y,i) be the aocation of task T i s jobs in J over a set of intervas I. Definition 16. Let Res h (I) be the amount of time that is not avaiabe on processor h at time instants in the set of intervas I. Definition 17. We ca a processor J -busy at time t if it executes a job in J or is unavaiabe. The tota time for which processors are J -busy within a set of intervas I is caed the J -aocation for I and is defined as ÂJ (I) = T A i τ J (T i,i)+ m h=1 Res h(i). The foowing definition is used to cacuate ÂJ ([t 0 (λ),r,q + Θ )). Definition 18. Let Γ λ [r,q λ+1,r,q + Θ ) be the set of intervas where a processors are J -busy as shown in Figure 5. Let Γ λ =[r,q λ+1,r,q + Θ ) \ Γ λ.we et Γ λ (respectivey, Γ λ ) denote the tota ength of the intervas in Γ λ (respectivey, Γ λ ). We next cacuate the J -aocations for Γ λ, [t 0 (λ),r,q λ+1 ), and Γ λ. Because a processors are J -busy within Γ λ, Â J (Γ λ ) = m Γ λ. We now consider the interva [t 0 (λ),r,q λ+1 ). Caim 4. A processors are J -busy within [t 0 (λ),r,q λ+1 ); that is, Â J ([t 0 (λ),r,q λ+1 )) = m (r,q λ+1 t 0 (λ)). Proof. Suppose that a processor is not J -busy at time t [t 0 (λ),r,q λ+1 ). Then it is either avaiabe and ide or executes a job that is not in J. If the processor is ide at time t, then, because the scheduer being anayzed is work-conserving, a tasks in τ p (t) execute at time t and thus τ p (t ) m 1. Thus, by (18), IS HP(t ) is fase, which vioates Definition 13. Aternativey, if, at time t, the processor executes

19 19 a job T x,y J, then, by Definition 14, T x,y T,q and, by (18), IS HP(t ) is fase, which aso vioates Definition 13. The given expression for ÂJ ([t 0 (λ),r,q λ+1 )) foows. Finay, we consider the interva set Γ λ. Caim 5. Task T executes at each time t Γ λ, and hence, A J (T, Γ λ )= Γ λ. Proof. Suppose to the contrary that T does not execute at some time t Γ λ.by Definition 18, there exists processor P that is not J -busy at time t. By Caim 3, task T is ready at each time t [r,q λ+1,r,q + Θ ).IfP is ide, then, because the scheduer is work-conserving, a tasks in τ p (t), incuding T execute at time t. IfP executes job T x,y J, then, by Definition 14 and (18), IS HP(t) is fase, which impies that a tasks in τ p (t), incuding T execute at time t. In either case, we have a contradiction. Lemma 3: Â J ([t 0 (λ),r,q + Θ )) m (r,q λ+1 t 0 (λ)) + m Γ λ + Γ λ. Proof. We sum up the J -aocations for intervas [t 0 (λ),r,q λ+1 ), Γ λ, and Γ λ (see Figure 5; note that r,q λ+1 = r,q here). Â J ([t 0 (λ),r,q + Θ )) = ÂJ ([t 0 (λ),r,q λ+1 )) + ÂJ (Γ λ )+ÂJ(Γ λ ) {by Caim 4 and Definition 18} = m (r,q λ+1 t 0 (λ)) + m Γ λ + ÂJ (Γ λ ) {by Definition 17} m (r,q λ+1 t 0 (λ)) + m Γ λ + A J (T, Γ λ ) {by Caim 5} = m (r,q λ+1 t 0 (λ)) + m Γ λ + Γ λ The vaues of Γ λ and Γ λ depend on the amount of competing work due to T,q s predecessors (incuding T,q itsef), which is determined as foows. Definition 19. Let W (T i,y,t) denote the remaining execution time for job T i,y (if any) at time t. Let W J (T i,t)= T i,y J W (T i,y,t). In Figure 5, W J (T,r,q λ+1 ) corresponds to the execution demand of job T,q and the unfinished work of job T,q 1 at time r,q. Caim 6. (Proved in an appendix.) W J (T,r,q λ+1 ) r,q + Θ r,q λ+1. The foowing emma estabishes constraints on the tota ength of the intervas Γ λ and Γ λ. Lemma 4. If the response-time bound for T,q is vioated (as we have assumed), then Γ λ = r,q +Θ r,q λ+1 W J (T,r,q λ+1 )+1+μ, where μ 0. (Note that, by Caim 6, this impies that Γ λ > 0). Additionay, Γ λ = W J (T,r,q λ+1 ) 1 μ.

20 20 Proof. Suppose, contrary to the statement of the emma, that the response-time bound for T,q is vioated and μ<0, i.e., Γ λ <r,q + Θ r,q λ+1 W J (T,r,q λ+1 )+1. (19) Under these conditions, the tota ength of the intervas in Γ λ, where at east one avaiabe processor is not J -busy, is r,q +Θ r,q λ+1 Γ λ > W J (T,r,q λ+1 ) {by (19)} 1. Thus, this tota ength is at east W J (T,r,q λ+1 ), as time is integra. By Caim 5, job T,q or one of its predecessors executes at each time t Γ λ. Thus, job T,q competes by time r,q + Θ, which is a contradiction. Hence, μ 0. Γ λ can be found as Γ λ = r,q + Θ r,q λ+1 Γ λ = W J (T,r,q λ+1 ) 1 μ. In the statement of Theorem 3, which defines a scheduabiity condition, the functions defined beow are used. Definition 20. Let E (k) be a finite function of k such that W J (T,r,q λ+1 ) E (λ). Definition 21. Let M (δ, τ, m) be a finite function of δ, τ, and m such that A J (T i, [t 0 (λ),r,q + Θ )) M (r,q t 0 (λ),τ,m). T i τ (As mentioned earier at the beginning of Section 6, M (δ, τ, m) and E (k) are cacuated in order to test whether the response-time bound of T is not vioated. Later, in Section 6.2, we expain how M (δ, τ, m) and E (k) are cacuated.) Definition 22. We require that there exists a constant H 0 such that, for a δ 0, M (δ, τ, m) U sum δ + H. (20) This requirement is reasonabe because the growth rate of the tota demand over the interva of interest, which has ength δ + Θ, cannot be arger than the tota ong-term utiization of the tasks in τ for arge vaues of δ. This aso aows us to upper-bound our test s computationa compexity. Henceforth, we omit the ast two arguments of M. Definition 23. Let δ max (k) = (H +(m 1) (E (k) 1)+Û σ tot Θ Û)/(Û U sum ). We next cacuate an upper bound on Res h ([t 0 (λ),r,q + Θ )). For processor h and the interva [t 0 (λ),r,q + Θ ), by Definition 5, Res h ([t 0 (λ),r,q + Θ )) = (r,q t 0 (λ)+θ ) suppy h (t 0 (λ),r,q t 0 (λ)+θ ). (21)

21 21 Summing (21) for a h, wehave m Res h ([t 0 (λ),r,q + Θ )) h=1 = m ( (r,q t 0 (λ)+θ ) suppy h (t 0 (λ),r,q t 0 (λ)+θ ) ) h=1 {by Definition 5} = m (r,q t 0 (λ)+θ ) Suppy(t 0 (λ),r,q t 0 (λ)+θ ) {by Definition 6} m (r,q t 0 (λ)+θ ) B(r,q t 0 (λ)+θ ). (22) The foowing theorem wi be used to define our scheduabiity test. Theorem 3. If the response-time bound Θ is vioated for T,q (as we have assumed), then, for k = λ and some δ [A 1 (λ 1),δ max (λ)] (such that δ = r,q t 0 (λ)), M (δ)+(m 1) (E (k) 1) B(δ+Θ ). (23) Proof. Consider job T,q, k = λ, and time instants r,q λ+1 and t 0 (λ) as defined in Definitions 11 and 13. To estabish (23), we consider the tota J -aocation within the interva [t 0 (λ),r,q + Θ ). By Definition 17 and Lemma 3, m A J (T i, [t 0 (λ),r,q + Θ )) + Res([t 0 (λ),r,q + Θ )) T i τ h=1 m (r,q λ+1 t 0 (λ)) + m Γ λ + Γ λ {by Lemma 4} = m (r,q λ+1 t 0 (λ))+m (r,q +Θ r,q λ+1 W J (T,r,q λ+1 )+1+μ) + W J (T,r,q λ+1 ) 1 μ = m (r,q t 0 (λ)+θ ) (m 1) (W J (T,r,q λ+1 ) 1) + (m 1) μ {because μ 0} m (r,q t 0 (λ)+θ ) (m 1) (W J (T,r,q λ+1 ) 1). (24) Setting (22) into (24), we have A J (T i, [t 0 (λ),r,q + Θ )) + m (r,q t 0 (λ)+θ ) B(r,q t 0 (λ)+θ ) T i τ m (r,q t 0 (λ)+θ ) (m 1) (W J (T,r,q λ+1 ) 1). Rearranging the terms in the above inequaity, we have A J (T i, [t 0 (λ),r,q +Θ ))+(m 1) (W J (T,r,q λ+1 ) 1) T i τ B(r,q t 0 (λ)+θ ).

22 22 Setting E (λ) and M (r,q t 0 (λ)) as defined in Definitions 20 and 21 into the inequaity above, we have M (r,q t 0 (λ))+(m 1) (E (λ) 1) B(r,q t 0 (λ)+θ ). Setting r,q t 0 (λ) =δ into the inequaity above, we get (23). Our remaining proof obigation is to estabish the stated range for δ. Note that, by Definition 13, δ = r,q t 0 (λ) r,q r,q λ+1 A 1 (λ 1), where the ast inequaity foows from Lemma 2. By (20) and (23), we have for k = λ, U sum δ + H +(m 1) (E (k) 1) B(δ + Θ ). (25) Appying (4) to (25), we have U sum δ + H +(m 1) (E (k) 1) max(0, Û (δ + Θ σ tot )) Û (δ + Θ σ tot ). Soving the atter inequaity for δ, wehaveδ (H +(m 1) (E (k) 1) + Û σ tot Θ Û)/(Û U sum). Because δ is integra (as r,q and t 0 (k) are integra), by Definition 23, δ δ max (k). The theorem foows. Coroary 1. (Scheduabiity Test) If, for each task T τ, (23) does not hod for each k [1,K ] and δ [A 1 (k 1),δ max (k)], then no response-time bound is vioated. Proof. The coroary foows from Theorem 3 and Definition 11, which impies λ [1,K ]. In Section 8, we improve the above scheduabiity test for fixed-job-priority preemptive scheduers such as GEDF and FIFO by repacing the term (m 1) (E (k) 1) in (23) with a smaer term proportiona to max(m F 1, 0) E (k), where F is the number of processors that are aways avaiabe (see Definition 6). This can be done because, under GEDF and FIFO, the probem job T,q and its predecessors cannot be preempted by other jobs after a certain time point uness the competing demand carried from previous time instants is sufficienty arge. 6.2 Step S3 (Cacuating M (δ) and E (k)) Note that we did not make any assumptions above about how jobs are schedued except that the jobs of each task execute sequentiay and jobs are prioritized as in Definition 7. Therefore, Coroary 1 is appicabe to a fixed job-priority scheduing poicies (these poicies incude preemptive variants of EDF, FIFO, static-priority poicies, and their various combinations; non-preemptive variants can be supported simiary as discussed ater in Section 6.3) provided the functions M (δ) (and its inear upper bound in Definition 22) and E (k) are known. M (δ) and E (k) can be derived for a particuar agorithm by extending techniques from previousy-pubished

23 23 papers on the scheduabiity of sporadic tasks (Baruah 2007; Leontyev and Anderson 2008) to incorporate more genera arriva and execution patterns. In this section, we derive the functions E (k) and M (δ) for a fuy preemptive prioritization scheme in which χ i,j = r i,j + D i, where D i is a constant (preemptive goba EDF and FIFO are the subcases of this scheme). Note that in this case the set J ony contains jobs with higher or equa priority than that of T,q. We first prove some properties about jobs in the set J. Definition 24. Let C i,k = D i D k. Lemma 5. If T,q vioates its response-time bound and job T a,b is in J, then T a,b T,q and r a,b r,q + C,a. Proof. Consider job T a,b J. Case 1: T a,b T,q. By Definition 7, r a,b + D a r,q + D. The required resut foows from Definition 24. Case 2: T a,b T,q. By Definition 14, T a,b executes at some time t [t 0 (λ),r,q + Θ ) and IS HP(t) hods. By (18), since T a,b executes at time t, there exists task T x τ p (t) such that job T x,y is ready at t, T x,y T,q T a,b and T x,y does not execute at t. This contradicts the assumption of fu preemptivity. Derivation of M (δ). To derive M (δ), we first note that, by Lemma 5, ony jobs T a,b T,q beong to J and can compete with T,q or its predecessors. Definition 25. Let T h,bh be the eariest pending job of T h at time t 0 (k). We separate the tasks that may compete with T,q into two disjoint sets: HC = {T h :: (T h,bh exists) (r h,bh <t 0 (k)) (T h,bh J)}; NC = {T h :: (r h,bh t 0 (k)) (T h,bh J)}. Here, HC denotes high-priority carry-in and NC denotes non-carry-in. Caim 7: HC m 1. Proof. By Definitions 12 and 25, HC τ p (t 0 (k) 1). By Definition 13, a tasks in τ p (t 0 (k) 1) execute at t 0 (k) 1 and τ p (t 0 (k) 1) m 1. Thus, HC m 1. Since the cumuative ength of [t 0 (k),r,q +Θ ), depends on the difference r,q t 0 (k), we use A NC (T i,r,q t 0 (k)) and A HC (T i,r,q t 0 (k)) to denote an upperbound on A J (T i, [t 0 (k),r,q + Θ )) for the case when T i is in NC and HC, respectivey. With this notation, we have A J (T i, [t 0 (λ),r,q + Θ )) T i τ A HC (T i,r,q t 0 (λ))+ A NC (T i,r,q t 0 (λ)). (26) T i HC T i NC We provide expressions for computing A NC (T i,δ) and A HC (T i,δ) in the foowing two emmas. Their proofs can be found in the appendix.