Evaluate and Compare Two Utilization-Based Schedulability-Test Frameworks for Real-Time Systems
EEvaluate and Compare Two Utilization-BasedSchedulability-Test Frameworks for Real-Time Systems
Jian-Jia Chen and Wen-Hung HuangDepartment of InformaticsTU Dortmund University, Germany Cong LiuDepartment of Computer ScienceThe University of Texas at Dallas
Abstract —This report summarizes two general frameworks,namely k Q and k U , that have been recently developed by us.The purpose of this report is to provide detailed evaluations andcomparisons of these two frameworks. These two frameworksshare some similar characteristics, but they are useful for dif-ferent application cases. These two frameworks together providecomprehensive means for the users to automatically convert thepseudo polynomial-time tests (or even exponential-time tests) intopolynomial-time tests with closed mathematical forms. With thequadratic and hyperbolic forms, k Q and k U frameworks canbe used to provide many quantitive features to be measured andevaluated, like the total utilization bounds, speed-up factors, etc.,not only for uniprocessor scheduling but also for multiprocessorscheduling. These frameworks can be viewed as “blackbox”interfaces for providing polynomial-time schedulability tests andresponse time analysis for real-time applications. We have alreadypresented their advantages for being applied in some models inthe previous papers. However, it was not possible to present amore comprehensive comparison between these two frameworks.We hope this report can help the readers and users clearlyunderstand the difference of these two frameworks, their uniquecharacteristics, and their advantages. We demonstrate theirdifferences and properties by using the traditional sporadic real-time task models in uniprocessor scheduling and multiprocessorglobal scheduling. To analyze the worst-case response time or to ensure thetimeliness of the system, for each of individual task models,researchers tend to develop dedicated techniques that resultin schedulability tests with different computation complexityand accuracy of the analysis. Although many successful resultshave been developed, after many real-time systems researchersdevoted themselves for many years, there does not exist ageneral framework that can provide efficient and effectiveanalysis for different task models.A very widely adopted case is the schedulability test of a(constrained-deadline) sporadic real-time task τ k under fixed-priority scheduling in uniprocessor systems, in which the time-demand analysis (TDA) developed in [21] can be adopted. Thatis, if ∃ t with < t ≤ D k and C k + (cid:88) τ i ∈ hp ( τ k ) (cid:24) tT i (cid:25) C i ≤ t, (1)then task τ k is schedulable under the fixed-priority schedulingalgorithm, where hp ( τ k ) is the set of tasks with higher prioritythan τ k , D k , C k , and T i represent τ k ’s relative deadline, worst-case execution time, and period, respectively. TDA requirespseudo-polynomial-time complexity to check the time pointsthat lie in (0 , D k ] for Eq. (1). However, it is not always necessary to test all possibletime points to derive a safe worst-case response time or toprovide sufficient schedulability tests. The general and keyconcept to obtain sufficient schedulability tests in k U in[10], [11] and k Q in [9], [12] is to test only a subset ofsuch points for verifying the schedulability. Traditional fixed-priority schedulability tests often have pseudo-polynomial-time(or even higher) complexity. The idea implemented in the k U and k Q frameworks is to provide a general k -pointschedulability test, which only needs to test k points under any fixed-priority scheduling when checking schedulability ofthe task with the k th highest priority in the system. Moreover,this concept is further extended in k Q to provide a safeupper bound of the worst-case response time. The responsetime analysis and the schedulability analysis provided by theframeworks can be viewed as “ blackbox ” interfaces that canresult in sufficient utilization-based analysis, in which theutilization of a task is its execution time divided by its period.The k U and k Q frameworks are in fact two differ-ent important components for building efficient and effectiveschedulability tests and response time analysis. Even thoughthey come from the same observations by testing only k effective points, they are in fact fundamentally different frommathematical formulations and have different properties. In k U , all the testings and formulations are based on the taskutilizations. In k Q , the testings are based not only on the taskutilizations, but also on the task execution times. The differentformulations of testings result in different types of solutions.The natural form of k U is a hyperbolic form for testing theschedulability of a task, whereas the natural form of k Q isa quadratic form for testing the schedulability or the responsetime. In general, if the k points can be effectively defined, k U has more precise results. However, if these k points cannot beeasily defined or there is some ambiguity to fine the effectivepoints, then k Q may be more suitable for such models.There have been several results in the literature with respectto utilization-based, e.g., [17]–[19], [23], [24] for the sporadicreal-time task model and its generalizations in uniprocessorsystems. The novelty of k U and k Q comes from a differ-ent perspective from these approaches [17]–[19], [23], [24].We do not specifically seek for the total utilization bound.Instead, we look for the critical value in the specified sufficientschedulability test while verifying the schedulability of task τ k .The natural condition to test the schedulability of task τ k is ahyperbolic bound when k U is adopted, whereas the naturecondition to test task τ k is a quadratic bound when k Q isadopted (to be shown in Section 3). The corresponding totalutilization bound can be obtained.1 a r X i v : . [ c s . D S ] S e p he generality of the k Q and k U frameworks hasbeen demonstrated in [9]–[12]. We believe that these twoframeworks, to be used for different cases, have great potentialin analyzing many other complex real-time task models, wherethe existing analysis approaches are insufficient or cumber-some. We have already presented their advantages for beingapplied in some models in [9]–[12]. However, it was notpossible to present a more comprehensive comparison betweenthese two frameworks in [9]–[12]. We hope this report canhelp the readers and users clearly understand the difference ofthese two frameworks, their unique characteristics, and theiradvantages. Since our focus in this report is only to demon-strate the similarity, the difference and the characteristics ofthese two frameworks, we will use the simplest setting, i.e.,the traditional sporadic real-time task models in uniprocessorscheduling and multiprocessor global scheduling.For the k Q and k U frameworks, their characteristics andadvantages over other approaches have been already discussedin [9]–[12]. However, between these two frameworks, weonly gave short sketches and high-level descriptions of theirdifferences and importance. These explanations may seemincomplete in [9]–[12] to explain whether both are neededor only one of them is important. Therefore, we would liketo present in this report to explain why both frameworks areneeded and have to be applied for different cases. Moreover,we would like to emphasize that both frameworks are im-portant. In general, the k U framework is more precise byusing only the utilization values of the higher-priority tasks.If we can formulate the schedulability tests into the k U framework, it is also usually possible to model it into the k Q framework. In such cases, the same pseudo-polynomial-time test is used. When we consider the worst-case quantitivemetrics like utilization bounds, resource augmentation bounds,or speedup factors, the result derived from the k U frameworkis better for such cases. However, there are also cases, inwhich formulating the test by using the k U framework isnot possible. These cases may even start from schedulabilitytests with exponential-time complexity. We have successfullydemonstrated three examples in [9] by using the k Q frame-work to derive polynomial-time tests. In those demonstratedcases, either the k U framework cannot be applied or withworse results (since different exponential-time or pseudo-polynomial-time schedulability tests are applied). Organizations.
The rest of this report is organized as follows: • The basic terminologies and models are presented inSection 2. • The two frameworks from [9]–[12] are summarized andpresented in Section 3. • We demonstrate two different comparisons between theframeworks by using sporadic task systems in uniproces-sor systems and multiprocessor systems.Note that this report does not intend to provide new theoreticalresults. All the omitted proofs are already provided in [9]–[12]. For some simple properties derived from the results in[9]–[12], we will explain how such results are derived.
This report will demonstrate the effectiveness and differ-ences of the two frameworks by using the sporadic real-time task model, even though the frameworks target at more generaltask models. We define the terminologies in this section forcompleteness. A sporadic task τ i is released repeatedly, witheach such invocation called a job. The j th job of τ i , denoted τ i,j , is released at time r i,j and has an absolute deadline attime d i,j . Each job of any task τ i is assumed to have executiontime C i . Here in this report, whenever we refer to the executiontime of a job, we mean for the worst-case execution time of thejob, since all the analyses we use are safe by only consideringthe worst-case execution time. The response time of a job isdefined as its finishing time minus its release time. Successivejobs of the same task are required to execute in sequence.Associated with each task τ i are a period T i , which specifiesthe minimum time between two consecutive job releases of τ i , and a deadline D i , which specifies the relative deadline ofeach such job, i.e., d i,j = r i,j + D i . The worst-case responsetime of a task τ i is the maximum response time among all itsjobs. The utilization of a task τ i is defined as U i = C i /T i .A sporadic task system τ is said to be an implicit-deadlinesystem if D i = T i holds for each τ i . A sporadic task system τ is said to be a constrained-deadline system if D i ≤ T i holdsfor each τ i . Otherwise, such a sporadic task system τ is anarbitrary-deadline system.A task is said schedulable by a scheduling policy if allof its jobs can finish before their absolute deadlines, i.e., theworst-case response time of the task is no more than its relativedeadline. A task system is said schedulable by a schedulingpolicy if all the tasks in the task system are schedulable. A schedulability test is to provide sufficient conditions to ensurethe feasibility of the resulting schedule by a scheduling policy.Throughout the report, we will focus on fixed-prioritypreemptive scheduling. That is, each task is associated witha priority level. More specifically, we will only use ratemonotonic (RM, i.e., tasks with smaller periods are with higherpriority levels) and deadline monotonic (DM, i.e., tasks withsmaller relative deadlines are with higher priority levels) inthis report. For a uniprocessor system, the scheduler alwaysdispatches the job with the highest priority in the ready queueto be executed. For a multiprocessor system, we considermultiprocessor global scheduling on M identical processors,in which each of them has the same computation power. Forglobal multiprocessor scheduling, there is a global queue and aglobal scheduler to dispatch the jobs. We consider only globalfixed-priority scheduling. At any time, the M -highest-priorityjobs in the ready queue are dispatched and executed on these M processors.Note that the above definitions are just for simplifying thepresentation flow in this report. The frameworks can still workfor non-preemptive scheduling and different types of fixed-priority scheduling.We will only present the schedulability test of a certaintask τ k , that is being analyzed, under the above assump-tion. For notational brevity, in the framework presentation,we will implicitly assume that there are k − tasks, says τ , τ , . . . , τ k − with higher-priority than task τ k . These k − higher-priority tasks are assumed to schedulable before wemove on to test task τ k . We will use hp ( τ k ) to denote the setof these k − higher priority tasks, when their orderings donot matter. Moreover, we only consider the cases when k ≥ ,2ince k = 1 is pretty trivial. k U and k Q Frameworks
This section presents the definitions and properties of the k U and k Q frameworks for testing the schedulability oftask τ k in a given set of real-time task. The construction ofthe frameworks requires the following definitions: Definition 1. A k -point effective schedulability test is a suf-ficient schedulability test of a fixed-priority scheduling pol-icy, that verifies the existence of t j ∈ { t , t , . . . t k } with < t ≤ t ≤ · · · ≤ t k such that C k + k − (cid:88) i =1 α i t i U i + j − (cid:88) i =1 β i t i U i ≤ t j , (2) where C k > , α i > , U i > , and β i > are dependentupon the setting of the task models and task τ i . Definition 2 (Last Release Time Ordering) . Let π be thelast release time ordering assignment as a bijective function π : hp ( τ k ) → { , , . . . , k − } to define the last release timeordering of task τ j ∈ hp ( τ k ) in the window of interest. Lastrelease time orderings are numbered from to k − , i.e., | hp ( τ k ) | , where 1 is the earliest and k − the latest. Definition 3. A k -point last-release schedulability test under agiven ordering π of the k − higher priority tasks is a sufficientschedulability test of a fixed-priority scheduling policy, thatverifies the existence of ≤ t ≤ t ≤ · · · ≤ t k − ≤ t k suchthat C k + k − (cid:88) i =1 α i t i U i + j − (cid:88) i =1 β i C i ≤ t j , (3) where C k > , for i = 1 , , . . . , k − , α i > , U i > , C i ≥ ,and β i > are dependent upon the setting of the task modelsand task τ i . Definition 4. A k -point last-release response time analysis isa safe response time analysis of a fixed-priority schedulingpolicy under a given ordering π of the k − higher-prioritytasks by finding the maximum t k = C k + k − (cid:88) i =1 α i t i U i + k − (cid:88) i =1 β i C i , (4) with ≤ t ≤ t ≤ · · · ≤ t k − ≤ t k and C k + k − (cid:88) i =1 α i t i U i + j − (cid:88) i =1 β i C i > t j , ∀ j = 1 , , . . . , k − , (5) where C k > , α i > , U i > , C i ≥ , and β i > aredependent upon the setting of the task models and task τ i . Throughout the report, we implicitly assume that < t k (cid:54) = ∞ when Definition 1 and Definition 3 are used, as t k is usuallyrelated to the given relative deadline requirement. Note that t k may still become ∞ when Definition 4 for response timeanalysis is used. Moreover, we only consider non-trivial cases,in which C k > , and α i > , β i > , C i ≥ , and < U i ≤ for i = 1 , , . . . , k − . The definition of the k -point last-release schedulabilitytest C k + (cid:80) k − i =1 α i t i U i + (cid:80) j − i =1 β i C i ≤ t j in Definition 3only slightly differs from the k -point effective schedulabilitytest C k + (cid:80) k − i =1 α i t i U i + (cid:80) j − i =1 β i t i U i ≤ t j in Definition 1.However, since the tests are different, they are used fordifferent situations and the resulting bounds and properties arealso different.In Definition 1, the k -point effective schedulability test isa sufficient schedulability test by testing only k time points,defined by the k − higher-priority tasks and task τ k . These k − points defined by the k − higher-priority tasks can bearbitrary as long as the corresponding α i > and β i > canbe defined. In Definition 3, the k − points defined by the k − higher-priority tasks have to be the last release times ofthe highest-priority tasks, and the k − higher-priority taskshave to be indexed according to their last release time before t k . In Definition 3, the last release time ordering π is assumedto be given. In some cases, this ordering can be easily obtained.However, in some of the cases in our demonstrated task modelsin [9], the last release ordering cannot be defined. It may seemthat we have to test all possible last release time orderingsand take the worst case. Fortunately, finding the worst-caseordering is not a difficult problem, which requires to sort the k − higher-priority tasks under a simple criteria. Therefore,before adopting the k Q framework, we have to know whetherwe can obtain the last release time ordering or we have toconsider a pessimistic ordering for the higher priority tasks.The frameworks assume that the corresponding coefficients α i and β i in Definitions 1, 3, and 4 are given. How toderive them depends on the task models and the schedulingpolicies. Provided that these coefficients α i , β i , C i , U i forevery higher priority task τ i ∈ hp ( τ k ) are given, we can findthe worst-case assignments of the values t i for the higher-priority tasks τ i ∈ hp ( τ k ) . Therefore, in case Definition 1 isadopted, changing t i affects the values α i t i U i and β i t i U i ; incase Definitions 3 and 4 are adopted, changing t i only affectsthe value α i t i U i . By using the above approach, we can analyze(1) the response time by finding the extreme case for a given C k (under Definition 4), or (2) the schedulability by findingthe extreme case for a given C k and t k (under Definitions 1and Definition 3).In Section 4, we will give a comparison about the differ-ence of Definition 1 and Definition 3 based on uniprocessorschedulability tests for sporadic tasks. k U By using the property defined in Definition 1, we can havethe following lemmas in the k U framework [10], [11]. Allthe proofs of the following lemmas are in [10], [11]. Lemma 1.
For a given k -point effective schedulability test of ascheduling algorithm, defined in Definition 1, in which < t k and < α i ≤ α , and < β i ≤ β for any i = 1 , , . . . , k − , task τ k is schedulable by the scheduling algorithm if thefollowing condition holds C k t k ≤ αβ + 1 (cid:81) k − j =1 ( βU j + 1) − αβ . (6)3 emma 2. For a given k -point effective schedulability test of ascheduling algorithm, defined in Definition 1, in which < t k and < α i ≤ α and < β i ≤ β for any i = 1 , , . . . , k − ,task τ k is schedulable by the scheduling algorithm if C k t k + k − (cid:88) i =1 U i ≤ ( k − α + β ) k −
1) + (( α + β ) k − α ) β . (7) Lemma 3.
For a given k -point effective schedulability test of ascheduling algorithm, defined in Definition 1, in which < t k and < α i ≤ α and < β i ≤ β for any i = 1 , , . . . , k − ,task τ k is schedulable by the scheduling algorithm if β k − (cid:88) i =1 U i ≤ ln( αβ + 1 C k t k + αβ ) . (8) Lemma 4.
For a given k -point effective schedulability test ofa fixed-priority scheduling algorithm, defined in Definition 1,task τ k is schedulable by the scheduling algorithm, in which < t k and < α i and < β i for any i = 1 , , . . . , k − , ifthe following condition holds < C k t k ≤ − k − (cid:88) i =1 U i ( α i + β i ) (cid:81) k − j = i ( β j U j + 1) . (9) k Q By using the property defined in Definition 3, we can havethe following lemmas in the k Q framework [9], [12]. All theproofs of the following lemmas are in [9], [12]. Lemma 5.
For a given k -point last-release schedulabilitytest, defined in Definition 3, of a scheduling algorithm, inwhich < α i , and < β i for any i = 1 , , . . . , k − , < t k , (cid:80) k − i =1 α i U i ≤ , and (cid:80) k − i =1 β i C i ≤ t k , task τ k isschedulable by the fixed-priority scheduling algorithm if thefollowing condition holds C k t k ≤ − k − (cid:88) i =1 α i U i − (cid:80) k − i =1 ( β i C i − α i U i ( (cid:80) k − (cid:96) = i β (cid:96) C (cid:96) )) t k . (10)It may seem at first glance that we need to test all thepossible orderings. Fortunately, with the following lemma, wecan safely consider only one specific ordering of the k − higher priority tasks. Lemma 6.
The worst-case ordering π of the k − higher-priority tasks under the schedulability condition in Eq. (10) inLemma 5 is to order the tasks in a non-increasing order of β i C i α i U i , in which < α i and < β i for any i = 1 , , . . . , k − ,and < t k . The analysis in Lemma 5 uses the execution time and theutilization of the tasks in hp ( τ k ) to build an upper bound of C k /t k for schedulability tests. It is also very convenient inreal-time systems to build schedulability tests only based onutilization of the tasks. We explain how to achieve that in thefollowing lemmas under the assumptions that < α i ≤ α ,and < β i C i ≤ βU i t k for any i = 1 , , . . . , k − . Theselemmas are useful when we are interested to derive utilization bounds, speed-up factors, resource augmentation factors, etc.,for a given scheduling policy by defining the coefficients α and β according to the scheduling policies independently from thedetailed parameters of the tasks. Since the property repeatsin all the statements, we make a formal definition beforepresenting the lemmas. Definition 5.
Lemmas 7 to 9 are based on the following k -point last-release schedulability test of a scheduling algorithm,defined in Definition 3, in which < α i ≤ α , and < β i C i ≤ βU i t k for any i = 1 , , . . . , k − , < t k , α (cid:80) k − i =1 U i ≤ ,and β (cid:80) k − i =1 U i ≤ . Lemma 7.
For a given k -point last-release schedulability testof a scheduling algorithm, with the properties in Definition 5,task τ k is schedulable by the scheduling algorithm if thefollowing condition holds C k t k ≤ − ( α + β ) k − (cid:88) i =1 U i + αβ k − (cid:88) i =1 U i ( k − (cid:88) (cid:96) = i U (cid:96) ) (11) =1 − ( α + β ) k − (cid:88) i =1 U i + 0 . αβ (cid:32) ( k − (cid:88) i =1 U i ) + ( k − (cid:88) i =1 U i ) (cid:33) (12) Lemma 8.
For a given k -point last-release schedulability testof a scheduling algorithm, with the properties in Definition 5,task τ k is schedulable by the scheduling algorithm if k − (cid:88) i =1 U i ≤ (cid:18) k − k (cid:19) α + β − (cid:113) ( α + β ) − αβ (1 − C k t k ) kk − αβ . (13) Lemma 9.
For a given k -point last-release schedulability testof a scheduling algorithm, with the properties in Definition 5,provided that α + β ≥ , then task τ k is schedulable by thescheduling algorithm if C k t k + k − (cid:88) i =1 U i ≤ (cid:0) k − k (cid:1) α + β − (cid:113) ( α + β ) − αβ kk − αβ , if k > ( α + β ) − α + β − and α + β >
11 + ( k − α + β − − ( α + β ) +0 . kαβ otherwise (14) The right-hand side of Eq. (14) (when α + β > )decreases with respect to k . Similarly, the right-hand sideof Eq. (13) also decreases with respect to k . Therefore, forevaluating the utilization bounds, it is alway safe to take k → ∞ as a safe upper bound. The right-hand side of Eq. (13)converges to α + β − (cid:113) α + β +2 αβ Cktk αβ when k → ∞ . The right-hand side of Eq. (14) (when α + β > ) converges to α + β − √ α + β αβ when k → ∞ .The following two lemmas are from the k -point last-releaseresponse time analysis, defined in Definition 4. Lemma 10.
For a given k -point last-release response timeanalysis of a scheduling algorithm, defined in Definition 4, inwhich < α i ≤ α , < β i ≤ β for any i = 1 , , . . . , k − , < t k and (cid:80) k − i =1 α i U i < , the response time to execute C k emonstrated Applications: Sec. 5: Uniprocessor sporadic tasksSec. 6: Multiprocessor global RMand several others in [9]–[12]. U i , ∀ i < kC i , ∀ i < kα i , ∀ i < kβ i , ∀ i < kC k t k (for Lemmas 5-9) Derive parametersby Definitions 3 or 4 k Q framework QuadraticboundOtherutilizationboundsResponse-time test d e fi n e t h e l a s t r e l ea s e ti m e o r d e r i ng π o r u s e L e mm a r L e m m a L e m m a s - L e mm a U i , ∀ i < kα i , ∀ i < kβ i , ∀ i < kC k , t k Derive parametersby Definition 1 k U framework HyperbolicboundOtherutilizationboundsExtremepoints test d e fi n e t i , ∀ i < k a ndo r d e r k − t a s k s L e m m a L e m m a s & L e mm a Fig. 1: The k U and k Q frameworks. for task τ k is at most C k + (cid:80) k − i =1 β i C i − (cid:80) k − i =1 α i U i ( (cid:80) k − (cid:96) = i β (cid:96) C (cid:96) )1 − (cid:80) k − i =1 α i U i . (15) Lemma 11.
The worst-case ordering π of the k − higher-priority tasks under the response bound in Eq. (15) inLemma 10 is to order the tasks in a non-increasing order of β i C i α i U i , in which < α i and < β i for any i = 1 , , . . . , k − , < t k . The k U and k Q frameworks can be used by a widerange of applications, as long as the users can properly specifythe corresponding task properties C i (in case of k Q ) and U i and the constant coefficients α i and β i of every higherpriority task τ i . The choice of the parameters α i and β i affectsthe quality of the resulting schedulability bounds. However,deriving the good settings of α i and β i is not the focus of theframeworks. The frameworks do not care how the parameters α i and β i are obtained. It simply derives the bounds accordingto the given parameters α i and β i , regardless of the settings of α i and β i . The correctness of the settings of α i and β i is notverified by the frameworks. Figure 1 provides an overview ofthe procedures.The ignorance of the settings of α i and β i actually leadsto the elegance and the generality of the frameworks, whichwork as long as Definitions 1, 3, or 4 can be successfullyconstructed for the sufficient schedulability test or the responsetime analysis. The other approaches in [8], [17], [19] also havesimilar observations by testing only several time points in theTDA schedulability analysis based on Eq. (1) in their problemformulations. However, as these approaches in [8], [17], [19] seek for the total utilization bounds, they have limited applica-tions and are less flexible. For example, they are typically notapplicable directly when considering sporadic real-time taskswith arbitrary deadlines or multiprocessor systems.The k U and k Q frameworks provide comprehensivemeans for the users to almost automatically convert the pseudopolynomial-time tests (or even exponential-time tests) intopolynomial-time tests with closed mathematical forms. Withthe availability of the k U and k Q frameworks, the hy-perbolic bounds, quadratic bounds, or speedup factors can bealmost automatically derived by adopting the provided lemmasin Section 3 as long as the safe upper bounds α and β to coverall the possible settings of α i and β i for the schedulability testor the response-time analysis can be derived, regardless of thetask model or the platforms.The above characteristics and advantages over other ap-proaches have been already discussed in [9]–[12]. However,between these two frameworks, it is unclear whether both areneeded or only one of them is important.As the simplest example, consider the test of task τ with T = 1 in an implicit-deadline sporadic task set in uniprocessorRM scheduling. Suppose that task τ has utilization U =0 . . If we only use the utilization of the higher-priority tasksas the means of testing, modeling the schedulability test inDefinition 3 is less precise since we may have to inflate andset C properly according to the given priority assignment.Using Definition 1 with t = 0 . leads to C = 0 . , butusing Definition 3 with any < t ≤ can only be feasibleif we set C to . . Therefore, for such cases, we can only besafe by putting C i = t k U i , and, therefore, using k Q is morepessimistic than using k U .In the above example, it may seem at first glance thatthe test in the k U framework is better than the test in the5 U T /T Lemma 1 - k2ULemma 5 - k2QLemma 7 - k2Q
Fig. 2: Adopting different tests from k U and k Q for RMuniprocessor scheduling with k = 2 . k Q framework. However, this observation can only hold ifa schedulability test can be applicable to satisfy Definition 1and Definition 3.We test the above case with different settings of T T with T > T when U is . . Figure 2 illustrates the maximumutilization of task τ by using different tests from the twoframeworks. In such a case, we can clearly define t as (cid:108) T T − (cid:109) T . Therefore, α is and β is set to (cid:108) T T − (cid:109) whenadopting Lemma 1 from k U . Moreover, α is and β isset to (cid:108) T T − (cid:109) when adopting Lemma 7 from k Q .As shown in Figure 2, when we adopt only utilizations ofthe higher-priority task, i.e., Lemma 1 from k U and Lemma7 from k Q , the results from k U are always better. However,the results of Lemma 1 from k U and Lemma 5 from k Q are not comparable.Therefore, there is no clear dominator between these twoframeworks. Moreover, there are also cases, in which formu-lating the test by using the k U framework is not possible (c.f.the results in Theorems 5 and 11). These cases may even startfrom schedulability tests with exponential-time complexity.We have successfully demonstrated three examples in [9] byusing the k Q framework to derive polynomial-time tests withapproximation guarantees. In those demonstrated cases, eitherthe k U framework cannot be applied or with worse results(since different exponential-time or pseudo-polynomial-timeschedulability tests are applied). This section examines the applicability of the k U and k Q frameworks to derive utilization-based schedulabilityanalysis and response-time analysis for sporadic task systemsin uniprocessor systems. We will consider constrained-deadlinesystems in Section 5.1 and arbitrary-deadline systems in Sec-tion 5.2. For a specified fixed-priority scheduling algorithm,let hp ( τ k ) be the set of tasks with higher priority than τ k . Wenow classify the task set hp ( τ k ) into two subsets: • hp ( τ k ) consists of the higher-priority tasks with periodssmaller than D k . • hp ( τ k ) consists of the higher-priority tasks with periodslarger than or equal to D k .For the rest of this section, we will implicitly assume C k > . For a constrained-deadline task τ k , the schedulability testin Eq. (1) is equivalent to the verification of the existence of < t ≤ D k such that C k + (cid:88) τ i ∈ hp ( τ k ) C i + (cid:88) τ i ∈ hp ( τ k ) (cid:24) tT i (cid:25) C i ≤ t. (16)We can then create a virtual sporadic task τ (cid:48) k with executiontime C (cid:48) k = C k + (cid:80) τ i ∈ hp ( τ k ) C i , relative deadline D (cid:48) k = D k ,and period T (cid:48) k = D k . It is clear that the schedulability test toverify the schedulability of task τ (cid:48) k under the interference ofthe higher-priority tasks hp ( τ k ) is the same as that of task τ k under the interference of the higher-priority tasks hp ( τ k ) . Fornotational brevity, suppose that there are k ∗ − tasks, indexedas , , . . . , k ∗ − , in hp ( τ k ) . Adopting k U : Setting t i = (cid:16)(cid:108) D k T i (cid:109) − (cid:17) T i for everytask τ i in hp ( τ k ) , and indexing the tasks in a non-decreasingorder of t i lead to the satisfaction of Definition 1 with α i = 1 and < β i ≤ . Therefore, we can apply Lemmas 1 and 2 toobtain the following theorem. Theorem 1.
Task τ k in a sporadic task system with constraineddeadlines is schedulable by the fixed-priority scheduling algo-rithm if ( C (cid:48) k D k + 1) (cid:89) τ j ∈ hp ( τ k ) ( U j + 1) ≤ (17) or C (cid:48) k D k + (cid:88) τ j ∈ hp ( τ k ) U j ≤ k ∗ (2 k ∗ − . (18) Corollary 1.
Task τ k in a sporadic task system with implicitdeadlines is schedulable by the RM scheduling algorithm ifLemmas 2 and 3 holds by setting C k t k as U k , α = 1 , and β = 1 . The above result in Corollary 1 leads to the utilizationbound ln 2 (by using Lemma 2 with α = 1 and β = 1 ) forRM scheduling, which is the same as the Liu and Laylandbound ln 2 [23]. It also leads to the hyperbolic bound for RMscheduling by Bini and Buttazzo [6] when adopting Theorem 1directly. Adopting k Q : Setting t i = (cid:16)(cid:108) D k T i (cid:109) − (cid:17) T i for everytask τ i in hp ( τ k ) , and indexing the tasks in a non-decreasingorder of t i leads to the satisfaction of Definition 3 with α i = 1 and β i = 1 . For such a case, the last release ordering is well-defined by the sorting of the tasks above. Therefore, we canuse Lemma 5 to derive the following theorem. Theorem 2.
Task τ k in a sporadic task system with constraineddeadlines is schedulable by the fixed-priority scheduling algo- ithm if (cid:80) k ∗ − i =1 C i D k ≤ and C (cid:48) k D k ≤ − k ∗ − (cid:88) i =1 U i − k ∗ − (cid:88) i =1 C i D k + (cid:80) k ∗ − i =1 U i ( (cid:80) k ∗ − (cid:96) = i C (cid:96) ) D k , (19) in which the k ∗ − higher priority tasks in hp ( τ k ) are indexedin a non-decreasing order of (cid:16)(cid:108) D k T i (cid:109) − (cid:17) T i . Corollary 2.
Task τ k in a sporadic task system with implicitdeadlines is schedulable by the RM scheduling algorithm ifLemmas 5, 7, 8, or 9 holds by setting C k t k as U k , α = 1 , and β = 1 . The above result in Corollary 2 leads to the utilizationbound − √ (by using Lemma 9 with α = 1 and β = 1 )for RM scheduling, which is worse than the existing Liu andLayland bound ln 2 [23].Moreover, the above utilization bound −√ has been alsoprovided by Abdelzaher et al. [1] for uniprocessor deadline-monotonic scheduling when an aperiodic task may generatedifferent task instances (jobs) with different combinations ofexecution times and minimum inter-arrival times. Such a modelis a more general model than the sporadic task model. Undersuch a setting, the k U framework cannot be used, whereasthe k Q framework is very suitable. The schedulability analysis for arbitrary-deadline sporadictask sets is to use a busy-window concept to evaluate the worst-case response time [20]. That is, we release all the higher-priority tasks together with task τ k at time and all thesubsequent jobs are released as early as possible by respectingto the minimum inter-arrival time. The busy window finisheswhen a job of task τ k finishes before the next release of ajob of task τ k . It has been shown in [20] that the worst-caseresponse time of task τ k can be found in one of the jobs oftask τ k in the busy window. For the h -th job of task τ k inthe busy window, let the finishing time R k,h is the minimum t such that hC k + k − (cid:88) i =1 (cid:24) tT i (cid:25) C i ≤ t, and, hence, its response time is R k,h − ( h − T k . The busywindow of task τ k finishes on the h -th job if R k,h ≤ hT k .A simpler sufficient schedulability test for a task τ k is totest whether the length of the busy window is within D k . Ifso, all invocations of task τ k released in the busy window canfinish before their relative deadline. Such an observation hasalso been adopted in [13]. Therefore, a sufficient test is toverify whether ∃ t with < t ≤ D k and (cid:24) tT k (cid:25) C k + (cid:88) τ i ∈ hp ( τ k ) (cid:24) tT i (cid:25) C i ≤ t. (20)If the condition in Eq. (20) holds, it implies that the busywindow (when considering task τ k ) is no more than D k , and,hence, task τ k has worst-case response time no more than D k .Similarly, we can use hp ( τ k ) and hp ( τ k ) , as in Sec-tion 5.1, and, then create a virtual sporadic task τ (cid:48) k with execution time C (cid:48) k = (cid:108) D k T k (cid:109) C k + (cid:80) τ i ∈ hp ( τ k ) C i , relativedeadline D (cid:48) k = D k , and period T (cid:48) k = D k . For notationalbrevity, suppose that there are k ∗ − tasks, indexed as , , . . . , k ∗ − , in hp ( τ k ) . Adopting k U : Setting t i = (cid:16)(cid:108) D k T i (cid:109) − (cid:17) T i , andindexing the tasks in a non-decreasing order of t i leads to thesatisfaction of Definition 1 with α i = 1 and β i ≤ . Therefore,we can apply Lemmas 1 and 2 to obtain the following theorem. Theorem 3.
Task τ k in a sporadic task system with arbitrarydeadlines is schedulable by the fixed-priority scheduling algo-rithm if ( C (cid:48) k D k + 1) (cid:89) τ j ∈ hp ( τ k ) ( U j + 1) ≤ (21) or C (cid:48) k D k + (cid:88) τ j ∈ hp ( τ k ) U j ≤ k ∗ (2 k ∗ − . (22) Adopting k Q : If we use the busy-window conceptto analyze the schedulability of task τ i by using Eq. (20), wecan reach the following theorem directly by Lemma 5. Theorem 4.
Task τ k in a sporadic task system is schedulableby the fixed-priority scheduling algorithm if (cid:80) k ∗ − i =1 C i D k ≤ and C (cid:48) k D k ≤ − k ∗ − (cid:88) i =1 U i − k ∗ − (cid:88) i =1 C i D k + (cid:80) k ∗ − i =1 U i ( (cid:80) k ∗ − (cid:96) = i C (cid:96) ) D k , (23) in which C (cid:48) k = (cid:108) D k T k (cid:109) C k + (cid:80) τ i ∈ hp ( τ k ) C i , and the k ∗ − higher priority tasks in hp ( τ k ) are indexed in a non-decreasing order of (cid:16)(cid:108) D k T i (cid:109) − (cid:17) T i . Analyzing the schedulability by using Theorem 4 can begood if D k T k is small. However, as the busy window may bestretched when D k T k is large, it may be too pessimistic. Supposethat t j = (cid:16)(cid:108) R k,h T j (cid:109) − (cid:17) T j for a higher priority task τ j . Weindex the tasks such that the last release ordering π of the k − higher priority tasks is with t j ≤ t j +1 for j = 1 , , . . . , k − .Therefore, we know that R k,h is upper bounded by finding themaximum t k = hC k + k − (cid:88) i =1 t i U i + k − (cid:88) i =1 C i , (24)with ≤ t ≤ t ≤ · · · ≤ t k − ≤ t k and hC k + k − (cid:88) i =1 t i U i + j − (cid:88) i =1 C i > t j , ∀ j = 1 , , . . . , k − . (25)Therefore, the above derivation of R k,h satisfies Definition 4with α i = 1 , and β i = 1 for any higher priority task τ i .However, it should be noted that the last release time ordering π is actually unknown since R k,h is unknown. Therefore, wehave to apply Lemma 11 for such cases to obtain the worst-case ordering. Lemma 12.
Suppose that (cid:80) k − i =1 U i ≤ . Then, for any h ≥ nd C k > , we have R k,h ≤ hC k + (cid:80) k − i =1 C i − (cid:80) k − i =1 U i ( (cid:80) k − (cid:96) = i C (cid:96) )1 − (cid:80) k − i =1 U i , (26) where the k − higher-priority tasks are ordered in a non-increasing order of their periods. The worst-case response time for such cases can be set to h = 1 , in which the detailed proof is in [9], [12]. Theorem 5.
Suppose that (cid:80) ki =1 U i ≤ . The worst-caseresponse time of task τ k is at most R k ≤ C k + (cid:80) k − i =1 C i − (cid:80) k − i =1 U i ( (cid:80) k − (cid:96) = i C (cid:96) )1 − (cid:80) k − i =1 U i , (27) where the k − higher-priority tasks are ordered in a non-increasing order of their periods. Corollary 3.
Task τ k in a sporadic task system is schedulableby the fixed-priority scheduling algorithm if (cid:80) ki =1 U i ≤ and C k D k ≤ − k − (cid:88) i =1 U i − (cid:80) k − i =1 C i D k + (cid:80) k − i =1 U i ( (cid:80) k − (cid:96) = i C (cid:96) ) D k , (28) where the k − higher-priority tasks are ordered in a non-increasing order of their periods. k Q and k U The utilization-based worst-case response-time analysis inTheorem 5 is analytically tighter than the best known result, R k ≤ C k + (cid:80) k − i =1 C i − (cid:80) k − i =1 U i C i − (cid:80) k − i =1 U i , by Bini et al. [7]. Lehoczky[20] also provides the total utilization bound of RM schedulingfor arbitrary-deadline systems. The analysis in [20] is basedon the Liu and Layland analysis [23]. The resulting utilizationbound is a function of ∆ = max τ i { D i T i } . When ∆ is , it isan implicit-deadline system. The utilization bound in [20] hasa closed-form when ∆ is an integer. However, calculating theutilization bound for non-integer ∆ is done asymptotically for k = ∞ with complicated analysis. Bini [5] provides a totalutilization bound of RM scheduling, based on the quadraticresponse time analysis in [7], that works for any arbitrary ratioof max τ i { D i T i } .For constrained-deadline sporadic task sets, since the sametest in Eq. (16) is used for constructing Definition 1 andDefinition 3, the result (with respect to the conditions inTheorem 1, Corollary 1, Theorem 5, and Corollary 2) byusing k U is superior to that by using k Q . The speedupfactor of the test in Eq. (17) in Theorem 3 has been provedto be . , which is also better than that in Eq. (19) inTheorem 4. However, the quadratic bound in Eq. (19) can bebetter than the hyperbolic bound in Eq. (17), as demonstratedin the evaluations.For arbitrary-deadline sporadic task sets, two differenttests are applied: one comes from Eq. (20) for constructingTheorem 3 and Theorem 4 and another comes from Eqs. (24) The speedup factor for the schedulability test by using Eq. (19) is . Thisis obtained by ignoring the last term in the right-hand-side of Eq. (19). Sincethis is not analytically superior, the analysis was not shown in [9]. and (25) for construction Theorem 5 and Corollary 3. It shouldbe clear that the test from Eqs. (24) and (25) is tighter thanthat from Eq. (20). Therefore, these results are not analyticallycomparable.Note that we can also use Lemma 4 by defining the valuesof α i and β i for each task τ i in hp ( τ k ) precisely to makethe hyperbolic bound in Eq. (17) and Eq. (21) more precisely.Their performance will be provided in the evaluation results. The rest of this section presents our evaluation results forthe above tests. We generated a set of sporadic tasks. Thecardinality of the task set was . The UUniFast-Discardmethod [14] was adopted to generate a set of utilizationvalues with the given goal. We used the approach suggestedby Davis et al. [15] to generate the task periods accordingto a uniform distribution in the range of the logarithm ofthe task periods (i.e., log-uniform distribution). The order ofmagnitude p to control the period values between the largestand smallest periods is parameterized in evaluations, (e.g., − ms for p = 1 , − ms for p = 2 , etc.). Weevaluate these tests in uniprocessor systems with p ∈ [1 , , .The priority ordering of the tasks is assigned according todeadline-monotonic (DM) scheduling. The execution time wasset accordingly, i.e., C i = T i U i .The metric to compare results is to measure the acceptanceratio of the above tests with respect to a given task setutilization. We generate 100 task sets for each utilization level.The acceptance ratio of a level is said to be the number of tasksets that are schedulable under the schedulability test dividedby the number of task sets for this level, i.e., 100. Task relative deadlines were uniformly drawn from theinterval [0 . T i , T i ] . The evaluated tests are as follows: • RTA : the exact response time test by Lehoczky et al. [21]. • Bini : the linear-time response time bound by Bini etal. [7]. • HP (from k U ): Eq. (17) in Theorem 1 in this report. • HP-EP (from k U ): using Lemma 4 (with a more preciseextreme point) by defining the values of α i and β i for eachtask τ i in hp ( τ k ) precisely in this report. This improvesHP. • QB (from k Q ): Eq. (19) in Theorem 2 in this report. Results.
Figure 3 shows that the performance of the abovetests in terms of acceptance ratios, for three different settingsof p . The tests by HP-EP, Bini, QB, and RTA are sensitiveto p : the larger the value of p is, the more the test sets theyadmit. In the case of p = 1 , the test by Bini (the QB test,respectively) can admit all task sets with their total utilizationsof up to ( , respectively), and its performance startsto decline at utilization ( , respectively). On the otherhand, the tests by HP and HP-EP can fully accept a task setwith around more utilizations, but acceptance ratio of HPdrops sharply and becomes completely ineffective at utilization .8 .0 0.2 0.4 0.6 0.8 1.0 Utilization A cc e p t a n c e R a t i o (a) p=1 HP HP-EP Bini QB RTA (b) p=2 (c) p=3 Fig. 3: Performance evaluation on uniprocessor systems in terms of acceptance ratio for constrained-deadline uniprocessor systemswhere D i T i ∈ [0 . , .In the case of p = 1 , we can also see that test HP derivedfrom k U and test QB derived from k Q are incomparable.HP itself becomes pessimistic since we do not take the differ-ent values of α i to have more precise tests, whereas HP-EP ismore precise. In general, for uniprocessor constrained-deadlinetask systems, we can observe that HP-EP outperforms the otherpolynomial-time tests. Due to the analytical dominance, wealso see that the QB test dominates the test by Bini. Task relative deadlines were uniformly drawn from theinterval [ T i , T i ] . The tests evaluated are shown as follows: • RTA : the exact response time test by Lehoczky [20]. • Bini : the linear-time response time bound by Bini etal. [7]. • HP-Busy (from k U ): Eq. (21) from Theorem 3 in thisreport • HP-EP (from k U ): using Lemma 4 (with a more preciseextreme point) by defining the values of α i and β i for eachtask τ i in hp ( τ k ) precisely in this report. This improvesHP-Busy. • QB-Busy (from k Q ): Eq. (23) in Theorem 4 in thisreport. • QB-Response (from k Q ): Eq. (28) in Corollary 3 in thisreport. Results.
Figure 4 compares the performance on arbitrary-deadline uniprocessor system where D i T i ∈ [1 , . Analytically,we know that test by QB is superior to that by Bini, whichis the best-known test for arbitrary-deadline uniprocessorsystems. The results shown in Figure 4 also support suchdominance.In the case of p = 1 , the acceptance ratio by Bini decreasessteadily from utilization to . On the other hand,the number of task sets accepted by QB-Response starts todecrease at utilization . Test QB-Response is able to admitmore task tests from utilization to , compared to testBini. With utilization more , Bini performs better thanQB-Busy. In the other cases, Bini outperforms QB-Busy.For arbitrary-deadline systems, since the test in Eq. (20) istoo pessimistic by checking whether the busy-window length is no more than D k , HP-Busy, HP-EP, and QB-Busy do notperform very well. In the above experimental results, thequadratic forms by using k Q are better than the hyperbolicforms by using k U in such cases. This is due to the fact thatthese two tests start from different pseudo-polynomial timetests. This section demonstrates the two frameworks for multi-processor global fixed-priority scheduling. We consider that thesystem has M identical processors. For global fixed-priorityscheduling, there is a global queue and a global schedulerto dispatch the jobs. We demonstrate the applicability forimplicit-deadline sporadic systems under global RM.Unfortunately, unlike uniprocessor systems, up to now,there is no exact schedulability test to verify whether task τ k isschedulable by global RM. Therefore, existing schedulabilitytests (in pseudo-polynomial time or exponential time) areonly sufficient tests. We will use three different tests fordemonstrating the use of the k U and k Q frameworks andcompare their results.One way to quantify the quality of the resulting schedu-lability test is to use the capacity augmentation factor [22].Suppose that the test is to verify whether the total utilization (cid:80) τ i U i M ≤ b and the maximum utilization max τ i U i ≤ b . Sucha factor b has been recently named as a capacity augmentationfactor [22].We only consider testing the schedulability of task τ k underglobal RM, where k > M . For k ≤ M , the global RMscheduling guarantees the schedulability of task τ k if U k ≤ .Without loss of generality, we limit our presentation to thecase that T i < T k for i = 1 , , . . . , k − , for the simplicity ofpresentation. We now present three different tests that require pseudo-polynomial-time or exponential-time complexity.9 .0 0.2 0.4 0.6 0.8 1.0
Utilization A cc e p t a n c e R a t i o (a) p=1HP-Busy HP-EP Bini QB-Busy QB-Response RTA (b) p=2 (c) p=3 Fig. 4: Performance evaluation on uniprocessor systems in terms of acceptance ratio for arbitrary-deadline uniprocessor systemswhere D i T i ∈ [1 , Greedy-Carry-In : The first one is based on a simple observa-tion to carry-in a job for each of the higher-priority tasks in thewindow of interest [16]. The following time-demand function W i ( t ) can be used for a simple sufficient schedulability test: W i ( t ) = (cid:18)(cid:24) tT i (cid:25) − (cid:19) C i + 2 C i . (29)That is, we allow the first release of task τ i to be inflated bya factor , whereas the other jobs of task τ i have the sameexecution time C i . Therefore, task τ k is schedulable underglobal RM on M identical processors, if ∃ t with < t ≤ T k and C k + (cid:88) τ i ∈ hp ( τ k ) W i ( t ) M ≤ t, (30) Bounded-Carry-In : The second test is based on the observa-tion by Guan et al. [16] that we only have to consider M − tasks with carry-in jobs, for constrained-deadline task sets. Forimplicit-deadline task sets, this means that we only need toset α i of some tasks to M , rather than all the k − tasks inEq. (29). More precisely, we can define two different time-demand functions, depending on whether task τ i is with acarry-in job or not: W carryi ( t ) = (cid:40) C i < t < C i C i + (cid:108) t − C i T i (cid:109) C i otherwise, (31)and W normali ( t ) = (cid:24) tT i (cid:25) C i . (32)Moreover, we can further over-approximate W carryi ( t ) , since W carryi ( t ) ≤ W normali ( t )+ C i . Therefore, a sufficient schedu-lability test for testing task τ k with k > M for global RM isto verify whether ∃ < t ≤ T k , C k + ( (cid:80) τ i ∈ T (cid:48) C i ) + ( (cid:80) k − i =1 W normali ( t )) M ≤ t. (33)for all T (cid:48) ⊆ hp ( τ k ) with | T (cid:48) | = M − . It is not necessary toenumerate all T (cid:48) with | T (cid:48) | = M − if we can construct the This is an over-approximation of the linear function used by Guan et al.[16]. task set T (cid:48) ⊆ hp ( τ k ) with the maximum (cid:80) τ i ∈ T (cid:48) C i . Forced-Forward : The third one is based on a reformulationof the forced-forward approach by Baruah et al. [3]. This isthe reformulation in [9] based on a simple revision of theforced-forward algorithm in [3]. Let U max k be max kj =1 { U j } .As shown and proved in [9], task τ k in a sporadic task systemwith implicit deadlines is schedulable by a global RM on M processors if ∀ y ≥ , ∀ ≤ ω i ≤ T i , ∀ τ i ∈ hp ( τ k ) , ∃ t with < t ≤ T k + yU max k · ( T k + y ) + (cid:80) k − i =1 ω i · U i + (cid:108) t − ω i T i (cid:109) C i M ≤ t. (34)The schedulability condition in Eq. (34) requires to test allpossible y ≥ and all possible settings of ≤ ω i ≤ T i forthe higher priority tasks τ i with i = 1 , , . . . , k − . Therefore,it needs exponential time (for all the possible combinations of ω i ). k U We now demonstrate how the k U framework can beadopted. Based on Greedy-Carry-In : Such a case is pretty clear bysetting < α i ≤ M and < β i ≤ M in Definition 1 for task τ i ∈ hp ( τ k ) . Therefore, by using Lemma 1 and Lemma 3, wehave the following theorem. Theorem 6.
Task τ k in a sporadic implicit-deadline tasksystem is schedulable by global RM on M processors if ( C k T k + 2) (cid:89) τ i ∈ hp ( τ k ) ( U i M + 1) ≤ , (35) or (cid:88) τ i ∈ hp ( τ k ) U i M ≤ ln 3 C k T k + 2 . (36) Based on Bounded-Carry-In : There are two ways to use k U . In the first case, we consider that C i for task τ i in hp ( τ k )
10s known. For such a case, we simply have to put the M − higher-priority tasks with the largest execution times into T (cid:48) .This can be imagined as if we increase the execution timeof task τ k from C k to C (cid:48) k = C k + (cid:80) τi ∈ T (cid:48) C i M . Therefore, westill have < α i ≤ M and < β i ≤ M for τ i ∈ hp ( τ k ) .Therefore, by using Lemma 1 and Lemma 3, we have thefollowing theorem: Theorem 7.
Task τ k in a sporadic implicit-deadline tasksystem is schedulable by global RM on M processors if ( C (cid:48) k T k + 1) (cid:89) τ i ∈ hp ( τ k ) ( U i M + 1) ≤ , (37) or (cid:88) τ i ∈ hp ( τ k ) U i M ≤ ln 2 C (cid:48) k T k + 1 , (38) where C (cid:48) k = C k + (cid:80) τi ∈ T (cid:48) C i M . In the second case, if only the task utilizations are given,we are not sure which tasks should be put into the carry-intask set T (cid:48) . That is, if we are testing the worst-case periodassignments of the higher-priority tasks in hp ( τ k ) , we needto enumerate T (cid:48) . Nevertheless, if T (cid:48) with | T (cid:48) | = M − isspecified, the translation to the k U framework is as follows:(1) the parameters are < α i ≤ M and < β i ≤ M by usingEq. (33) if τ i is in T (cid:48) , and (2) the parameters are α i = M and < β i ≤ M by using Eq. (33) if τ i is not in T (cid:48) . It may seemat first glance that we have to check all possible permutationsof T (cid:48) . Fortunately, with the analysis in [10], [11], the worstpermutation of T (cid:48) is to the M − higher-priority tasks with thelargest utilization into T (cid:48) . This leads to the following theoremby extending Lemma 4. Theorem 8.
Task τ k in a sporadic implicit-deadline tasksystem is schedulable by global RM on M processors if < U k ≤ − k − (cid:88) i =1 U i ( α i + M ) (cid:81) k − j = i ( M U j + 1) , (39) by indexing the k − higher-priority tasks in a non-decreasingorder of U i and assigning α , α , . . . , α k − M to M and α k − M +1 , α k − M +2 , . . . , α k − to M . Based on Forced-Forward : Formulating the test in Eq. (34)into the k U framework is problematic. Suppose that T k is .Assume that y is set to , ω i is set to . , and T i is set to . .Under the above setting, t i is . , and α i is , β i is . . Infact, we even cannot safely set β i to any possible value except ∞ if t i is small enough. Therefore, constructing parametersbased on Definition 1 is not possible (or non-trivial). k Q Based on Greedy-Carry-In : This is possible by setting <α i ≤ M and < β i ≤ M and applying Lemma 5. However,since the results are not superior to the one with bounded-carry-in, we omit it. Based on Bounded-Carry-In : To use k Q , we are certainabout which tasks should be put into the carry-in task set T (cid:48) by assuming that C i and T i are both given. That is, we simply have to put the M − higher-priority tasks with the largestexecution times into T (cid:48) . This can be imagined as if we increasethe execution time of task τ k from C k to C (cid:48) k = C k + (cid:80) τi ∈ T (cid:48) C i M .This leads to the following theorem by using Lemma 5. Theorem 9.
Task τ k in a sporadic implicit-deadline tasksystem is schedulable by global RM on M processors if (cid:80) k − i =1 C i ≤ M T k and U k ≤ − (cid:80) τ i ∈ T (cid:48) C i MT k − k − (cid:88) i =1 U i M − (cid:80) k − i =1 C i MT k + (cid:80) k − i =1 ( U i (cid:80) k − (cid:96) = i C (cid:96) ) M T k . (40) by indexing the k − higher-priority tasks in a non-decreasingorder of ( (cid:108) T k T i (cid:109) − T i and by putting the M − higher-prioritytasks with the largest execution times into T (cid:48) . We can of course revise the statement in Theorem 9 byadopting Lemma 7 and Lemma 8 to construct schedulabilitytests by using only task utilizations.
Based on Forced-Forward : We present the correspondingpolynomial-time schedulability tests for global fixed-priorityscheduling. By using the forced-forward test, we can adoptthe k Q framework by setting α i = M and β i = M . Due tothe fact that T i ≤ T k for any task τ i ∈ hp ( τ k ) , i.e., C i ≤ U i T k ,under global RM, we can reach the following theorems andcorollary, where the proofs are in [9]. Theorem 10.
Let U max k be max kj =1 U j . Task τ k in a sporadictask system with implicit deadlines is schedulable by globalRM on M processors if U max k ≤ − k − (cid:88) i =1 U i M − (cid:80) k − i =1 C i M T k + (cid:80) k − i =1 U i ( (cid:80) k − (cid:96) = i C (cid:96) )) M T k , (41) by ordering the k − higher-priority tasks in a non-increasingorder of T i . Theorem 11.
Let U max k be max kj =1 U j . Task τ k in a sporadictask system with implicit deadlines is schedulable by globalRM on M processors if U max k ≤ − M k − (cid:88) i =1 U i + 0 . M (cid:32) ( k − (cid:88) i =1 U i ) + ( k − (cid:88) i =1 U i ) (cid:33) (42) or (cid:80) k − j =1 U j M ≤ (cid:18) k − k (cid:19) (cid:32) − (cid:114) U max k kk − (cid:33) . (43) Corollary 4.
The capacity augmentation factor of global RMfor a sporadic system with implicit deadlines is √ ≈ . . k Q and k U The utilization-based worst-case response-time analysis inTheorem 11 and Corollary 4 is analytically tighter than thebest known result by Bertogna et al. [4] with linear-time tests.Moreover, our polynomial-time schedulability test extendedto handle deadline-monotonic scheduling for constrained-deadline task sets based on the forced-forward analysis in [9]has the same speedup factor as the best known result in pseudo-polynomial time by Baruah et al. [3].11 .0 0.2 0.4 0.6 0.8 1.0 U Σ /M A cc e p t a n c e R a t i o (a) p=1BCLBAK FFHP-GC HP-BCHP-BC-EP HP-BC2QB-BC QB-FFQB-FF2 Guan (b) p=2 (c) p=3 Fig. 5: Acceptance ratio comparison on implicit-deadline 8 multiprocessor systems.With respect to the capacity augmentation factors, thetest derived from k Q by using the forced-forward approachobtains the best one, whereas the tests from bounded carry-inare worse. As shown in the above examples, different schedu-lability tests may lead to different quality of the schedulabilitytests. Therefore, these results are not analytically comparable.We will have to compare these results in the evaluations.
In this section, we conduct experiments using synthesizedtask sets for evaluating the proposed tests on multiprocessorsystems. We first generated a set of sporadic tasks. The car-dinality of the task set was times the number of processors,e.g., 40 tasks on 8 multiprocessor systems. The task sets weregenerated in a similar manner in Section 5.4. Tasks’ relativedeadlines were equal to their periods.The evaluated tests are as follows: • BCL : the linear-time test in Theorem 4 in [4]. • FF : the pseudo-polynomial-time force-forward (FF) anal-ysis in Eq. (5) in [3]. • BAK : the O ( n ) test in Theorem 11 in [2]. • Guan : the pseudo-polynomial-time response time analysis[16]. • HP-GC (from k U ): Eq. (35) from Theorem 6 based ongreedy carry-in (GC) in this report. • HP-BC (from k U ): Eq. (37) from Theorem 7 based onbounded carry-in (BC) in this report. • HP-BC-EP (from k U ): using Lemma 4 (with a moreprecise extreme point) by defining the values of α i and β i for each task τ i in hp ( τ k ) precisely in this report.This improves HP-BC from Theorem 7 based on boundedcarry-in (BC) in this report. • HP-BC2 (from k U ): Eq. (39) from Theorem 8 based onbounded carry-in (BC) in this report. They can be easily obtained by setting < α i ≤ M and < β i ≤ M . • QB-BC (from k Q ): Eq. (40) in Theorem 9 based onbounded carryin (BC) in this report. • QB-FF (from k Q ): Eq. (41) from Theorem 10 based onforce-forward (FF) in this report. • QB-FF2 (from k Q ): Eq. (42) from Theorem 11 basedon force-forward (FF) in this report.Among the above tests, BCL, HP-GC, HP-BC2, QB-FF and QB-FF2 can be implemented in linear time. Ourother tests (HB-BC, HP-BC-EP, QB-BC) require to sort thehigher-priority tasks to define the proper last release ordering;therefore, their time complexity is O ( n log n ) for a task setwith n tasks. Results.
Figure 5 depicts the result of the performance compar-ison. In all the cases, we can see that QB-BC and HP-BC-EPare superior to almost all the other polynomial-time tests. Itmay seem that QB-FF is superior to QB-BC when we inspecttheir schedulability tests. However, the way how we formulatedthe force-forward algorithm in Eq. (34) is also pessimisticby introducing U max k instead of just U k . Such inflation from U k to U max k makes the analysis for the worst-case capacity-augmentation factor tighter, but also makes QB-FF with lessacceptance ratio when testing tasks with utilization larger thanthe threshold . . Therefore, if U max k > U k + (cid:80) τ i ∈ T (cid:48) C i MT k ,then QB-FF is worse than QB-BC.The greedy carry-in in HP-GC makes it too pessimistic.However, HP-BC is comparable with BAK. Among the linear-time tests, QB-FF outperforms the others in all the cases.Among the above tests, it is difficult to compare QB-BC andHP-BC-EP, since they perform very closely. Overall, most ofthe tests derived by using the two frameworks perform verywell with low time complexity. We assume that the priority ordering is given. We just have to use thereversed order in Theorem 10. Conclusion
This report presents the similarly, difference, and the char-acteristics of the k U and k Q frameworks. These two frame-works have great potential to be used for deriving polynomial-time schedulability tests almost automatically , as soon as thecorresponding parameters in Definitions 1, 3, and 4 can be con-structed. In the past, exponential-time schedulability tests weretypically not recommended and most of time ignored, as thisrequires very high complexity. However, by adopting these twoframeworks, we have successfully shown that exponential-timeschedulability tests may lead to good polynomial-time testsby using the k U and k Q frameworks. Both frameworksare needed and have to be applied for different cases. Withthese two frameworks, some difficult schedulability test andresponse time analysis problems may be solved by building agood (or exact) exponential-time test and applying these twoframeworks.These two frameworks are both useful and needed fordifferent cases and applications. We have demonstrated theirdifferences in details and present evaluation results for theschedulability tests derived from these two frameworks. Forsome cases, k U is better, and for some cases, k Q is better. Acknowledgement : This paper has been supported by DFG, aspart of the Collaborative Research Center SFB876 (http://sfb876.tu-dortmund.de/), and the priority program ”Dependable EmbeddedSystems” (SPP 1500 - http://spp1500.itec.kit.edu).
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