Techniques for Multiprocessor Global Schedulability Analysis ∗ Sanjoy Baruah The University of North Carolina at Chapel Hill Abstract algorithm is then applied locally in each cluster, to the jobs generated by the tasks that are assigned to the cluster. It is assumed that inter-processor communication within a clus- The scheduling of sporadic task systems upon multipro- ter incurs no overhead. We focus here on the two extremes cessor platforms is considered, when inter-processor migra- of clustering. In partitioned scheduling , each processor is a tion is permitted. It is known that current schedulability cluster of size one. That is, each task is assigned to one pro- tests for such systems perform quite poorly when compared cessor and all the jobs generated by a task are constrained to schedulability tests for partitioned scheduling. Limita- to execute only upon the processor to which the task has tions of current tests are identified, which may be responsi- been assigned. In global scheduling , by contrast, there is ble for the unsatisfactory performance of these tests. A new only one cluster containing all the processors. A job may test that overcomes some of these limitations is proposed execute upon any processor, and a preempted job may later and proved correct. resume execution upon the same processor as, or a differ- ent processor from, the one on which it had previously been executing. However, each job may execute on at most one 1 Introduction and Motivation processor at each instant in time. A real-time system is often modelled as a finite collec- Current state of the art. Currently, the partitioned tion of independent recurring tasks, each of which gener- scheduling of sporadic task systems is much better un- ates a potentially infinite sequence of jobs . Every job is derstood than global scheduling. Sufficient schedulabil- characterized by an arrival time, an execution requirement, ity tests of polynomial time-complexity have been de- and a deadline, and it is required that a job complete execu- signed [7, 17, 8] for various commonly-used scheduling tion between its arrival time and its deadline. Different for- algorithms (such as Earliest Deadline First ( EDF ) [22, 14] mal models for recurring tasks place different restrictions and Deadline Monotonic (DM) [21]). Worst-case resource- on the values of the parameters of jobs generated by each augmentation bounds, that provide a quantitative measure task. One of the more commonly used formal models is the of how effective these tests are, have been obtained. These sporadic task model [23, 9], which will be considered in this schedulability tests have also been extensively evaluated via paper. simulations [4, 6], and shown to have much better average- Scheduling is the allocation of processor time to jobs, case behavior than indicated by their worst-case guaran- and a scheduling algorithm is used for determining such al- tees. In contrast, we are not aware of non-trivial theoreti- location. A schedulability test for a given scheduling algo- cal bounds on the performance of known sufficient schedu- rithm accepts as input the specifications of a real-time sys- lability tests [2, 3, 10] for global scheduling (other than tem, and determines whether the scheduling algorithm can a relatively naive resource-augmentation bound on global guarantee to schedule the system such that all jobs of all DM [16]), and simulation experiments have tended to in- tasks will meet all deadlines, under all permissible combi- dicate that they perform poorly in comparison to the parti- nations of job-arrival sequences by the different tasks com- tioned schedulability tests. prising the system. In this paper, we study the scheduling of systems of sporadic tasks upon a platform comprised of several identical processors. In scheduling a task system This research. This research is aimed at obtaining a bet- upon such a platform, it is possible to divide the processors ter understanding of global schedulability for sporadic task into clusters and assign each task to a cluster. A scheduling systems. After formally defining the task and machine mod- els in Section 2, we start out in Section 3 by highlighting ∗ Supported in part by NSF Grant Nos. CNS-0408996, CCF-0541056, what appears to be one of the root causes of difficulty in and CCR-0615197, ARO Grant No. W911NF-06-1-0425, and funding such analysis: our failure thus far to come up with tech- from the Intel Corporation.
niques for characterizing the “worst case” behavior of such Schedulability. A real-time system implemented on a particular computing platform is said to be A -schedulable systems. In Section 4, we demonstrate that global and with respect to a given scheduling algorithm A , if the al- partitioned scheduling are incomparable for priority-based gorithm A schedules the system such that all jobs of all scheduling algorithms that are not allowed to dynamically change the priorities of jobs; this extends a result of Leung tasks will meet all deadlines, under all permissible (also and Whitehead [21], who obtained a similar incomparabil- called legal ) combinations of job-arrival sequences by the ity result for algorithms in which all jobs of each task are different tasks comprising the system. A schedulability test required to have the same priority. In Section 5, we briefly for scheduling algorithm A (also called an A -schedulability summarize prior tests for global EDF -schedulability analy- test ) accepts as input the specifications of a real-time sys- sis, and highlight their features and disadvantages. In Sec- tem, and determines whether the system is A -schedulable tion 6, we derive and analyze a new global- EDF schedulabil- or not. An A -schedulability test is said to be exact if it cor- ity test that overcomes some of the disadvantages of these rectly identifies all A -schedulable systems, and sufficient if prior tests. We conclude in Section 7 with a summary of the it may fail to identify some A -schedulable systems (it must guarantee, though, that all identified systems are indeed A - main results in this paper. schedulable.) For the sake of concreteness, we have chosen to fo- cus in this paper upon a specific global scheduling algo- rithm – EDF . However, the issues are similar for many Task and system characteristics. The concepts of task other scheduling algorithms, and many of the techniques and system utilization and density prove useful in the analy- that are applicable to global EDF scheduling are likely sis of sporadic task systems on multiprocessors. These con- to be applicable to these other scheduling algorithms as cepts are defined as follows: well. (For example, Baker has used similar techniques to Utilization: The utilization u i of a task τ i is the ratio C i /T i study both global EDF -schedulability [2, 3] and global DM- of its execution time to its period. The total utilization schedulability [2, 5], and Bertogna et al. have extended u sum ( τ ) and the largest utilization u max ( τ ) of a task their global EDF -schedulability results [10] to apply to DM scheduling [11].) system τ are defined as follows: � def def u sum ( τ ) = u i ; u max ( τ ) = max τ i ∈ τ ( u i ) 2 Model τ i ∈ τ Density: The density δ i of a task τ i is the ratio C i /D i of A sporadic task τ i = ( C i , D i , T i ) is characterized by a its execution time to its relative deadline. The total worst-case execution requirement C i , a (relative) deadline density δ sum ( τ ) and the largest density δ max ( τ ) of a D i , and a minimum inter-arrival separation T i , which is, task system τ are defined as follows: for historical reasons, also referred to as the period of the task. Such a sporadic task generates a potentially infinite � def def δ sum ( τ ) = δ i ; δ max ( τ ) = max τ i ∈ τ ( δ i ) sequence of jobs, with successive job-arrivals separated by τ i ∈ τ at least T i time units. Each job has a worst-case execution requirement equal to C i and a deadline that occurs D i time An additional concept that plays a critical role in the units after its arrival time. We refer to the interval, of size schedulability analysis of sporadic task systems is that of D i , between such a job’s arrival instant and deadline as its demand bound function . For any interval length t , the scheduling window. We assume a fully preemptive execu- demand bound function DBF ( τ i , t ) of a sporadic task τ i tion model: any executing job may be interrupted at any bounds the maximum cumulative execution requirement by instant in time, and its execution resumed later with no cost jobs of τ i that both arrive in, and have deadlines within, any or penalty. A sporadic task system is comprised of several interval of length t . It has been shown [9] that such sporadic tasks. Let τ denote a system of such sporadic tasks: τ = { τ 1 , τ 2 , . . . τ n } , with τ i = ( C i , D i , T i ) for all � � t − D i � � DBF ( τ i , t ) = max 0 , ( + 1) C i i , 1 ≤ i ≤ n . Task system τ is said to be a constrained T i sporadic task system if it is guaranteed that each task τ i ∈ τ has its relative deadline parameter no larger than its period: A load parameter, based upon the DBF function, may be D i ≤ T i , and an implicit-deadline sporadic task system if defined for any sporadic task system τ as follows: D i = T i for all τ i ∈ τ . (Implicit-deadline systems are also �� τ i ∈ τ DBF ( τ i , t ) � known as Liu and Layland [22] task systems.) In this pa- def LOAD ( τ ) = max per, we restrict our attention to constrained and implicit- t t> 0 deadline task systems.
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