Paper Presentation. Amo Guangmo Tong. University of Taxes at Dallas January 24, 2014

Transcription

1 Paper Presentation Amo Guangmo Tong University of Taxes at Dallas January 24, 2014 Amo Guangmo Tong (UTD) January 24, / 30

2 Overview 1 Tardiness Bounds under Global EDF Scheduling on a Multiprocessor 2 An O(m) Analysis Technique for Supporting Real-Time Self-Suspending Task Systems Amo Guangmo Tong (UTD) January 24, / 30

3 Tardiness Bounds under Global EDF Scheduling on a Multiprocessor Tardiness Bounds under Global EDF Scheduling on a Multiprocessor UmaMaheswari C. Devi and James H. Anderson Amo Guangmo Tong (UTD) January 24, / 30

4 Tardiness Bounds under Global EDF Scheduling on a Multiprocessor Abstact The major topic in this paper is the scheduling of soft real-time sporadic task systems on a multiprocessor. This paper derives the tardiness bound under preemptive or non-preemptive global earliest-deadline-first (EDF) schedule when the total utilization is less than or even equal to the number of processors. Amo Guangmo Tong (UTD) January 24, / 30

5 System Model Soft real-time system T n sporadic tasks T n m identical processors,m > 1 minimum inter-arrival time or period, p i > 0 execution cost e i < p i relative deadline D i = p i utilization u i = e i /p i U sum = u i m U max (k) denotes the subset of k tasks with highest utilizations in T E max (k) denotes the subset of k tasks with highest executions in T Amo Guangmo Tong (UTD) January 24, / 30

6 Schedule Algorithms Processor Sharing(PS) schedule In a PS schedule, each job of T i is allocated a fraction u i of a processor at each instant. Every job will complete executing exactly at its deadline. As long as U sum = u i m, a valid PS schedule exists. Earliest-Deadline-First(EDF) schedule In a EDF schedule, the job with earlier deadline has higher priority. Non-preemptive Earliest-Deadline-First(NP-EDF) schedule In a NP-EDF schedule, the job with earlier deadline has higher priority and a job cannot be paused until its completion. Amo Guangmo Tong (UTD) January 24, / 30

7 LAG(T, t) For any given soft real-time system T A(PS, T i, t) denotes the allocation of resource to task T i within the interval [0, t] under PS schedule; A(EDF, T i, t) denotes the allocation of resource to task T i within interval [0, t] under EDF schedule. lag(t i, t) = A(PS, T i, t) A(EDF, T i, t) LAG(T, t) = i n lag(t i, t) Amo Guangmo Tong (UTD) January 24, / 30

8 How to derive the tardiness bound of job T i,j Step 1 Let t d = d i,j denotes the deadline of T i,j. Notice that jobs with lower priority have no influence on the schedule of jobs with higher priority. We ignore the jobs with priority lower than T i,j. In this case, LAG(T, t d ) represents the workload remaining to be done after t d in order to finish all the execution. Intuitively, the tardiness bound of T i,j is directly related to LAG(T, t d ) Amo Guangmo Tong (UTD) January 24, / 30

9 How to derive the tardiness bound of job T i,j Step 2 Lemma 1 Let the tardiness of T k with a deadline less than t d be at most x + e k, where x 0, for all 1 i n under the EDF schedule for system T on m processors. If LAG(T, t d ) mx + e i. Then, the tardiness of T i,j is at most x + e i. We denote mx + e i as Lower Bound. Amo Guangmo Tong (UTD) January 24, / 30

10 How to derive the tardiness bound of job T i,j Step 3 Lemma 2 Let the tardiness of T k with a deadline less than t d be at most x + e k, where x 0, for all 1 i n under the EDF schedule for system T on m processors. And U sum m. Then LAG(T, t d ) x T i U max (m 1) u i + T i E max (m 1) e i We denote x T i U max (m 1) u i + T i E max (m 1) e i as Upper Bound. Amo Guangmo Tong (UTD) January 24, / 30

11 How to derive the tardiness bound of job T i,j Step 4 Let the UpperBound not more than LowerBound (x T i U max (m 1) u i + T i E max (m 1) e i) (mx + e i ) Solve this inequation. Then, x [ ( T i E max (m 1) e i) e min m T i U max (m 1) u ], i in which case the condition in Lemma 1 can be satisfied. According to the Lemma 1, the tardiness of job T i,j in task T i is at most [ ( T i E max (m 1) e i) e min m T i U max (m 1) u ] + e i. By induction, the tardiness i bound of all of the jobs in task T i is [ ( T i E max (m 1) e i) e min m T i U max (m 1) u ] + e i. i Amo Guangmo Tong (UTD) January 24, / 30

12 Conclusion The tardiness bounds under global EDF for task T i in a sporadic real-time systems on multiprocessors is [ ( T i E max (m 1) e i) e min m T i U max (m 1) u ] + e i, when the total utilization of a task i system is not more than the number of processors, m, which means the resource can be fully utilized in the long run. The tardiness bound under non-preemptive global EDF can be derived by the similar framework with slight difference in the prove of UpperBound. Amo Guangmo Tong (UTD) January 24, / 30

13 Strength Multiprocessor Multiprocessor-based design has been used widely in the present-day applications. For example, virtual-reality systems, systems that track people and machines, and signal-processing systems such as synthetic-aperture. In this case, the previous theories on uniprocessor are insufficient. Soft real-time system Then, not all real-time systems require that the deadline of job has to be met. In other words, such as in multimedia and gaming systems, we can tolerant tardiness as long as it is bounded with limitation. Non-preemptive Finally, this paper derives a tardiness bound under non-preemptive global EDF, which provides a theory foundation for the case in practice that tasks cannot be paused before completion. Amo Guangmo Tong (UTD) January 24, / 30

14 An O(m) Analysis Technique for Supporting Real-Time Self-Suspending Task Systems An O(m) Analysis Technique for Supporting Real-Time Self-Suspending Task Systems Liu Cong and James H. Anderson Amo Guangmo Tong (UTD) January 24, / 30

15 An O(m) Analysis Technique for Supporting Real-Time Self-Suspending Task Systems Abstact This paper considers soft real-time self-suspending task systems under the preemptive global EDF and derives a new schedulability test with O(m) suspension-related utilization loss. Similar to the first paper, this paper also analyze the tardiness bound by using the allocation difference between PS schedule and EDF schedule Amo Guangmo Tong (UTD) January 24, / 30

16 System Model Soft real-time Self-Suspending system T Self-suspending: jobs alternate between computation and suspension phase n sporadic tasks T n m identical processors,m > 1 minimum inter-arrival time or period, p i > 0 execution cost e i ; suspension costs i ; e i + s i < p i relative deadline D i = p i utilization u i = e i /p i ; suspension ratio v i = s i /p i ; u = u i + v i U sum = u i m Amo Guangmo Tong (UTD) January 24, / 30

17 Schedule Algorithms Processor Sharing(PS) schedule In a PS schedule, each job of T i is allocated a fraction u i of a processor at each instant. Every job will complete executing exactly at its deadline. As long as U sum = u i m, a valid PS schedule exists. Earliest-Deadline-First(EDF) schedule In a EDF schedule, the job with earlier deadline has higher priority. Amo Guangmo Tong (UTD) January 24, / 30

18 How to deal with suspension In the first paper, the critical point is if t is non-busy time instant there are at most m-1, tasks have tardy jobs at t. Otherwise, t cannot be non-busy. However, when it comes to self-suspending system, there could be at most n tasks have tardy jobs at a non-busy instant t. Amo Guangmo Tong (UTD) January 24, / 30

19 How to deal with suspension Convert all the suspension into execution? Then, e i = e i + s i. To guarantee a valid PS schedule, we need U sum = U sum + i n s i p i m It works but pessimistic, because resource will be wasted severely Amo Guangmo Tong (UTD) January 24, / 30

20 How to deal with suspension If,by converting at most m tasks suspensions into execution at any instant t in PS, we can guarantee that any non-busy instant t in EDF there are at most m 1 tasks have tardy jobs. In this case, to guarantee a valid PS schedule, we need U sum + V max(m) v i m It works better but how to convert the suspension into execution? Amo Guangmo Tong (UTD) January 24, / 30

21 Transformation Consider processor M k. Let t f be the finish time of the system. From t f to the left, let t h denote the first encountered non-busy time instant on M k where at least one task T i has an tardy job T i,v suspending at t h. Let v c denote the minimum job index of T i such that all job from T i,v c to T i,v are tardy. Then [A 1 k, B1 k ] = [r i,v c, t h + 1]. In this way, from r i,v c we can get [A 2 k, B2 k ]. Until time 0, we will get [Aq k, Bq k ] transformation intervals respect to M k. Note that each job in a transformation interval has its release time and deadline time within the interval. Move Switch Convert Do the same things to all processors. Amo Guangmo Tong (UTD) January 24, / 30

22 Transformation Check At any non-busy time instant t [0, t f ] in the transformed schedule, at most m 1 tasks can have enabled tardy jobs with deadlines at or before t. A valid PS schedule exists when U sum + V max(m) v i m. We denote the new schedule as PS and EDF Amo Guangmo Tong (UTD) January 24, / 30

23 LAG(T, t) For any given soft real-time system T A(PS, T i, t) denotes the allocation of resource to task T i within the interval [0, t] under PS schedule; A(EDF, T i, t) denotes the allocation of resource to task T i within interval [0, t] under EDF schedule. lag(t i, t) = A(PS, T i, t) A(EDF, T i, t) LAG(T, t) = i n lag(t i, t) Amo Guangmo Tong (UTD) January 24, / 30

24 How to derive the tardiness bound of job T i,j Step 1 Let t d = d i,j denotes the deadline of T i,j. Notice that jobs with lower priority have no influence on the schedule of jobs with higher priority. We ignore the jobs with priority lower than T i,j. In this case, LAG(T, t d ) represents the workload remaining to be done after t d in order to finish all the execution. Intuitively, the tardiness bound of T i,j is directly related to LAG(T, t d ) Amo Guangmo Tong (UTD) January 24, / 30

25 How to derive the tardiness bound of job T i,j Step 2 Lemma 1 Let the tardiness of T k with a deadline less than t d be at most x + e k + s k, where x 0, for all 1 i n under the EDF schedule for system T on m processors. If LAG(T, t d ) mx. Then, the tardiness of T i,j is at most x + e i + s i. We denote mx as Lower Bound. Amo Guangmo Tong (UTD) January 24, / 30

26 How to derive the tardiness bound of job T i,j Step 3 Lemma 2 Let the tardiness of T k with a deadline less than t d be at most x + e k + s k, where x 0, for all 1 i n under the EDF schedule for system T on m processors. And U sum m. Let E denotes the largest value of the expression τ i τ (e i + s i ) + τ i ψ u e i, where ψ denotes any set of m 1 tasks. Let U m 1 be the sum of the m 1 largest u i Then LAG(T, t d ) x U m 1 + E We denote x U m 1 + E as Upper Bound. Amo Guangmo Tong (UTD) January 24, / 30

27 How to derive the tardiness bound of job T i,j Step 4 Let the UpperBound not more than LowerBound x U m 1 + E (mx + e i + s i ). Solve this inequation. Then, x [ E e i s i m U m 1 ], in which case, the condition in Lemma 1 can be satisfied. Amo Guangmo Tong (UTD) January 24, / 30

28 Conclusion The tardiness bounds under global EDF for task T i in a sporadic real-time systems on multiprocessors is [ E e i s i m U m 1 ] + e i, when the total utilization of a task system is not more than the number of processors, m, which means the resource can be fully utilized in the long run. The tardiness bound under non-preemptive global EDF can be derived by the similar framework with slight difference in the prove of UpperBound. Amo Guangmo Tong (UTD) January 24, / 30

29 Strength Suspension As the paper says, suspension delays may occur in many systems due to the cases such as the interaction with external devices and resource sharing. Then this paper derives an tardiness bound with O(m) utilization loss. This result is much better than the previous result in that when the number of tasks is very large then the O(n) utilization loss can be highly significant. Amo Guangmo Tong (UTD) January 24, / 30

30 The End Amo Guangmo Tong (UTD) January 24, / 30