Fixed-Priority Multiprocessor Scheduling. Real-time Systems. N periodic tasks (of different rates/periods) i Ji C J. 2 i. ij 3

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1 0//0 Fixed-Priority Multiprocessor Scheduling Real-time Systems N periodic tasks (of different rates/periods) r i T i C i T i C C J i Ji i ij i r i r i r i Utilization/workload: How to schedule the jobs to avoid deadline miss? T i

2 0//0 On Single-processors Liu and Layland s Utilization Bound [9] (the 9 th most cited paper in computer science) number of tasks Scheduled by RMS (Rate Monotonic Scheduling) Quetion (since 9) Find a multiprocessor scheduling algorithm that can achieve Liu and Layland s utilization bound? number of processors

3 0//0 Multiprocessor Scheduling Global Scheduling Partitioned Scheduling Partitioned Scheduling with Task Splitting new task waiting queue 9 cpu cpu cpu cpu cpu cpu cpu cpu cpu Best Known Results % [OPODIS 0] From Uppsala RTAS [TPDS 0] [ECRTS 0] [RTSS 0] Liu and Layland s Utilization Bound 9. [RTCSA 0] Fixed Priority Dynamic Priority Fixed Priority Dynamic Priority Fixed Priority Dynamic Priority Task Splitting Global Partitioned Multiprocessor Scheduling

4 0//0 Multiprocessor Scheduling Global Scheduling new task Would fixed-priority scheduling e.g. RMS work? waiting queue cpu cpu cpu Multiprocessor Scheduling Global Scheduling new task waiting queue Would fixed-priority scheduling e.g. RMS work? Unfortunately RMS suffers from the Dhall s anomali Utilization may be 0% cpu cpu cpu

5 0//0 Dhall s anomali Task Task Task 0 ε + ε Dhall s anomali CPU CPU Deadline miss 0 ε + ε Schedule the tasks on CPUs using RMS

6 0//0 Dhall s anomali (M+ tasks and M processors) /(ε+) ε/ ε/ ε/ # # #M #M+ P P P M U M*ε + /(+ ε) M 0 Multiprocessor Scheduling Partitioned Scheduling 9 cpu cpu cpu

7 0//0 Multiprocessor Scheduling Partitioned Scheduling Resource utilization may be limited to 0% 9 cpu cpu cpu Partitioning Partitioned Scheduling 9 P P P

8 0//0 Partitioned Scheduling Partitioning P P P 9 Partitioned Scheduling Scheduling reduced to single-processor scheduling on each processor P 00%.9% Yes, schedulable! P P P 9

9 0//0 Partitioned Scheduling The Partitioning Problem is similar to Bin-packing Problem (NP-hard) Limited Resource Usage, 0% necessary condition to guarantee schedulability # # #M #M+ 0%+ ε P P P M Partitioned Scheduling The Partitioning Problem is similar to Bin-packing Problem (NP-hard) Limited Resource Usage necessary condition to guarantee schedulability #M+ 0%+ ε P P P M # # #M 9

10 0//0 Partitioned Scheduling The Partitioning Problem is similar to Bin-packing Problem (NP-hard) Limited Resource Usage necessary condition to guarantee schedulability #M+, 0%+ ε #M+, P P P M # # #M Partitioned Scheduling The Partitioning Problem is similar to Bin-packing Problem (NP-hard) Limited Resource Usage necessary condition to guarantee schedulability 0%+ ε P P P M #M+, #M+, # # #M 0

11 0//0 Multiprocessor Scheduling Partitioned Scheduling with Task Splitting cpu cpu cpu Partitioning Partitioned Scheduling 9 P P P

12 0//0 Bin-Packing with Item Splitting Resource can be fully (better) utilized Bin Bin Bin Depth-First Partitioning Algorithms [Kato et al. IPDPS 0] [Kato et al. RTAS 09] [Lakshmanan et al. ECRTS 09] Sort the tasks e.g. in increasing priority order Select a processor, and assign as many tasks as possible P

13 0//0 Lakshmanan s Algorithm [ECRTS 09] Sort all tasks in decreasing order of utilization highest util. lowest util. Lakshmanan s Algorithm [ECRTS 09] Pick up one processor, and assign as many tasks as possible highest util. lowest util. P

14 0//0 Lakshmanan s Algorithm [ECRTS 09] Pick up one processor, and assign as many tasks as possible highest util. P lowest util. Lakshmanan s Algorithm [ECRTS 09] Pick up one processor, and assign as many tasks as possible highest util. P lowest util.

15 0//0 Lakshmanan s Algorithm [ECRTS 09] Pick up one processor, and assign as many tasks as possible highest util. lowest util. P Lakshmanan s Algorithm [ECRTS 09] Pick up one processor, and assign as many tasks as possible highest util. lowest util. P P

16 0//0 Lakshmanan s Algorithm [ECRTS 09] Pick up one processor, and assign as many tasks as possible highest util. P P lowest util. Lakshmanan s Algorithm [ECRTS 09] Pick up one processor, and assign as many tasks as possible highest util. P P lowest util.

17 0//0 Lakshmanan s Algorithm [ECRTS 09] Pick up one processor, and assign as many tasks as possible highest util. P P lowest util. Lakshmanan s Algorithm [ECRTS 09] Pick up one processor, and assign as many tasks as possible highest util. P P lowest util.

18 0//0 Lakshmanan s Algorithm [ECRTS 09] Pick up one processor, and assign as many tasks as possible highest util. P P lowest util. Lakshmanan s Algorithm [ECRTS 09] Pick up one processor, and assign as many tasks as possible highest util. P P P lowest util.

19 0//0 Lakshmanan s Algorithm [ECRTS 09] Pick up one processor, and assign as many tasks as possible highest util. P P P lowest util. Lakshmanan s Algorithm [ECRTS 09] Pick up one processor, and assign as many tasks as possible highest util. P P P lowest util. key feature: depth-first partitioning with decreasing utilization order 9

20 0//0 Lakshmanan s Algorithm [ECRTS 09] Pick up one processor, and assign as many tasks as possible highest util. P P P Utilization Bound: lowest util. % Breadth-First Partitioning Algorithms [RTAS 00] Sort all tasks in increasing priority order 0

21 0//0 Breadth-First Partitioning Algorithms Select the processor on which the assigned utilization is the lowest P P P Breadth-First Partitioning Algorithms Select the processor on which the assigned utilization is the lowest P P P

22 0//0 Breadth-First Partitioning Algorithms Select the processor on which the assigned utilization is the lowest P P P Breadth-First Partitioning Algorithms Select the processor on which the assigned utilization is the lowest P P P

23 0//0 Breadth-First Partitioning Algorithms Select the processor on which the assigned utilization is the lowest P P P Breadth-First Partitioning Algorithms Select the processor on which the assigned utilization is the lowest P P P

24 0//0 Breadth-First Partitioning Algorithms Select the processor on which the assigned utilization is the lowest P P P Breadth-First Partitioning Algorithms Select the processor on which the assigned utilization is the lowest P P P

25 0//0 Breadth-First Partitioning Algorithms Select the processor on which the assigned utilization is the lowest P P P Breadth-First Partitioning Algorithms Select the processor on which the assigned utilization is the lowest P P P

26 0//0 Comparison Why is the breadth algorithm better? P breadth-first & increasing priority order P P depth-first & decreasing utilization order P P P Fact for Light Tasks whose utilization less than 0. For a task set in which each task satisfies we have

27 0//0 Solution for Heavy Tasks whose utilization larger than 0. Pre-assigning the heavy tasks (that may have low priorities) 9 P P P Solution for Heavy Tasks Pre-assigning the heavy tasks (that may have low priorities) 9 P P P

28 0//0 Solution for Heavy Tasks Pre-assigning the heavy tasks (that may have low priorities) P P P 9 Solution for Heavy Tasks Pre-assigning the heavy tasks (that may have low priorities) P P P 9

29 0//0 Solution for Heavy Tasks Pre-assigning the heavy tasks (that may have low priorities) P P P 9 Solution for Heavy Tasks Pre-assigning the heavy tasks (that may have low priorities) P P P 9 9

30 0//0 Solution for Heavy Tasks Pre-assigning the heavy tasks (that may have low priorities) P P P 9 Solution for Heavy Tasks Pre-assigning the heavy tasks (that may have low priorities) P P P 9 0

31 0//0 Solution for Heavy Tasks Pre-assigning the heavy tasks (that may have low priorities) P P P 9 Solution for Heavy Tasks Pre-assigning the heavy tasks (that may have low priorities) P P P 9

32 0//0 Solution for Heavy Tasks Pre-assigning the heavy tasks (that may have low priorities) P P P 9 Solution for Heavy Tasks Pre-assigning the heavy tasks (that may have low priorities) P P P 9 avoid to split heavy tasks (that may have low priorities)

33 0//0 FACT By introducing the pre-assignment mechanism, we have Liu and Layland s utilization bound for all task sets! Overhead In both previous algorithms and ours The number of task splitting is at most M- task splitting -> extra migration/preemption Our algorithm on average has less task splitting Ours: width-first P P P depth-first P P P

34 0//0 Implementation Easy! One timer for each split task Implemented as task migration P P as being preempted i C i as being resumed i until finished higher prio tasks task i lower prio tasks Further Improvement P P P Schedulable? 9 00%

35 0//0 Uisng Liu and Layland s Utilization Bound P P P 00% Yes, schedulable by our algorithm Utilization Bound is Pessimistic The Liu and Layland utilization bound is sufficient but not necessary many task sets are actually schedulable even if the total utilization is larger than the bound P (, ) 0.9 (, ) (, ) (, )

36 0//0 Exact Analysis Exact Analysis: Response Time Analysis [Lehoczky_9] pseudo-polynomial (, ) (, ) (, ) (, ) R k task k is schedulable iff R k <= T k Utilization Bound v.s. Exact Analysis On single processors Utilization bound Test for RMS P Exact Analysis for RMS P 00% 00% % [Lehoczky_9]

37 0//0 On Multiprocessors Can we do something similar on multiprocessors? Utilization bound Test the algorithm introduced above? P P P P P P 00% 00% An Improved Algorithm Based on the similar idea RMS priority assignment worst-fit partitioning increasing priority order Employ Response Time Analysis to determine the maximal workload on each processor more flexible behavior (more difficult to prove ) Same utilization bound Liu and Layland s utilization bound Much better performance Pseudo-polynomial

38 0//0 Summary Real-time Systems infinite, multi-task, multi-rate Summary Real-time Systems infinite, multi-task, multi-rate On Single-processors Liu and Layland s Utilization bound % Liu and Layland s Utilization Bound

39 0//0 Summary Real-time Systems infinite, multi-task, multi-rate On Single-processors Liu and Layland s Utilization bound On Multiprocessors % 0 0 Liu and Layland s Utilization Bound Multiprocessor Scheduling Summary Real-time Systems infinite, multi-task, multi-rate On Single-processors Liu and Layland s Utilization bound On Multiprocessors Global Scheduling Dhall s Effect % 0 0 Liu and Layland s Utilization Bound 0 P 0 0 P P deadline miss [OPODIS 0] Global 0 ε + ε Multiprocessor Scheduling 9

40 0//0 Summary Real-time Systems infinite, multi-task, multi-rate On Single-processors Liu and Layland s Utilization bound On Multiprocessors Global Scheduling Dhall s Effect Partitioned Scheduling Bin-packing % Liu and Layland s Utilization Bound [OPODIS 0] 0 [ECRTS 0] 0 Global Partitioned Multiprocessor Scheduling Summary Real-time Systems infinite, multi-task, multi-rate On Single-processors Liu and Layland s Utilization bound On Multiprocessors Global Scheduling Dhall s Effect Partitioned Scheduling Bin-packing Our Recent Work Partitioned with Task Splitting split tasks with high priorities Liu and Layland s utilization bound % [OPODIS 0] Global Multiprocessor Scheduling 0 [ECRTS 0] Partitioned Liu and Layland s Utilization Bound [Our Work] Splitting 0

41 0//0 References [Liu9] C.L. Liu and James Layland, Scheduling Algorithms for Multiprogramming in a Hard-Real-Time Environment [Andersson0ECRTS] Bjorn Andersson, Jan Jonsson: The Utilization Bounds of partitioned and Pfair Static-Priority Scheduling on multiprocessors are 0%. ECRTS 00 [Andersson0OPODIS] Bjorn Andersson: Global Static-Priority Preemptive Multiprocessor Scheduling with Utilization Bound %. OPODIS 00 [Andersson0RTCSA] Bjorn Andersson, Eduardo Tovar: Multiprocessor Scheduling with Few preemption. RTCSA 00 [Andersson0RTSS] Bjorn Andersson, Sanjoy K. Baruah, Jan Jonsson: Static-Priority Scheduling on multiprocessors. RTSS 00 [Baker0TPDS] Theodore P. Baker: An Analysis of EDF Schedulability on a Multiprocessor. IEEE Trans. Parallel Distrib. Syst. (): 0- (00) [Lakshmanan09ECRTS] Karthik Lakshmanan, Ragunathan Rajkumar, John Lehoczky Partitioned Fixed-Priority Preemptive Scheduling for Multi-core Processors. ECRTS 000 [Lopez0RTSS] J. M. Lopez, J. L. Diaz, and D. F. Garcia, Utilization Bounds for EDF Scheduling on Real-Time Multiprocessor Systems, RTSS 00. [Oh9] D. Oh and T. P. Baker. Utilization bounds for n-processor Rate Monotone scheduling with static processor assignment. In Real-Time Systems, 99. [Kato0IPDPS] Shinpei Kato, Nobuyuki Yamasaki: Portioned static-priority scheduling on multiprocessors. IPDPS 00 [Kato09RTAS] Shinpei Kato, Nobuyuki Yamasaki: Semi-partitioned Fixed-Priority Scheduling on Multiprocessors. RTAS 009. [Guan et al0rtas] Fixed-Priority Multiprocessor Scheduling with Liu & Layland's Utilization Bound. Nan Guan, Martin Stigge, Wang Yi and Ge Yu. In the proc. of RTAS0, the th IEEE Real-Time and Embedded Technology and Applications Symposium Stockholm, Sweden, April -, 00.

Fixed-Priority Multiprocessor Scheduling

Fixed-Priority Multiprocessor Scheduling Real-time Systems N periodic tasks (of different rates/periods) r i T i C i T i C C J i Ji i ij i r i r i r i Utilization/workload: How to schedule the jobs to