Fixed-Priority Multiprocessor Scheduling

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1 Fixed-Priority Multiprocessor Scheduling

2 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

3 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)

4 Rate Monotonic Scheduling Priority assignment: shorter period higher prio. Run-time schedule: the highest priority first high priority mediate priority low priority Run-time schedule How to check whether all deadlines are met?

5 Liu and Layland s Utilization Bound Schedulability Analysis P Schedulable? 00%.9% U i U i = C i / T i Liu and Layland s bound:

6 Liu and Layland s Utilization Bound Schedulability Analysis CPU 00%.9% Yes, schedulable! Liu and Layland s bound:

7 Multiprocessor (multicore) Scheduling Significantly more difficult: Timing anomalies Hard to identify the worst-case scenario Bin-packing/NP-hard problems Multiple resources e.g. caches, bandwidth

8 Quetion (since 9) Find a multiprocessor scheduling algorithm that can achieve Liu and Layland s utilization bound? number of processors

9 Multiprocessor Scheduling Global Scheduling Partitioned Scheduling Partitioned Scheduling with Task Splitting new task waiting queue cpu cpu cpu cpu cpu cpu cpu cpu cpu

10 Best Known Results 80 0 % From Uppsala RTAS 00 Liu and Layland s Utilization Bound [OPODIS 08] [TPDS 0] [ECRTS 0] [RTSS 0] 6 66 [RTCSA 06] Fixed Priority Dynamic Priority Fixed Priority Dynamic Priority Fixed Priority Dynamic Priority Task Splitting Global Partitioned Multiprocessor Scheduling

11 Multiprocessor Scheduling Global Scheduling new task Would fixed-priority scheduling e.g. RMS work? waiting queue 8 6 cpu cpu cpu

12 Multiprocessor Scheduling Global Scheduling new task waiting queue 8 Would fixed-priority scheduling e.g. RMS work? Unfortunately RMS suffers from the Dhall s anomali Utilization may be 0% 6 cpu cpu cpu

13 Dhall s anomali CPU CPU CPU 0 ε + ε

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

15 Dhall s anomali (M+ tasks and M processors) /(ε+) ε/ ε/ ε/ # # #M #M+ P P P M U M*ε + /(+ ε) M 0

16 Multiprocessor Scheduling Partitioned Scheduling cpu cpu cpu

17 Multiprocessor Scheduling Partitioned Scheduling Resource utilization may be limited to 0% cpu cpu cpu

18 Partitioning Partitioned Scheduling P P P

19 Partitioning Partitioned Scheduling P P P 6 8 9

20 Scheduling Partitioned Scheduling reduced to single-processor scheduling on each processor P 00%.9% Yes, schedulable! P P P 6 8 9

21 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

22 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

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

24 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

25 Multiprocessor Scheduling Partitioned Scheduling with Task Splitting 6 cpu cpu cpu

26 Partitioning Partitioned Scheduling P P P

27 Bin-Packing with Item Splitting Resource can be fully (better) utilized Bin Bin Bin 8 6 8

28 Depth-First Partitioning Algorithms [Kato et al. IPDPS 08] [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 lowest priority 8 P highest priority 6

29 Lakshmanan s Algorithm [ECRTS 09] Sort all tasks in decreasing order of utilization highest util. 8 6 lowest util.

30 Lakshmanan s Algorithm [ECRTS 09] Pick up one processor, and assign as many tasks as possible highest util. lowest util. 8 6 P

31 Lakshmanan s Algorithm [ECRTS 09] Pick up one processor, and assign as many tasks as possible highest util. P 6 8 lowest util.

32 Lakshmanan s Algorithm [ECRTS 09] Pick up one processor, and assign as many tasks as possible highest util. P 6 8 lowest util.

33 Lakshmanan s Algorithm [ECRTS 09] Pick up one processor, and assign as many tasks as possible highest util. 6 P 6 8 lowest util.

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

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

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

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

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

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

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

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

42 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

43 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. 6%

44 Breadth-First Partitioning Algorithms [RTAS 00] Sort all tasks in increasing priority order lowest priority 6 highest priority

45 Breadth-First Partitioning Algorithms Select the processor on which the assigned utilization is the lowest lowest priority highest priority 6 P P P

46 Breadth-First Partitioning Algorithms Select the processor on which the assigned utilization is the lowest lowest priority 6 P P P highest priority

47 Breadth-First Partitioning Algorithms Select the processor on which the assigned utilization is the lowest lowest priority P P P 6 highest priority

48 Breadth-First Partitioning Algorithms Select the processor on which the assigned utilization is the lowest lowest priority P P P 6 highest priority

49 Breadth-First Partitioning Algorithms Select the processor on which the assigned utilization is the lowest lowest priority P P P 6 highest priority

50 Breadth-First Partitioning Algorithms Select the processor on which the assigned utilization is the lowest lowest priority P P P 6 highest priority

51 Breadth-First Partitioning Algorithms Select the processor on which the assigned utilization is the lowest lowest priority P P P highest priority 6

52 Breadth-First Partitioning Algorithms Select the processor on which the assigned utilization is the lowest lowest priority P P P 6 highest priority

53 Breadth-First Partitioning Algorithms Select the processor on which the assigned utilization is the lowest lowest priority P P P 6 highest priority

54 Breadth-First Partitioning Algorithms Select the processor on which the assigned utilization is the lowest P P P 6

55 Comparison Why is the breadth algorithm better? 8 P breadth-first & increasing priority order P P 6 depth-first & decreasing utilization order P P P 6 8 6

56 Split Task Consider an extreme scenario: suppose each subtask has the highest priority schedulable anyway, we do not need to worry about their deadlines 8 6 The difficult case is when the tail task is not on the top the key point is to ensure the tail task is schedulable

57 Why the tail task is schedulable? The typical case: two CPUs and task is split to two sub-tasks U Y U As we always select the CPU with the lowest load assigned, we know X X Y + U <= U Y <= U - U That is, the blocking factor for the tail task is bounded.

58 Theorem For a task set in which each task satisfies we have

59 Theorem For a task set in which each task satisfies get rid of this constraint we have

60 Problem of Heavy Tasks lowest priority P P P highest priority

61 Problem of Heavy Tasks lowest priority 8 6 P P P highest priority 9

62 Problem of Heavy Tasks lowest priority P P P highest priority 6 9 8

63 Problem of Heavy Tasks lowest priority 6 P P P highest priority 9 8

64 Problem of Heavy Tasks lowest priority P P P highest priority

65 Problem of Heavy Tasks lowest priority P P P highest priority

66 Problem of Heavy Tasks lowest priority P P P highest priority

67 Problem of Heavy Tasks lowest priority P P P highest priority

68 Problem of Heavy Tasks lowest priority P P P highest priority

69 Problem of Heavy Tasks lowest priority P P P highest priority

70 Problem of Heavy Tasks 6 9 P P P 6 8

71 Problem of Heavy Tasks the heavy tasks tail task may have too low priority level 6 9 P P P 6 8

72 Solution for Heavy Tasks Pre-assigning the heavy tasks (that may have low priorities) lowest priority P P P highest priority

73 Solution for Heavy Tasks Pre-assigning the heavy tasks (that may have low priorities) lowest priority 9 8 P P P highest priority 6

74 Solution for Heavy Tasks Pre-assigning the heavy tasks (that may have low priorities) lowest priority 8 P P P highest priority 6 9

75 Solution for Heavy Tasks Pre-assigning the heavy tasks (that may have low priorities) lowest priority P P P highest priority

76 Solution for Heavy Tasks Pre-assigning the heavy tasks (that may have low priorities) lowest priority P P P highest priority

77 Solution for Heavy Tasks Pre-assigning the heavy tasks (that may have low priorities) lowest priority P P P highest priority

78 Solution for Heavy Tasks Pre-assigning the heavy tasks (that may have low priorities) lowest priority P P P highest priority

79 Solution for Heavy Tasks Pre-assigning the heavy tasks (that may have low priorities) lowest priority P P P highest priority

80 Solution for Heavy Tasks Pre-assigning the heavy tasks (that may have low priorities) lowest priority P P P highest priority

81 Solution for Heavy Tasks Pre-assigning the heavy tasks (that may have low priorities) lowest priority 6 P P P 9 8 highest priority

82 Solution for Heavy Tasks Pre-assigning the heavy tasks (that may have low priorities) 6 P P P 9 8

83 Solution for Heavy Tasks Pre-assigning the heavy tasks (that may have low priorities) 6 P P P 9 8 avoid to split heavy tasks (that may have low priorities)

84 Theorem By introducing the pre-assignment mechanism, we have Liu and Layland s utilization bound for all task sets!

85 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

86 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

87 Further Improvement P P P Schedulable? % 8

88 Uisng Liu and Layland s Utilization Bound P P P 00% Yes, schedulable by our algorithm

89 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 (, ) 0.69 (, ) (, 8) (, )

90 Exact Analysis Exact Analysis: Response Time Analysis [Lehoczky_89] pseudo-polynomial (, ) (, ) (, 8) (, ) R k task k is schedulable iff R k <= T k

91 Utilization Bound v.s. Exact Analysis On single processors Utilization bound Test for RMS P Exact Analysis for RMS P 00% 00% 88% [Lehoczky_89]

92 On Multiprocessors Can we do something similar on multiprocessors? Utilization bound Test? the algorithm introduced above P P P P P P 00% 00%

93 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

94 Summary Real-time Systems infinite, multi-task, multi-rate

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

96 Summary Real-time Systems infinite, multi-task, multi-rate On Single-processors Liu and Layland s Utilization bound On Multiprocessors % Liu and Layland s Utilization Bound Multiprocessor Scheduling

97 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 % 80 0 Liu and Layland s Utilization Bound 60 P 0 0 P P deadline miss [OPODIS 08] 0 ε + ε Global Multiprocessor Scheduling

98 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 08] 0 [ECRTS 0] 0 Global Partitioned Multiprocessor Scheduling

99 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 0 Our Recent Work Partitioned with Task Splitting split tasks with high priorities Liu and Layland s utilization bound [OPODIS 08] Global 0 [ECRTS 0] Partitioned [Our Work] Splitting Multiprocessor Scheduling

100 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 [Andersson08OPODIS] Bjorn Andersson: Global Static-Priority Preemptive Multiprocessor Scheduling with Utilization Bound 8%. OPODIS 008 [Andersson06RTCSA] Bjorn Andersson, Eduardo Tovar: Multiprocessor Scheduling with Few preemption. RTCSA 006 [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. 6(8): (00) [Lakshmanan09ECRTS] Karthik Lakshmanan, Ragunathan Rajkumar, John Lehoczky Partitioned Fixed-Priority Preemptive Scheduling for Multi-core Processors. ECRTS 0006 [Lopez0RTSS] J. M. Lopez, J. L. Diaz, and D. F. Garcia, Utilization Bounds for EDF Scheduling on Real-Time Multiprocessor Systems, RTSS 00. [Oh98] D. Oh and T. P. Baker. Utilization bounds for n-processor Rate Monotone scheduling with static processor assignment. In Real-Time Systems, 998. [Kato08IPDPS] Shinpei Kato, Nobuyuki Yamasaki: Portioned static-priority scheduling on multiprocessors. IPDPS 008 [Kato09RTAS] Shinpei Kato, Nobuyuki Yamasaki: Semi-partitioned Fixed- Priority Scheduling on Multiprocessors. RTAS 009.

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

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