New Schedulability Tests for Real- Time task sets scheduled by Deadline Monotonic on Multiprocessors Marko Bertogna, Michele Cirinei, Giuseppe Lipari Scuola S.Anna, Pisa, Italy

Overview Real-time multiprocessing Deadline-Monotonic (DM) for multiprocessors Existing schedulability tests for RM/DM An improved test for DM Existing schedulability bounds Improving the bound for fixed priority global scheduling

Real-time scheduling for multiprocessor platforms Platform: identical, uniform or heterogeneous Migration and priorities: MIGRATION PRIORITY FullAt job boundariesNot allowed (partitioning) StaticRM-global, DM-global, … RM-global, DM-global, … RM-FFDU, DM-WFIU, … Job-level dynamicEDF-global, fpEDF, … EDF, fpEDF, … EDF-FFDU, EDF-WFIU … Unrestricted dynamic pfair algorithms, LLF, … not useful

Multiprocessor DM CPU1 CPU2 CPU3 Global queue (ordered by relative deadline)    The first m tasks are scheduled upon the m CPUs     

 Multiprocessor DM CPU1 CPU2 CPU3 Global queue (ordered by relative deadline)   When a task finishes its execution, the next one in the queue is scheduled on the available CPU           

Multiprocessor DM CPU1 CPU2 CPU3 Global queue (ordered by relative deadline)   When a higher priority task arrives, it preempts the task with highest deadline among the executing tasks              

Multiprocessor DM CPU1 CPU2 CPU3 Global queue (ordered by relative deadline)   When another task ends its execution, the preempted task can resume its execution                Task    “migrated” from CPU3 to CPU1

Why fixed priority global scheduling? Advantages: Load balancing Number of preemptions Simple implementation Easy rescheduling Reclaiming Disadvantages: Cache affinity: HW mitigates migration cost Utilization bound lower than pfair algorithms

RM for uniprocessor systems Optimality among fixed priority systems Bounded number of preemptions Efficient implementations Easy sufficient schedulability test:

RM uniprocessor: necessary and sufficient test Response Time Analysis: Repeat: Until: Pseudopolynomial complexity

RM on multiprocessors Low utilization bound (Dhall ’ s effect) Bounded number of preemptions/migrations Good performances on average Schedulability tests (sufficient conditions): Andersson, Baruah, Jonsson (2002)  ABJ test Baker (2003)  BAK test Efficient implementations

T Dhall ’ s effect Example: m processors, n=m+1 tasks, D i = T i  1,…,  m = (1,T-1)  m+1 = (T,T) RM can fail at very low utilizations DEADLINE MISS

The ABJ test For implicit deadline systems (D i = T i ) using RM Linear complexity A task set is schedulable with RM on a platform with m identical processors if: Total utilization

The BAK test For constrained deadline systems (D i  T i ) Quadratic complexity A task set is schedulable with EDF on a platform with m identical processors if:  i = f(  i,  k ) k = C k /D k

Toward a better schedulability test for RM/DM Improve BAK when heavy tasks are considered Extend the ABJ test: for arbitrary task utilizations for constrained deadline systems

Can BAK be improved? I k > (D k -C k ) kk kk kk DEADLINE MISS CPU1 CPU2 CPU3 rkrk r k +D k I k = Total interference suffered by task  k I 2,k I 1,k I 2,k I 3,k I 4,k I 5,k I 6,k I 8,k I 5,k I 3,k I 7,k I 3,k I i,k = Interference of task  i on task  k

The BCL test  I i,k > m(D k -C k ) kk kk kk DEADLINE MISS CPU1 CPU2 CPU3 rkrk r k +D k I 2,k I 1,k I 2,k I 3,k I 4,k I 5,k I 6,k I 8,k I 5,k I 3,k I 7,k I 3,k IDEA: It is sufficient to consider at most the portion D k -C k of each term I i,k in the sum for all i,k: I i,k ≤ I k

The BCL test for DM A task set is schedulable with DM on m processors if and only if, for every task  k : Computing each I i,k requires exponential time To reduce the complexity: bound the interference with the load give an upper bound on the load Derive a sufficient condition to be checked for every task

The BCL test for DM  i = f(  i,D k ) k = C k /D k Complexity is O(n 2 ) A task set is schedulable with DM on m processors if, for every task  k :

Can ABJ be improved? New analysis for constrained deadline systems and priorities according to DM Improvement over ABJ: Preperiod deadline systems Arbitrary individual task utilization Higher global utilization Introduce to a better schedulability bound for the fixed priority global scheduling class of algorithms

Density and utilization based test for RM/DM A task set with constrained deadlines is schedulable with DM on m ≥ 2 identical processors if: A task set with implicit deadlines is schedulable with RM on m ≥ 2 identical processors if:

Improvement over existing bounds Bound more general than ABJ: taking we have as ABJ. Corrected (and extended) Baker ’ s bound [RTSS ’ 03]

Existing schedulability bounds for SMPs M=number of processors U=worst-case total utilization [Carpenter, Funk, Holman, Srinivasan, Anderson, Baruah]

Hybrid algorithms Treat differently heavy and light tasks Allow to overcome Dhall ’ s effect ALGORITHM RM-US[U th ] - if (U i >U th )  task has maximum priority - else task has priority according to RM ALGORITHM DM-DS[λ th ] - if (λ i >λ th )  task has maximum priority - else task has priority according to DM

RM-US[1/3] and DM-DS[1/3] A task set with constrained deadlines is schedulable with DM-DS[1/3] on m ≥ 2 identical processors if: A task set with implicit deadlines is schedulable with RM-US[1/3] on m ≥ 2 identical processors if:

Existing schedulability bounds for SMPs M=number of processors U=worst-case total utilization [Carpenter, Funk, Holman, Srinivasan, Anderson, Baruah]

Conclusions Extended BAK test for DM: BCL test that better behaves with heavy tasks Improved ABJ test: generalized to constrained deadline systems extended to arbitrary task utilizations/densities increased the schedulability bound for RM/DM Proposed hybrid algorithms (RM-US, DM-DS): improved the schedulability bound of the fixed priority global scheduling class of algorithms

THE END Marko: Michele: Peppe: