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Insup Lee PRECISE Center Department of Computer and Information Science University of Pennsylvania

Simulation Relations, Interface Complexity, and Resource Optimality for Real-Time Hierarchical Systems. Insup Lee PRECISE Center Department of Computer and Information Science University of Pennsylvania November 15, 2009 RePP Workshop, ESWeek 2009

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Insup Lee PRECISE Center Department of Computer and Information Science University of Pennsylvania

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  1. Simulation Relations, Interface Complexity, and Resource Optimality for Real-Time Hierarchical Systems Insup Lee PRECISE Center Department of Computer and Information Science University of Pennsylvania November 15, 2009 RePP Workshop, ESWeek 2009 Join work with M. Anand, A. Easwaran, L. Phan, A. Philippou, O. Sokolsky

  2. Hierarchical scheduling framework Interface Interface Interface Global scheduler 2 1 n R R R m 2 1 Local scheduler Local Local Local scheduler scheduler scheduler Subsystem Subsystem Subsystem 1 2 n HRT HRT HRT 2 i 1 [Behnam] RePP Workshop

  3. Abstraction and Composition Sporadic (10,2,10) Sporadic (15,2,15) Component interface EDF • Abstraction Problem: Abstract resource demand of real-time component into an interface • Minimize resource usage: Identify minimum amount of resource required to guarantee schedulability of real-time component RePP Workshop

  4. Resource Satisfiability Analysis • Given a component and a resource model, resource satisfiability analysis is to determine if, for every time interval, (maximum possible) resource demand that the component’s task set needs under its scheduling algorithm (minimum possible) resource supply that resource model provides ≤ RePP Workshop

  5. Motivation • Compositional schedulability analysis • Resource model based analysis • Periodic resource models Periodic resource model EDP resource model Incremental analysis Resource optimality Multiprocessor clustering Conclusions RePP Workshop

  6. Resource Demand Bound • Resource demand bound during an interval of length t • dbf(W,A,t) computes the maximum possibleresource demandthat Wrequires under algorithm A during atime interval of lengtht • Periodic task model T(p,e) [Liu & Layland, ’73] • characterizes the periodic behavior of resource demand with period p and execution time e • Ex: T(3,2) demand t 0 1 2 3 4 5 6 7 8 9 10 RePP Workshop

  7. Demand Bound - EDF • For a periodic workload set W = {Ti(pi,ei)}, • dbf (W,A,t) for EDF algorithm [Baruah et al.,‘90] demand t 0 1 2 3 4 5 6 7 8 9 10 RePP Workshop

  8. Task (resource demand) representations RePP Workshop

  9. Resource Supply Bound • Resource supply during an interval of length t • sbfR(t) : the minimumpossible resource supply by resourceR over all intervalsof lengtht • For a single periodic resource model, i.e.,Γ(3,2) • we can identify the worst-case resource allocation supply t 0 1 2 3 4 5 6 7 8 9 10 RePP Workshop

  10. Demand-based Schedulability Analysis • A periodic task set is schedulable under EDF if and only if dbf(t) ≤ t ≤ over the periodic resource modelΓ(P,Q) lsbf(t) [Shin and Lee, 2003] [Baruah, et. al., ‘90] t lsbf(t) resource dbf(t) t RePP Workshop

  11. Resource Models as Interfaces • Characterization of resource supply • Underlying component’s perspective: Virtualizes scheduling hierarchy represented by all higher-level components • Parent component’s perspective: Characterizes resource supply to underlying component • Fractional resource model • b.t units of resource in every t time units (0 < b <= 1) • Not realizable (processor cannot be shared fractionally) Fraction b 0 1 2 3 4 5 6 7 8 9 10 RePP Workshop

  12. Component Abstraction T11(25,4) T12(40,5) T21(25,4) T22(40,5) R2(?, ?) R1(?, ?) R1(10, 3.01) R2(10, 4.34) EDF R(?, ?) EDF RM RePP Workshop

  13. Compositional Real-Time Guarantees T2(10, 4.4) T1(10, 3.1) T11(25,4) T12(40,5) T21(25,4) T22(40,5) R2(10, 4.4) R1(10, 3.1) R1(?, ?) R2(?, ?) EDF R(5, 4.4) R(?, ?) EDF RM RePP Workshop

  14. Resource Supply Models ACSR+ Recurring branching resource supply model EDP model Tree schedule Bounded-delay Resource model Periodic model Cyclic Executive RePP Workshop

  15. EDP Resource Model • Explicit Deadline Periodic resource model:  = (,,) •  resource units in  time units • Repeat supply every  time units • Properties • Time-partitioned, periodic resource allocation behavior • Benefits of realizability and implementability • Blackout interval in EDP depends on  and , for fixed  • Interval can be controlled using  without changing bandwidth • Smaller bandwidth required to schedule the same component, when compared to periodicresource models       h = (5,3,4) 0 1 2 3 4 5 6 7 8 9 10 RePP Workshop

  16. Motivation Periodic resource model EDP resource model Incremental analysis Characterization of optimality in compositional schedulability analysis Resource optimality Multiprocessor clustering Conclusions RePP Workshop

  17. What is the Problem? Sporadic tasks C1 C4 C3 C5 EDF EDF EDF DM C2 DM Sporadic tasks Sporadic tasks • Existing models (periodic and EDP) have resource overheads • At least in comparison to total demand of elementary components • Is total elementary workload a good measure? • What about overhead of DM? • How to account for DM overhead in component C5 • Depends on interfaces of C4 and C3 • Do we really need resource models for this analysis? • Final goal is to abstract components into tasks and jobs RePP Workshop

  18. Assumptions and Notations • Assumptions • Workload comprised of constrained deadline periodic tasks • W = {i = (Ti,Ci,Di)i=1…n}, with Di ≤ Ti for all i • Ignore preemption related overheads • Notations • Schedulability load of component C = {W, EDF} • LOADC = maxt>0 {dbfC(t)/t} • Schedulability load of component C = {W, DM} • LOADC = maxi mint≤Di {rbfC,i(t)/t} • Feasibility load of workload W • LOADW = LOADC, where C = {W, EDF} RePP Workshop

  19. Load Optimal Interfaces • Match feasibility load of interface Cwith schedulability load of component C • C = (1,LOADC,1)is a periodic task • Release of first job in C is synchronized with release of first job in  t × LOADC Slope of line LOADC = LOAD{,S} C C S dbfC Interface set  dbf of periodic task C RePP Workshop

  20. Significance of Load Optimality • Proportionate fair scheduling of components • Comparison to resource model based interfaces • Periodic model =(,) is load optimal only when =0 • EDP model =(,,) is load optimal when • = and  is GCD of deadlines and periods of all tasks in the system sbf -+- (-+-) dbfC dbfC  + - (+-) RePP Workshop t × LOADC Slope of line LOADC = LOAD{,S} dbf of periodic task C

  21. Are Load Optimal Interfaces Really Optimal? • Consider • C1 has workload {(6,1,6),(12,1,12)} and uses EDF • C2 has workload {(5,1,3),(10,1,7)} and uses EDF • C3 has workload {C1,C2} and uses EDF • argmaxt dbfC1(t)/t ≠ argmaxt dbfC2(t)/t t × (LOADC1 + LOADC2) 7 6 4 2 1 dbfC1+ dbfC2 5 6 10 12 15 RePP Workshop

  22. Example (1) • Zero slack assumption in open systems • Let 1=(7,1,7), 2=(9,1,9), C1={(1,2), DM}, C3={(C1,C2), EDF} for some C2 • Since C2 is unknown to C1, assume max. interference for 1 and 2 from C2 • Suppose we abstract1 using periodic task  = (O,T,C,D), where O is release offset • O and O+D must be an entry in table • For all integers k, O+kT and O+kT+D must also be entries in the table • Only possible when T = 63, which is LCM of periods of tasks 1 and 2 RePP Workshop

  23. Example (2) RePP Workshop

  24. Questions • Hardness of achieving demand optimal interfaces • Can we classify the hardness? • Classification may lead us to some approximation schemes • Load optimal interfaces and resource model based techniques offer one extreme solution • Abstract component into a single periodic task • Can we trade interface size for resource utilization? • What about logarithmically or polynomially large interfaces? RePP Workshop

  25. Task (resource demand) representations RePP Workshop

  26. Non-composable periodic models? • What are right abstraction levels for real-time components? (period, execution time) • P1 = (p1,e1); e.g., (3,1) • P2 = (p2,e2); e.g., (7,1) • What is P1 || P2? • (LCM(p1,p2), e1*n1 + e2*n2); e.g., (21,10) where n1*p1 = n2*p2 = LCM(p1,p2) • What is the problem? • beh(P1) || beh(P2) = beh(P1||P2)? • Compositionality • (P1 || P2) || P3 = P || P3, where P = P1 || P2 RePP Workshop

  27. ACSR RePP Workshop

  28. ACSR+ for supply partition specification • Notion of “schedulable under” • T_1 is schedulable under S_1 (2) T_2 is schedulable under S_2 (3) T_1 is not schedulable under S_2 RePP Workshop

  29. Current work • Simulation and schedulability Relations (preliminary results with Anna Philippou) • Between demand and supply • Between demand processes • Between supplies • Semantic characterization of demand-supply based schedulability analysis RePP Workshop

  30. Motivation Periodic resource model EDP resource model • Virtual processor cluster-based • scheduling on multiprocessors • Need for virtual clustering • MPR model based scheduling • Optimal virtual clustering for • implicit deadline task systems Incremental analysis Resource optimality Multiprocessor clustering CARTS July 23, 2008 AFOSR PI Meeting RePP Workshop 30

  31. Multicore Processor Virtualization CPU CPU CPU interface interface interface • Compositional analysis of hierarchical multiprocessor real-time systems, through component interfaces • Using virtualization to develop new component interface for multiprocessor platforms Virtual CPU Scheduler S S S Task Task Task Task Task Task July 23, 2008 AFOSR PI Meeting RePP Workshop 31

  32. Virtual Clusters • Use platform virtualization to provide a trade-off between resource utilization and scheduling complexity • Cluster interface: (,m) •  is the resource model, m is the maximum number of physical processors available • Inter-cluster scheduling is optimal partitionedscheduling:low utilization,easy to compute cluster-basedscheduling:small clusters => partitioned,large clusters => global globalscheduling:high utilization,hard to compute RePP Workshop

  33. Virtual Clustering Interface 1 2 . . . m Physical processors . . . 1(m1) 2(m2) k(mk) Virtual processors . . . x1 x2 xk Task clusters For each i, mi (<= m) is maximum number of physical processors that can be assigned to i at any instant RePP Workshop 33

  34. Virtual Clustering • Task set and number of processors • ====(3,2,3), =(6,4,6), and =(6,3,6), m=4 • Schedule under clustered scheduling • 1, 2, 3 scheduled on processors 1 and 2 • 4, 5, 6 scheduled on processors 3 and 4 1 2 1 1 2 1 Processor 1 2 3 3 2 3 3 Processor 2 4 4 6 4 4 Processor 3 5 5 5 5 6 6 Processor 4 0 1 2 3 4 5 6 RePP Workshop

  35. Multiprocessor Periodic Resource (MPR) model •  = (, ,,m’) •  units of resource supply guaranteed in every  time units, with concurrency at most m’ in any time instant • Consider  = (5, 12,3) • Why MPR model? • Periodicity enables transformation of MPR model to periodic tasks which can be scheduled using standard algorithms           Processor 2         Processor 3       Processor 4 0 1 2 3 4 5 6 7 8 9 10 RePP Workshop

  36. Virtual Cluster-based Scheduling • Split task set  into clusters x1, …, xk • Abstract xi into MPR interface i (for cluster VCi) • Transform each i into periodic tasks • Enables inter-cluster scheduler to schedule i RePP Workshop

  37. Resource Satisfiability Condition • Workload upper bound for Type (1) intervals • Workload upper bound for Type(2) intervals • assuming no carry-in demand at tidle [BCL05] mi-1 largest carry-in demand values at tidle (at most mi-1 tasks are active at tidle) [Baruah07] • Pseudo-polynomialupper bound for Al[Thm. 2 in SEL08] RePP Workshop

  38. Inter-cluster Scheduling • Suppose • i (=) is GCD of periods/deadlines of all tasks in W • Each model i=(, i, mi) is transformed to some periodic tasks all having period and deadline  • McNaughton’s algorithm is used for inter-cluster scheduling 1 2 1 2 Processor 1 2 3 4 2 3 4 Processor 2 4 4 Processor 3 0  2 • Inter-cluster scheduling is optimal • Successive jobs of the same task are scheduled in identical intervals RePP Workshop

  39. Questions • Open issues • Efficient clustering techniques for constrained and arbitrary deadline task systems • Interface optimality for multiprocessor systems • Hierarchical/compositional multi-mode systems with resource constraints RePP Workshop

  40. References • A Compositional Scheduling Framework for Digital Avionics Systems, Arvind Easwaran, Insup Lee, Oleg Sokolsky, Steve Vestal, IEEE RTCSA 2009. • Optimal Virtual Cluster-based Multiprocessor Scheduling, Arvind Easwaran, Insik Shin, and Insup Lee, to be published in Real-Time Systems Journal (RTSJ). • On the complexity of generating optimal interfaces for hierarchical systems, Arvind Easwaran, Madhukar Anand, Insup Lee, Oleg Sokolsky, Workshop on Compositional Theory and Technology for Real-Time Embedded Systems (CRTS 2008). • Hierarchical Scheduling Framework for Virtual Clustering of Multiprocessors, Insik Shin, Arvind Easwaran, Insup Lee, ECRTS, Prague, Czech Republic, July 2-4, 2008 (Runner-up in the best paper award) • Robust and Sustainable Schedulability Analysis of Embedded Software, Madhukar Anand and Insup Lee, LCTES, Tucson, AZ, Jun 12-13, 2008 • Compositional Feasibility Analysis for Conditional Task Models, Madhukar Anand, Arvind Easwaran, Sebastian Fischmeister, and Insup Lee, ISORC, Orlando, Florida, May 5-7, 2008 • Compositional Real-Time Scheduling Framework with Periodic Model, Insik Shin and Insup Lee, ACM Transactions on Embedded Computing Systems (TECS), vol 7, no 3, April 2008 • Interface Algebra for Analysis of Hierarchical Real-Time Systems, Arvind Easwaran, Insup Lee, Oleg Sokolsky, Foundations of Interface Technologies (FIT) workshop, Budapest, Hungary, April 5, 2008 • Compositional Analysis Framework using EDP Resource Models, Arvind Easwaran, Madhukar Anand, and Insup Lee, Tucson, Arizona, Dec 3-6, 2007 • Resources in process algebra, Insup Lee, Anna Philippou, Oleg Sokolsky, Journal of Logic and Algebraic Programming, Vol. 72, pp. 98-122,2007 • Incremental Schedulability Analysis of Hierarchical Real-Time Components, Arvind Easwaran, Insik Shin, Oleg Sokolsky, Insup Lee, ACM EMSOFT 2006. This work was supported in part by AFOSR and NSF. RePP Workshop

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