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

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

Abstraction and Composition 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 3 Sporadic (10,2,10) EDF Sporadic (15,2,15) Component interface

Resource Satisfiability Analysis Given a component and a resource model, resource satisfiability analysis is to determine if, for every time interval, RePP Workshop 4 (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 Resource optimality Periodic resource model Conclusions EDP resource model Incremental analysis Multiprocessor clustering Compositional schedulability analysis Resource model based analysis Periodic resource models

RePP Workshop 6 Resource Demand Bound Resource demand bound during an interval of length t –dbf(W,A,t) computes the maximum possible resource demand that W requires under algorithm A during a time interval of length t 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) t demand

RePP Workshop 7 For a periodic workload set W = {Ti(pi,ei)}, –dbf (W,A,t) for EDF algorithm [Baruah et al.,‘90] Demand Bound - EDF t demand

RePP Workshop 8 Task (resource demand) representations

RePP Workshop 9 Resource Supply Bound Resource supply during an interval of length t –sbf R (t) : the minimum possible resource supply by resource R over all intervals of length t For a single periodic resource model, i.e., Γ (3,2) –we can identify the worst-case resource allocation t supply

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

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) RePP Workshop Fraction b

RePP Workshop 12 Component Abstraction T 11 (25,4) T 12 (40,5) T 21 (25,4) T 22 (40,5) R(?, ?) EDF RM R 2 (?, ?)R 1 (?, ?)R 1 (10, 3.01)R 2 (10, 4.34)

RePP Workshop 13 Compositional Real-Time Guarantees T 11 (25,4) T 12 (40,5) T 21 (25,4) T 22 (40,5) R(?, ?) EDF RM R 2 (?, ?)R 1 (?, ?)R 1 (10, 3.1)R 2 (10, 4.4) T 1 (10, 3.1)T 2 (10, 4.4) R(5, 4.4)

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

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 periodic resource models RePP Workshop  (5,3,4)  

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

What is the Problem? 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 C 5 –Depends on interfaces of C 4 and C 3 Do we really need resource models for this analysis? –Final goal is to abstract components into tasks and jobs RePP Workshop 17 C1C1 EDF C2C2 DM C4C4 EDF C3C3 C5C5 DM Sporadic tasks

Assumptions and Notations Assumptions –Workload comprised of constrained deadline periodic tasks W = {  i = (T i,C i,D i ) i=1…n }, with D i ≤ T i for all i –Ignore preemption related overheads Notations –Schedulability load of component C = {W, EDF} LOAD C = max t>0 {dbf C (t)/t} –Schedulability load of component C = {W, DM} LOAD C = max i min t≤D i {rbf C,i (t)/t} –Feasibility load of workload W LOAD W = LOAD C, where C = {W, EDF} RePP Workshop 18

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

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 RePP Workshop 20 t × LOAD C dbf C dbf of periodic task  C Slope of line L OAD C = LOAD { ,S} dbf C sbf   +  -  (  +  -  ) -+- (-+-)-+- (-+-)

Are Load Optimal Interfaces Really Optimal? Consider –C 1 has workload {(6,1,6),(12,1,12)} and uses EDF –C 2 has workload {(5,1,3),(10,1,7)} and uses EDF –C 3 has workload {C 1,C 2 } and uses EDF argmax t dbf C 1 (t)/t ≠ argmax t dbf C 2 (t)/t RePP Workshop 21 dbf C 1 + dbf C 2 t × (LOAD C 1 + LOAD C 2 )

Example (1) Zero slack assumption in open systems –Let  1 =(7,1,7),  2 =(9,1,9), C 1 ={(  1,  2 ), DM}, C 3 ={(C 1,C 2 ), EDF} for some C 2 –Since C 2 is unknown to C 1, assume max. interference for  1 and  2 from C RePP Workshop 22 ReleaseDeadlineReleaseDeadline Task  1 Task  2 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

Example (2) RePP Workshop 23

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 24

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” (1)T_1 is schedulable under S_1 (2) T_2 is schedulable under S_2 (3) T_1 is not schedulable under S_2

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 29

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

July 23, 2008 AFOSR PI Meeting 31 Multicore Processor Virtualization Scheduler CPU 1.Compositional analysis of hierarchical multiprocessor real-time systems, through component interfaces 2.Using virtualization to develop new component interface for multiprocessor platforms Task S interface Task S interface Task S interface CPU Virtual CPU RePP Workshop 31

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 partitioned scheduling: low utilization, easy to compute global scheduling: high utilization, hard to compute cluster-based scheduling: small clusters => partitioned, large clusters => global RePP Workshop 32

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

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 RePP Workshop Processor 1 Processor 2 Processor 3 Processor 4 11 11 55 55 22 22 33 33 44 44 11 11 22 22 33 33 44 44 55 55 66 66 66 34

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        RePP Workshop 35

Virtual Cluster-based Scheduling 1.Split task set  into clusters  x 1, …,  x k 2.Abstract  x i into MPR interface  i (for cluster VC i ) 3.Transform each  i into periodic tasks Enables inter-cluster scheduler to schedule  i RePP Workshop 36

Resource Satisfiability Condition RePP Workshop Workload upper bound for Type (1) intervals Workload upper bound for Type(2) intervals assuming no carry-in demand at t idle [BCL05] m i -1 largest carry-in demand values at t idle (at most m i -1 tasks are active at t idle ) [Baruah07] Pseudo-polynomial upper bound for A l [Thm. 2 in SEL08] 37

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

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 39

References A Compositional Scheduling Framework for Digital Avionics Systems, Arvind Easwaran, Insup Lee, Oleg Sokolsky, Steve Vestal, IEEE RTCSA 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 ,2007 Incremental Schedulability Analysis of Hierarchical Real-Time Components, Arvind Easwaran, Insik Shin, Oleg Sokolsky, Insup Lee, ACM EMSOFT RePP Workshop 40 This work was supported in part by AFOSR and NSF.