ΗΜΥ 653 Ενσωματωμένα Συστήματα και Συστήματα Πραγματικού Χρόνου Εαρινό Εξάμηνο 2017

Transcription

1 ΗΜΥ 653 Ενσωματωμένα Συστήματα και Συστήματα Πραγματικού Χρόνου Εαρινό Εξάμηνο 2017 ΔΙΑΛΕΞΕΙΣ 7-8: Neural Network Build-Up and Training Mapping Applications Scheduling - Optimization Algorithms -- Mapping Tools ΧΑΡΗΣ ΘΕΟΧΑΡΙΔΗΣ Επίκουρος Καθηγητής ΗΜΜΥ (ttheocharides@ucy.ac.cy) Slides adopted from Prof. Peter Marwedel, Microsoft

2 Embedded System Software Application Knowledge 2: Specification 3: ES-hardware 4: system software (RTOS, middleware, ) Design repository 6: Application mapping 7: Optimization 5: Validation & Evaluation (energy, cost, performance, ) Design 8: Test ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.2 Θεοχαρίδης, ΗΜΥ, 2017

3 Mapping of Applications to Platforms Renesas, Thiele ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.3 Θεοχαρίδης, ΗΜΥ, 2017

4 Distinction between mapping problems ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.4 Θεοχαρίδης, ΗΜΥ, 2017

5 Problem Description q Given q Find A set of applications Scenarios on how these applications will be used A set of candidate architectures comprising - (Possibly heterogeneous) processors - (Possibly heterogeneous) communication architectures - Possible scheduling policies A mapping of applications to processors Appropriate scheduling techniques (if not fixed) A target architecture (if DSE is included) q Objectives Keeping deadlines and/or maximizing performance Minimizing cost, energy consumption ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.5 Θεοχαρίδης, ΗΜΥ, 2017

6 Related Work Mapping to EXUs in automotive design Scheduling theory: Provides insight for the mapping task start times Hardware/software partitioning: Can be applied if it supports multiple processors High performance computing (HPC) Automatic parallelization, but only for - single applications, - fixed architectures, - no support for scheduling, - memory and communication model usually different High-level synthesis Provides useful terms like scheduling, allocation, assignment Optimization theory ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.6 Θεοχαρίδης, ΗΜΥ, 2017

7 Scope of mapping algorithms Useful terms from hardware synthesis: Resource Allocation Decision concerning type and number of available resources Resource Assignment Mapping: Task (Hardware) Resource xx to yy binding: Describes a mapping from behavioral to structural domain, e.g. task to processor binding, variable to memory binding Scheduling Mapping: Tasks Task start times Sometimes, resource assignment is considered being included in scheduling. ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.7 Θεοχαρίδης, ΗΜΥ, 2017

8 Classes of mapping algorithms (considered in this course) Classical scheduling algorithms Mostly for independent tasks & ignoring communication, mostly for mono- and homogeneous multiprocessors Hardware/software partitioning Dependent tasks, heterogeneous systems, focus on resource assignment Dependent tasks as considered in architectural synthesis Initially designed in different context, but applicable Design space exploration using genetic algorithms Heterogeneous systems, incl. communication modeling ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.8 Θεοχαρίδης, ΗΜΥ, 2017

9 Real-time scheduling q Assume that we are given a task graph G=(V,E). q Def.: A schedule s of G is a mapping V T of a set of tasks V to start times from domain T. G=(V,E) V1 V2 V3 V4 s T t Typically, schedules have to respect a number of constraints, incl. resource constraints, dependency constraints, deadlines. Scheduling = finding such a mapping. ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.9 Θεοχαρίδης, ΗΜΥ, 2017

10 Hard and soft deadlines q Def.: A time-constraint (deadline) is called hard if not meeting that constraint could result in a catastrophe [Kopetz, 1997]. q All other time constraints are called soft. q We will focus on hard deadlines. ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.10 Θεοχαρίδης, ΗΜΥ, 2017

11 Periodic and aperiodic tasks q Def.: Tasks which must be executed once every p units of time are called periodic tasks. p is called their period. Each execution of a periodic task is called a job. q All other tasks are called aperiodic. q Def.: Tasks requesting the processor at unpredictable times are called sporadic, if there is a minimum separation between the times at which they request the processor. ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.11 Θεοχαρίδης, ΗΜΥ, 2017

12 Preemptive and non-preemptive scheduling Non-preemptive schedulers: Tasks are executed until they are done. Response time for external events may be quite long. Preemptive schedulers: To be used if - some tasks have long execution times or - if the response time for external events to be short. ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.12 Θεοχαρίδης, ΗΜΥ, 2017

13 Centralized and distributed scheduling Centralized and distributed scheduling: Multiprocessor scheduling either locally on 1 or on several processors. Mono- and multi-processor scheduling: - Simple scheduling algorithms handle single processors, - more complex algorithms handle multiple processors. algorithms for homogeneous multi-processor systems algorithms for heterogeneous multi-processor systems (includes HW accelerators as special case). ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.13 Θεοχαρίδης, ΗΜΥ, 2017

14 Dynamic/online scheduling Dynamic/online scheduling: Processor allocation decisions (scheduling) at run-time; based on the information about the tasks arrived so far. ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.14 Θεοχαρίδης, ΗΜΥ, 2017

15 Static/offline scheduling Static/offline scheduling: Scheduling taking a priori knowledge about arrival times, execution times, and deadlines into account. Dispatcher allocates processor when interrupted by timer. Timer controlled by a table generated at design time. ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.15 Θεοχαρίδης, ΗΜΥ, 2017

16 Time-triggered systems (1) q In an entirely time-triggered system, the temporal control structure of all tasks is established a priori by off-line support-tools. This temporal control structure is encoded in a Task-Descriptor List (TDL) that contains the cyclic schedule for all activities of the node. This schedule considers the required precedence and mutual exclusion relationships among the tasks such that an explicit coordination of the tasks by the operating system at run time is not necessary... q The dispatcher is activated by the synchronized clock tick. It looks at the TDL, and then performs the action that has been planned for this instant [Kopetz]. ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.16 Θεοχαρίδης, ΗΜΥ, 2017

17 Time-triggered systems (2) q pre-run-time scheduling is often the only practical means of providing predictability in a complex system. [Xu, Parnas]. q It can be easily checked if timing constraints are met. The disadvantage is that the response to sporadic events may be poor. ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.17 Θεοχαρίδης, ΗΜΥ, 2017

18 Schedulability q Set of tasks is schedulable under a set of constraints, if a schedule exists for that set of tasks & constraints. q Exact tests are NP-hard in many situations. q Sufficient tests: sufficient conditions for schedule checked. (Hopefully) small probability of not guaranteeing a schedule even though one exists. q Necessary tests: checking necessary conditions. Used to show no schedule exists. There may be cases in which no schedule exists & we cannot prove it. sufficient schedulable necessary ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.18 Θεοχαρίδης, ΗΜΥ, 2017

19 Cost functions q Cost function: Different algorithms aim at minimizing different functions. q Def.: Maximum lateness = max all tasks (completion time deadline) Is <0 if all tasks complete before deadline. T1 T2 t ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.19 Θεοχαρίδης, ΗΜΥ, 2017

20 Aperiodic scheduling - Scheduling with no precedence constraints - q Let {T i } be a set of tasks. Let: c i be the execution time of T i, d i be the deadline interval, that is, the time between T i becoming available and the time until which T i has to finish execution. l i be the laxity or slack, defined as l i = d i - c i f i be the finishing time. i d i c i l i t ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.20 Θεοχαρίδης, ΗΜΥ, 2017

21 Uniprocessor with equal arrival times q Preemption is useless. q Earliest Due Date (EDD): Execute task with earliest due date (deadline) first. f i f i f i EDD requires all tasks to be sorted by their (absolute) deadlines. Hence, its complexity is O(n log(n)). ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.21 Θεοχαρίδης, ΗΜΥ, 2017

22 Optimality of EDD q EDD is optimal, since it follows Jackson's rule: Given a set of n independent tasks, any algorithm that executes the tasks in order of non-decreasing (absolute) deadlines is optimal with respect to minimizing the maximum lateness. q Proof (See Buttazzo, 2002): Let s be a schedule produced by any algorithm A If A ¹ EDD $ T a, T b, d a d b, T b immediately precedes T a in s. Let s' be the schedule obtained by exchanging T a and T b. ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.22 Θεοχαρίδης, ΗΜΥ, 2017

23 Exchanging T a and T b cannot increase lateness q Max. lateness for T a and T b in s is L max (a,b)=f a -d a Max. lateness for T a and T b in s' is L' max (a,b)=max(l' a,l' b ) Two possible cases 1. L' a L' b : L' max (a,b) = f' a d a < f a d a = L max (a,b) since T a starts earlier in schedule s'. 2. L' a L' b : L' max (a,b) = f' b d b = f a d b f a d a = L max (a,b) since f a =f' b and d a d b F L' max (a,b) L max (a,b) s s' T b T a T a T b f a =f' b ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.23 Θεοχαρίδης, ΗΜΥ, 2017

24 EDD is optimal F Any schedule s with lateness L can be transformed into an EDD schedule s n with lateness L n L, which is the minimum lateness. F EDD is optimal (q.e.d.) ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.24 Θεοχαρίδης, ΗΜΥ, 2017

25 Earliest Deadline First (EDF) - Horn s Theorem - q Different arrival times: Preemption potentially reduces lateness. q Theorem [Horn74]: Given a set of n independent tasks with arbitrary arrival times, any algorithm that at any instant executes the task with the earliest absolute deadline among all the ready tasks is optimal with respect to minimizing the maximum lateness. ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.25 Θεοχαρίδης, ΗΜΥ, 2017

26 Earliest Deadline First (EDF) - Algorithm - q Earliest deadline first (EDF) algorithm: Each time a new ready task arrives: It is inserted into a queue of ready tasks, sorted by their absolute deadlines. Task at head of queue is executed. If a newly arrived task is inserted at the head of the queue, the currently executing task is preempted. q Straightforward approach with sorted lists (full comparison with existing tasks for each arriving task) requires run-time O(n 2 ); (less with binary search or bucket arrays). Sorted queue Executing task ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.26 Θεοχαρίδης, ΗΜΥ, 2017

27 Earliest Deadline First (EDF) - Example - Earlier deadline F preemption Later deadline F no preemption ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.27 Θεοχαρίδης, ΗΜΥ, 2017

28 Optimality of EDF q To be shown: EDF minimizes maximum lateness. q Proof (Buttazzo, 2002): Let s be a schedule produced by generic schedule A Let s EDF : schedule produced by EDF Preemption allowed: tasks executed in disjoint time intervals s divided into time slices of 1 time unit each Time slices denoted by [t, t+1) Let s(t): task executing in [t, t+1) Let E(t): task which, at time t, has the earliest deadline Let t E (t): time (³t) at which the next slice of task E(t) begins its execution in the current schedule ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.28 Θεοχαρίδης, ΗΜΥ, 2017

29 Optimality of EDF (2) q If s ¹ s EDF, then there exists time t: s(t) ¹ E(t) q Idea: swapping s(t) and E(t) cannot increase max. lateness. T 1 t=4; s(t)=4; E(t)=2; t E =6; s(t E )=2 t T 2 t T 3 t T t T 1 s(t)=2; s(t E )=4 t T 2 T 3 T 4 t t t If s(t) starts at t=0 and D=max i {d i } then s EDF can be obtained from s by at most D transpositions. [Buttazzo, 2002] ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.29 Θεοχαρίδης, ΗΜΥ, 2017

30 Optimality of EDF (3) q Algorithm interchange: q { for (t=0 to D-1) { Using the same argument as in the proof of Jackson s algorithm, it is easy to show that swapping cannot increase maximum lateness; hence EDF is optimal. q if (s(t) ¹ E(t)) { q s(t E ) = s(t); q s(t) = E(t); }}} Does interchange preserve schedulability? 1. task E(t) moved ahead: meeting deadline in new schedule if meeting deadline in s 2. task s(t) delayed: if s(t) is feasible, then (t E +1) d E, where d E is the earliest deadline. Since d E d i for any i, we have t E +1 d i, which guarantees schedulability of the delayed task. [Buttazzo, 2002] q.e.d. ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.30 Θεοχαρίδης, ΗΜΥ, 2017

31 Least laxity (LL), Least Slack Time First (LST) q Priorities = decreasing function of the laxity (the less laxity, the higher the priority); dynamically changing priority; preemptive. l l l l l l l l l l l l ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.31 Θεοχαρίδης, ΗΜΥ, 2017

32 Properties Not sufficient to call scheduler & re-compute laxity just at task arrival times. Overhead for calls of the scheduler. Many context switches. Detects missed deadlines early. LL is also an optimal scheduling for mono-processor systems. Dynamic priorities F cannot be used with a fixed prio OS. LL scheduling requires the knowledge of the execution time. ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.32 Θεοχαρίδης, ΗΜΥ, 2017

33 Scheduling without preemption (1) Lemma: If preemption is not allowed, optimal schedules may have to leave the processor idle at certain times. Proof: Suppose: optimal schedulers never leave processor idle. ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.33 Θεοχαρίδης, ΗΜΥ, 2017

34 Scheduling without preemption (2) q T1: periodic, c 1 = 2, p 1 = 4, d 1 = 4 q T2: occasionally available at times 4*n+1, c 2 = 1, d 2 = 1 q T1 has to start at t=0 q F deadline missed, but schedule is possible (start T2 first) q F scheduler is not optimal F contradiction! q.e.d. ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.34 Θεοχαρίδης, ΗΜΥ, 2017

35 Scheduling without preemption q Preemption not allowed: F optimal schedules may leave processor idle to finish tasks with early deadlines arriving late. F Knowledge about the future is needed for optimal scheduling algorithms FNo online algorithm can decide whether or not to keep idle. q EDF is optimal among all scheduling algorithms not keeping the processor idle at certain times. q If arrival times are known a priori, the scheduling problem becomes NP-hard in general. B&B typically used. ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.35 Θεοχαρίδης, ΗΜΥ, 2017

36 Scheduling with precedence constraints q Task graph and possible schedule: ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.36 Θεοχαρίδης, ΗΜΥ, 2017

37 Simultaneous Arrival Times: The Latest Deadline First (LDF) Algorithm q LDF [Lawler, 1973]: reads the task graph and among the tasks with no successors inserts the one with the latest deadline into a queue. It then repeats this process, putting tasks whose successor have all been selected into the queue. q At run-time, the tasks are executed in the generated total order. q LDF is non-preemptive and is optimal for mono-processors. If no local deadlines exist, LDF performs just a topological sort. ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.37 Θεοχαρίδης, ΗΜΥ, 2017

38 Asynchronous Arrival Times: Modified EDF Algorithm q This case can be handled with a modified EDF algorithm. The key idea is to transform the problem from a given set of dependent tasks into a set of independent tasks with different timing parameters [Chetto90]. q This algorithm is optimal for mono-processor systems. q If preemption is not allowed, the heuristic algorithm developed by Stankovic and Ramamritham can be used. ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.38 Θεοχαρίδης, ΗΜΥ, 2017

39 L. Thiele, 2006 ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.39 Θεοχαρίδης, ΗΜΥ, 2017

40 Summary q Definition mapping terms Resource allocation, assignment, binding, scheduling Hard vs. soft deadlines Static vs. dynamic FTT-OS Schedulability q Classical scheduling Aperiodic tasks - No precedences Simultaneous (FEDD) & Asynchronous Arrival Times (FEDF, LL) - Precedences Simultaneous Arrival Times (F LDF) Asynchronous Arrival Times (F medf) ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.40 Θεοχαρίδης, ΗΜΥ, 2017

41 The need to support heterogeneous architectures Energy efficiency a key constraint, e.g. for mobile systems Unconventional architectures close to IPE Hugo De Man/Philips, 2007 Renesas, MPSoC 07 F How to map to these architectures? ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.41 Θεοχαρίδης, ΗΜΥ, 2017

42 Practical problem in automotive design Which processor should run the software? ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.42 Θεοχαρίδης, ΗΜΥ, 2017

43 A Simple Classification? Architecture fixed/ Auto-parallelizing Fixed Architecture Architecture to be designed Starting from given task graph Take a job and put it on a fixed processor, i.e. CELL. Design a custom architecture for the job. Auto-parallelizing Devise an algorithm to parallelize the task on fixed resources. Design an algorithm AND a custom architecture to parallelize the job! ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.43 Θεοχαρίδης, ΗΜΥ, 2017

44 Example: System Synthesis L. Thiele, ETHZ ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.44 Θεοχαρίδης, ΗΜΥ, 2017

45 Basic Model Problem Graph L. Thiele, ETHZ ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.45 Θεοχαρίδης, ΗΜΥ, 2017

46 Basic Model: Specification Graph L. Thiele, ETHZ ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.46 Θεοχαρίδης, ΗΜΥ, 2017

47 Design Space Communication Templates Computation Templates Cipher FPGA SDRAM RISC DSP µe LookUp Scheduling/Arbitration TDMA EDF proportional WFQ share FCFS dynamic fixed priority static Which architecture is better suited for our application? Architecture # 1 Architecture # 2 LookUp Cipher RISC DSP EDF TDMA Priority WFQ µe µe µe µe µe µe static L. Thiele, ETHZ ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.47 Θεοχαρίδης, ΗΜΥ, 2017

48 Evolutionary Algorithms for Design Space Exploration (DSE) L. Thiele, ETHZ ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.48 Θεοχαρίδης, ΗΜΥ, 2017

49 Challenges L. Thiele, ETHZ ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.49 Θεοχαρίδης, ΗΜΥ, 2017

50 Application Model q Example of a simple stream processing task structure: L. Thiele, ETHZ ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.50 Θεοχαρίδης, ΗΜΥ, 2017

51 Exploration Case Study (1) L. Thiele, ETHZ ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.51 Θεοχαρίδης, ΗΜΥ, 2017

52 Exploration Case Study (2) L. Thiele, ETHZ ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.52 Θεοχαρίδης, ΗΜΥ, 2017

53 Exploration Case Study (3) L. Thiele, ETHZ ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.53 Θεοχαρίδης, ΗΜΥ, 2017

54 More Results Performance for encryption/decryption Performance for RT voice processing L. Thiele, ETHZ ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.54 Θεοχαρίδης, ΗΜΥ, 2017

55 Design Space Exploration with SystemCoDesigner (Teich, Erlangen) System Synthesis comprises: Idea: - Resource allocation - Actor binding - Channel mapping - Transaction modeling - Formulate synthesis problem as 0-1 ILP - Use Pseudo-Boolean (PB) solver to find feasible solution - Use multi-objective Evolutionary algorithm (MOEA) to optimize Decision Strategy of the PB solver J. Teich, U. Erlangen-Nürnberg ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.55 Θεοχαρίδης, ΗΜΥ, 2017

56 A 3rd approach based on evolutionary algorithms: SYMTA/S: q [R. Ernst et al.: A framework for modular analysis and exploration of heteterogenous embedded systems, Real-time Systems, 2006, p. 124] ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.56 Θεοχαρίδης, ΗΜΥ, 2017

57 A fixed architecture approach: Map CELL q Martino Ruggiero, Luca Benini: Mapping task graphs to the CELL BE processor, 1st Workshop on Mapping of Applications to MPSoCs, Rheinfels Castle, 2008 ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.57 Θεοχαρίδης, ΗΜΥ, 2017

58 Partitioning into Allocation and Scheduling R Ruggiero, Benini, 2008 ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.58 Θεοχαρίδης, ΗΜΥ, 2017

59 JPEG/JPEG2000 case study q Example architecture instances for a single-tile JPEG encoder: 16KB 2KB 32KB 32KB Vin,DCT Q,VLE,Vout Vin,Q,VLE,Vout 4KB DCT 2 MicroBlaze processors (50KB) 1 MicroBlaze, 1HW DCT (36KB) 8KB Vin DCT, Q DCT, Q 4x2KB 32KB VLE, Vout 2KB Vin 8KB DCT 2KB Q 8KB 32KB 2KB VLE, Vout 4x2KB DCT, Q DCT, Q 4x16KB 2KB DCT 2KB 8KB Q 2KB 6 MicroBlaze processors (120KB) 4 MicroBlaze, 2HW DCT (68KB) E. Deprettere, U. Leiden ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.59 Θεοχαρίδης, ΗΜΥ, 2017

60 Introduction of Memory Architecture-Aware Optimization The MACC PMS (Processor/Memory/Switch) Model Explicit memory architecture API provides access to memory information MACC_System C code CPU1 CPU2 CPU3 SPM SPM SPM L1$ L1$ BUS1 MM1 MM2 BRI L2$ MM3 BUS2 ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.60 Θεοχαρίδης, ΗΜΥ, 2017

61 MaCC Modeling Example via GUI ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.61 Θεοχαρίδης, ΗΜΥ, 2017

62 Toolflow Detailed View (Sequential C Source Code) START MACC Eco-System (1) Dynamic Data Type Optimizations (2) Map source code to task graphs (3) Parallelization Implem. MPSoC Parallelization Assistant (MPA) Memory Hierarchy (MH) (4) Dynamic Memory Management Optimizations MNEMEE Toolflow 1. Optimization of dynamic data structures 2. Extraction of potential parallelism 3. Implementation of parallelism; placement of static data 4. Placement of dynamic data ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.62 Θεοχαρίδης, ΗΜΥ, 2017 Page 62

63 Toolflow Detailed View (5) Scenario Based Mapping (5) Memory Aware Mapping (6) RTLIB Mapping (7) Scratchpad Memory Optimizations per PE MNEMEE Toolflow 5. Perform mapping to processing elements Scenario based Memory aware 6. Transform the code to implement the mapping 7. Perform scratchpad memory optimizations for each processing element Platform DB END (Optimized Source Code) ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.63 Θεοχαρίδης, ΗΜΥ, 2017 Page 63

64 MAPS-TCT Framework Leupers, Sheng, 2008 Rainer Leupers, Weihua Sheng: MAPS: An Integrated Framework for MPSoC Application Parallelization, 1st Workshop on Mapping of Applications to MPSoCs, Rheinfels Castle, 2008 ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.64 Θεοχαρίδης, ΗΜΥ, 2017

65 Summary Clear trend toward multi-processor systems for embedded systems, there exists a large design space Using architecture crucially depends on mapping tools Mapping applications onto heterogeneous MP systems needs allocation (if hardware is not fixed), binding of tasks to resources, scheduling Two criteria for classification - Fixed / flexible architecture - Auto parallelizing / non-parallelizing Introduction to proposed Mnemee tool chain q Evolutionary algorithms currently the best choice ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.65 Θεοχαρίδης, ΗΜΥ, 2017

66 Integer (linear) programming models q Ingredients: Cost function Constraints Involving linear expressions of integer variables from a set X Cost function C = å a x ÎX i i x i with a i ÎR, x Î N i N (1) Constraints: " j Î J : åbi, j xi ³ c j with bi, j, c j Î R x ÎX i R (2) Def.: The problem of minimizing (1) subject to the constraints (2) is called an integer (linear) programming (ILP) problem. If all x i are constrained to be either 0 or 1, the IP problem said to be a 0/1 integer (linear) programming problem. ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.66 Θεοχαρίδης, ΗΜΥ, 2017

67 Example C = 5x + x x x 1 1 +, x x2 4 x 2, x + 3 x 3 3 ³ 2 Î {0,1} C Optimal ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.67 Θεοχαρίδης, ΗΜΥ, 2017

68 Remarks on integer programming Maximizing the cost function: just set C =-C Integer programming is NP-complete. Running times depend exponentially on problem size, but problems of >1000 vars solvable with good solver (depending on the size and structure of the problem) The case of x i Î R is called linear programming (LP). Polynomial complexity, but most algorithms are exponential, in practice still faster than for ILP problems. The case of some x i Î R and some x i Î N is called mixed integer-linear programming. ILP/LP models good starting point for modeling, even if heuristics are used in the end. Solvers: lp_solve (public), CPLEX (commercial), ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.68 Θεοχαρίδης, ΗΜΥ, 2017

69 Simulated Annealing General method for solving combinatorial optimization problems. Based the model of slowly cooling crystal liquids. Some configuration is subject to changes. Special property of Simulated annealing: Changes leading to a poorer configuration (with respect to some cost function) are accepted with a certain probability. This probability is controlled by a temperature parameter: the probability is smaller for smaller temperatures. ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.69 Θεοχαρίδης, ΗΜΥ, 2017

70 Simulated Annealing Algorithm procedure SimulatedAnnealing; var i, T: integer; begin i := 0; T := MaxT; configuration:= <some initial configuration>; while not terminate(i, T) do begin while InnerLoop do begin NewConfig := variation(configuration); delta := evaluation(newconfig,configuration); if delta < 0 then configuration := NewConfig; else if SmallEnough(delta, T, random(0,1)) then configuration := Newconfiguration; end; T:= NewT(i,T); i:=i+1; end; end; ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.70 Θεοχαρίδης, ΗΜΥ, 2017

71 Explanation Initially, some random initial configuration is created. Current temperature is set to a large value. Outer loop: - Temperature is reduced for each iteration - Terminated if (temperature lower limit) or (number of iterations ³ upper limit). Inner loop: For each iteration: - New configuration generated from current configuration - Accepted if (new cost cost of current configuration) - Accepted with temperature-dependent probability if (cost of new config. > cost of current configuration). ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.71 Θεοχαρίδης, ΗΜΥ, 2017

72 Behavior for actual functions q 130 steps 200 steps [people.equars.com/~marco/poli/phd/node57.html] ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.72 Θεοχαρίδης, ΗΜΥ, 2017

73 Jigsaw puzzles Intuitive usage of Simulated Annealing Given a jigsaw puzzle such that one has to obtain the final shape using all pieces together. Starting with a random configuration, the human brain unconditionally chooses certain moves that tend to the solution. However, certain moves that may or may not lead to the solution are accepted or rejected with a certain small probability. The final shape is obtained as a result of a large number of iterations. ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.73 Θεοχαρίδης, ΗΜΥ, 2017

74 Performance This class of algorithms has been shown to outperform others in certain cases [Wegener, 2005]. Demonstrated its excellent results in the TimberWolf layout generation package [Sechen] Many other applications ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.74 Θεοχαρίδης, ΗΜΥ, 2017

75 Evolutionary Algorithms (1) Evolutionary Algorithms are based on the collective learning process within a population of individuals, each of which represents a search point in the space of potential solutions to a given problem. The population is arbitrarily initialized, and it evolves towards better and better regions of the search space by means of randomized processes of - selection (which is deterministic in some algorithms), - mutation, and - recombination (which is completely omitted in some algorithmic realizations). [Bäck, Schwefel, 1993] ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.75 Θεοχαρίδης, ΗΜΥ, 2017

76 Evolutionary Algorithms (2) The environment (given aim of the search) delivers a quality information (fitness value) of the search points, and the selection process favours those individuals of higher fitness to reproduce more often than worse individuals. The recombination mechanism allows the mixing of parental information while passing it to their descendants, and mutation introduces innovation into the population [Bäck, Schwefel, 1993] ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.76 Θεοχαρίδης, ΗΜΥ, 2017

77 Evolutionary Algorithms Principles of Evolution Selection Cross-over Mutation Thiele ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.77 Θεοχαρίδης, ΗΜΥ, 2017

78 ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.78 Θεοχαρίδης, ΗΜΥ, 2017

79 An Evolutionary Algorithm in Action max. y2 hypothetical trade-off front min. y1 Thiele ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.79 Θεοχαρίδης, ΗΜΥ, 2017

80 ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.80 Θεοχαρίδης, ΗΜΥ, 2017

81 A Generic Multiobjective EA population archive evaluate sample vary update truncate new population new archive Thiele ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.81 Θεοχαρίδης, ΗΜΥ, 2017

82 ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.82 Θεοχαρίδης, ΗΜΥ, 2017

83 Summary q Integer (linear) programming Integer programming is NP-complete Linear programming is faster Good starting point even if solutions are generated with different techniques q Simulated annealing Modeled after cooling of liquids Overcomes local minima q Evolutionary algorithms Maintain set of solutions Include selection, mutation and recombination ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.83 Θεοχαρίδης, ΗΜΥ, 2017

84 Classes of mapping algorithms considered in this course Classical scheduling algorithms Mostly for independent tasks & ignoring communication, mostly for mono- and homogeneous multiprocessors Dependent tasks as considered in architectural synthesis Initially designed in different context, but applicable Hardware/software partitioning Dependent tasks, heterogeneous systems, focus on resource assignment Design space exploration using genetic algorithms Heterogeneous systems, incl. communication modeling ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.84 Θεοχαρίδης, ΗΜΥ, 2017

85 Classification of Scheduling Problems Scheduling Independent Tasks Dependent Tasks EDD, EDF, LLF, RMS 1 Proc. Resource constrained Time constrained Unconstrained LDF LS FDS ASAP, ALAP ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.85 Θεοχαρίδης, ΗΜΥ, 2017

86 Dependent tasks The problem of deciding whether or not a schedule exists for a set of dependent tasks and a given deadline is NP-complete in general [Garey/Johnson]. q Strategies: 1. Add resources, so that scheduling becomes easier 2. Split problem into static and dynamic part so that only a minimum of decisions need to be taken at run-time. 3. Use scheduling algorithms from high-level synthesis ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.86 Θεοχαρίδης, ΗΜΥ, 2017

87 Taskgraph a b c d e f g h i j k l m q Assumption: execution time = 1 q for all tasks n z ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.87 Θεοχαρίδης, ΗΜΥ, 2017

88 As soon as possible (ASAP) scheduling q ASAP: All tasks are scheduled as early as possible q Loop over (integer) time steps: Compute the set of unscheduled tasks for which all predecessors have finished their computation Schedule these tasks to start at the current time step. ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.88 Θεοχαρίδης, ΗΜΥ, 2017

89 As soon as possible (ASAP) scheduling: Example s=0 a s=1 s=2 s=3 b c d e f g h i j k l m s=4 s=5 n z ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.89 Θεοχαρίδης, ΗΜΥ, 2017

90 As-late-as-possible (ALAP) scheduling q ALAP: All tasks are scheduled as late as possible Start at last time step*: Schedule tasks with no successors and tasks for which all successors have already been scheduled. * Generate a list, starting at its end ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.90 Θεοχαρίδης, ΗΜΥ, 2017

91 As-late-as-possible (ALAP) scheduling: Example s=0 a s=1 b c d e f g s=2 s=3 h i j k l m Start s=4 s=5 n z ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.91 Θεοχαρίδης, ΗΜΥ, 2017

92 (Resource constrained) List Scheduling q List scheduling: extension of ALAP/ASAP method q Preparation: Topological sort of task graph G=(V,E) Computation of priority of each task: Possible priorities u: - Number of successors - Longest path - Mobility = s (ALAP schedule)- s (ASAP schedule) Source: Teich: Dig. HW/SW Systeme ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.92 Θεοχαρίδης, ΗΜΥ, 2017

93 Mobility as a priority function s=0 a s=0 a s=1 b c d e f g s=1 b c d e f g s=2 h i j s=2 h i j s=3 k l m s=3 k l m s=4 s=5 n z s=4 s=5 n z Mobility is not very precise ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.93 Θεοχαρίδης, ΗΜΥ, 2017

94 Algorithm List(G(V,E), B, u){ i :=0; repeat { Compute set of candidate tasks A i ; Compute set of not terminated tasks G i ; Select S i Í A i of maximum priority r such that S i + G i B (*resource constraint*) may be repeated for different task/ processor classes foreach (v j Î S i ): s (v j ):=i; (*set start time*) i := i +1; } until (all nodes are scheduled); return (s); } Complexity: O( V ) ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.94 Θεοχαρίδης, ΗΜΥ, 2017

95 Example q Assuming B =2, unit execution time and u : path length q u(a)= u(b)=4 u(c)= u(f)=3 u(d)= u(g)= u(h)= u(j)=2 u(e)= u(i)= u(k)=1 " i : G i =0 s=0 s=1 s=2 s=3 a c d b f g h j a b f h j s=4 e i c g i k s=5 k d e Modified example based on J. Teich ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.95 Θεοχαρίδης, ΗΜΥ, 2017

96 (Time constrained) Force-directed scheduling q Goal: balanced utilization of resources q Based on spring model; q Originally proposed for high-level synthesis (HLT) ACM * [Pierre G. Paulin, J.P. Knight, Force-directed scheduling in automatic data path synthesis, Design Automation Conference (DAC), 1987, S ] ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.96 Θεοχαρίδης, ΗΜΥ, 2017

97 Phase 1: Generation of ASAP and ALAP Schedule s=0 a s=0 a s=1 b c d e f g s=1 b c d e f g s=2 h i j s=2 h i j s=3 k l m s=3 k l m s=4 s=5 n z s=4 s=5 n z ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.97 Θεοχαρίδης, ΗΜΥ, 2017

98 Next: computation of forces Direct forces push each task into the direction of lower values of D(i). Impact of direct forces on dependent tasks taken into account by indirect forces Balanced resource usage» smallest forces For our simple example and time constraint=6: result = ALAP schedule s=0 a 0 s=1 b c d e f g s=2 h i j s=3 k l m s=4 n s=5 z i F More precisely ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.98 Θεοχαρίδης, ΗΜΥ, 2017

99 1. Compute time frames R(j) 2. Compute probability P(j,i) of assignment j i R(j)={ASAP-control step ALAP-control step} if 0 otherwise ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.99 Θεοχαρίδης, ΗΜΥ, 2017

100 3. Compute distribution D(i) (# Operations in control step i) P(j,i) D(i) ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.100 Θεοχαρίδης, ΗΜΥ, 2017

101 4. Compute direct forces (1) DP i ( j,i ): D for force on task j in time step i, if j is mapped to time step i. The new probability for executing j in i is 1; the previous was P ( j, i). i The new probability for executing j in i ¹ i is 0; the previous was P (j, i). F if otherwise ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.101 Θεοχαρίδης, ΗΜΥ, 2017

102 4. Compute direct forces (2) SF(j, i) is the overall change of direct forces resulting from the mapping of j to time step i. if otherwise Example ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.102 Θεοχαρίδης, ΗΜΥ, 2017

103 4. Compute direct forces (3) q Direct force if task/operation 1 is mapped to time step 2 ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.103 Θεοχαρίδης, ΗΜΥ, 2017

104 5. Compute indirect forces (1) Mapping task 1 to time step 2 D implies mapping task 2 to time step 3 F Consider predecessor and successor forces: j Î predecessor of j j Î successor of j DP j, i (j,i ) is the D in the probability of mapping j to i resulting from the mapping of j to i ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.104 Θεοχαρίδης, ΗΜΥ, 2017

105 5. Compute indirect forces (2) j Î predecessor of j j Î successor of j Example: Computation of successor forces for task 1 in time step 2 ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.105 Θεοχαρίδης, ΗΜΥ, 2017

106 Overall forces q The total force is the sum of direct and indirect forces: In the example: The low value suggests mapping task 1 to time step 2 ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.106 Θεοχαρίδης, ΗΜΥ, 2017

107 Overall approach q procedure forcedirectedscheduling; begin end AsapScheduling; AlapScheduling; while not all tasks scheduled do begin select task T with smallest total force; schedule task T at time step minimizing forces; recompute forces; end; May be repeated for different task/ processor classes Not sufficient for today's complex, heterogeneous hardware platforms ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.107 Θεοχαρίδης, ΗΜΥ, 2017

108 Evaluation of HLS-Scheduling Focus on considering dependencies Mostly heuristics, few proofs on optimality Not using global knowledge about periods etc. Considering discrete time intervals Variable execution time available only as an extension Includes modeling of heterogeneous systems ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.108 Θεοχαρίδης, ΗΜΥ, 2017

109 Classes of mapping algorithms considered in this course Classical scheduling algorithms Mostly for independent tasks & ignoring communication, mostly for mono- and homogeneous multiprocessors Hardware/software partitioning Dependent tasks, heterogeneous systems, focus on resource assignment Dependent tasks as considered in architectural synthesis Initially designed in different context, but applicable Design space exploration using genetic algorithms Heterogeneous systems, incl. communication modeling ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.109 Θεοχαρίδης, ΗΜΥ, 2017

110 Periodic scheduling T1 T2 q For periodic scheduling, the best that we can do is to design an algorithm which will always find a schedule if one exists. F A scheduler is defined to be optimal iff it will find a schedule if one exists. ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.110 Θεοχαρίδης, ΗΜΥ, 2017

111 Periodic scheduling - Scheduling with no precedence constraints - q Let {T i } be a set of tasks. Let: p i be the period of task T i, c i be the execution time of T i, d i be the deadline interval, that is, the time between T i becoming available and the time until which T i has to finish execution. l i be the laxity or slack, defined as l i = d i - c i f i be the finishing time. p i i d i c i l i t ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.111 Θεοχαρίδης, ΗΜΥ, 2017

112 Average utilization q Average utilization: µ = n å i= 1 c i p i Necessary condition for schedulability (with m=number of processors): µ m ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.112 Θεοχαρίδης, ΗΜΥ, 2017

113 Independent tasks: Rate monotonic (RM) scheduling q Most well-known technique for scheduling independent periodic tasks [Liu, 1973]. q Assumptions: All tasks that have hard deadlines are periodic. All tasks are independent. d i =p i, for all tasks. c i is constant and is known for all tasks. The time required for context switching is negligible. For a single processor and for n tasks, the following equation holds for the average utilization µ: n µ = å i= 1 ci p i n(2 1/ n -1) ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.113 Θεοχαρίδης, ΗΜΥ, 2017

114 Rate monotonic (RM) scheduling - The policy - RM policy: The priority of a task is a monotonically decreasing function of its period. At any time, a highest priority task among all those that are ready for execution is allocated. Theorem: If all RM assumptions are met, schedulability is guaranteed. ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.114 Θεοχαρίδης, ΗΜΥ, 2017

115 Maximum utilization for guaranteed schedulability Maximum utilization as a function of the number of tasks: µ = n å i= 1 lim( n(2 n ci p i 1/ n n(2 1/ n -1) -1) = ln(2) ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.115 Θεοχαρίδης, ΗΜΥ, 2017

116 Example of RM-generated schedule t T1 preempts T2 and T3. T2 and T3 do not preempt each other. ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.116 Θεοχαρίδης, ΗΜΥ, 2017

117 Case of failing RM scheduling Task 1: period 5, execution time 2 Task 2: period 7, execution time 4 µ=2/5+4/7=34/35» (2 1/2-1)» t Missed deadline Missing computations scheduled in the next period levirts animation ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.117 Θεοχαρίδης, ΗΜΥ, 2017

118 Intuitively: Why does RM fail? Switching to T 1 too early, despite early deadline for T 2 T 1 tt T 2 No problem if p 2 = m p 1, mîn : should be completed T 1 fits t T 2 ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.118 Θεοχαρίδης, ΗΜΥ, 2017

119 Critical instants q Definition: A critical instant of a task is the time at which the release of a task will produce the largest response time. Lemma: For any task, the critical instant occurs if that task is simultaneously released with all higher priority tasks. Proof: Let T={T 1,,T n }: periodic tasks with "i: p i p i +1. Source: G. Buttazzo, Hard Real-time Computing Systems, Kluwer, 2002 ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.119 Θεοχαρίδης, ΗΜΥ, 2017

120 Critical instances (1) q Response time of T n is delayed by tasks T i of higher priority: T n T i c n +2c i t Delay may increase if T i starts earlier T n T i c n +3c i t Maximum delay achieved if T n and T i start simultaneously. ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.120 Θεοχαρίδης, ΗΜΥ, 2017

121 Critical instants (2) q F Repeating the argument for all i = 1, n-1: The worst case response time of a task occurs when it is released simultaneously with all higher-priority tasks. q.e.d. F F F Schedulability is checked at the critical instants. If all tasks of a task set are schedulable at their critical instants, they are schedulable at all release times. Observation helps designing examples ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.121 Θεοχαρίδης, ΗΜΥ, 2017

122 The case "i: p i+1 = m i p i q Lemma*: If each task period is a multiple of the period of the next higher priority task, then schedulability is also guaranteed if µ 1. Proof: Assume schedule of T i is given. Incorporate T i+1 : T i+1 fills idle times of T i ; T i+1 completes in time, if µ 1. T i T i+1 t F T i+1 Used as the higher priority task at the next iteration. * wrong in the book of 2007 ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.122 Θεοχαρίδης, ΗΜΥ, 2017

123 Proof of the RM theorem q Let T={T 1, T 2 } with p 1 < p 2. q Assume RM is not used à prio(t 2 ) is highest: More in-depth: p 1 T 1 c 1 T 2 c 2 t Schedule is feasible if c 1 +c 2 p 1 (1) Define F= ëp 2 /p 1 û: # of periods of T 1 fully contained in T 2 ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.123 Θεοχαρίδης, ΗΜΥ, 2017

124 Case 1: c 1 p 2 Fp 1 q Assume RM is used à prio(t 1 ) is highest: Case 1*: c 1 p 2 F p 1 (c 1 small enough to be finished before 2nd instance of T 2 ) T 1 T 2 t Fp 1 p 2 Schedulable if (F +1) c 1 + c 2 p 2 (2) * Typos in [Buttazzo 2002]: < and mixed up] ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.124 Θεοχαρίδης, ΗΜΥ, 2017

125 Proof of the RM theorem (3) q Not RM: schedule is feasible if c 1 +c 2 p 1 (1) q RM: schedulable if (F+1) c 1 + c 2 p 2 (2) q From (1): Fc 1 +Fc 2 Fp 1 q Since F ³ 1: Fc 1 +c 2 Fc 1 +Fc 2 Fp 1 q Adding c 1 : (F+1)c 1 +c 2 Fp 1 +c 1 q Since c 1 p 2 Fp 1 : (F+1)c 1 +c 2 Fp 1 +c 1 p 2 q Hence: if (1) holds, (2) holds as well q FFor case 1: Given tasks T 1 and T 2 with p 1 < p 2, then if the schedule is feasible by an arbitrary (but fixed) priority assignment, it is also feasible by RM. ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.125 Θεοχαρίδης, ΗΜΥ, 2017

126 Case 2: c 1 > p 2 Fp 1 q Case 2: c 1 > p 2 Fp 1 (c 1 large enough not to finish before 2 nd instance of T 2 ) T 1 T 2 t Fp 1 p 2 Schedulable if F c 1 + c 2 F p 1 (3) c 1 +c 2 p 1 (1) Multiplying (1) by F yields F c 1 + F c 2 F p 1 Since F ³ 1: F c 1 + c 2 F c 1 + Fc 2 F p 1 F Same statement as for case 1. ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.126 Θεοχαρίδης, ΗΜΥ, 2017

127 Calculation of the least upper utilization bound q Let T={T 1, T 2 } with p 1 < p 2. q Proof procedure: compute least upper bound U lup as follows Assign priorities according to RM Compute upper bound U up by setting computation times to fully utilize processor Minimize upper bound with respect to other task parameters q As before: F= ëp 2 /p 1 û q c 2 adjusted to fully utilize processor. ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.127 Θεοχαρίδης, ΗΜΥ, 2017

128 ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.128 Θεοχαρίδης, ΗΜΥ, 2017 Case 1: c 1 p 2 Fp 1 q Largest possible value of c 2 is c 2 = p 2 c 1 (F+1) q Corresponding upper bound is T 1 T 2 t p 2 Fp 1 þ ý ü + - î í ì + = = + + = + = 1) ( F p p p c p F c p c p F c p p c p c p c U ub ) ( ) ( { } is <0 à U ub monotonically decreasing in c 1 Minimum occurs for c 1 = p 2 Fp 1

129 ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.129 Θεοχαρίδης, ΗΜΥ, 2017 Case 2: c 1 ³ p 2 Fp 1 q Largest possible value of c 2 is c 2 = (p 1 -c 1 )F q Corresponding upper bound is: T 1 T 2 t p 2 Fp 1 þ ý ü - î í ì + = - + = + = + = F p p p c F p p F p c p c F p p p F c p p c p c p c U ub ) ( { } is ³ 0 à U ub monotonically increasing in c 1 (independent of c 1 if {}=0) Minimum occurs for c 1 = p 2 Fp 1, as before.

130 ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.130 Θεοχαρίδης, ΗΜΥ, 2017 Utilization as a function of G=p 2 /p 1 -F q For minimum value of c 1 : ( ) ïî ï í ì þ ý ü ø ö ç ç è æ - ø ö ç ç è æ - + = ø ö ç ç è æ = ø ö ç ç è æ - + = F p p F p p F p p F p p p F p p F p p F p p p c F p p U ub Þ - = Let ; 1 2 F p p G ( ) ( ) ( ) ( ) ( ) ( ) ( ) G F G G G F G G G F G F G F F F p p G F p p G F G F p p U ub = = + + = = + = + = 1 1 ) ( / / Since 0 G< 1: G(1-G) ³ 0 à U ub increasing in F à Minimum of U ub for min(f): F=1 à G G U ub + =

131 Proving the RM theorem for n=2 end 2 1+ G Uub = 1+ G Using derivative to find minimum of U du dg ub = 2G(1 + G) - (1 + G 2 (1 + G) 2 ) = ub 2 G + 2G -1 2 (1 + G) : = 0 G 1 = -1-2; Considering only G G 2 2 = -1+, since 0 2; G < 1: U lub = 1+ ( 1+ ( 2-1) 2 2-1) = = 2( 2-1) = q This proves the RM theorem for the special case of n=2 ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.131 Θεοχαρίδης, ΗΜΥ, 2017

132 Properties of RM scheduling RM scheduling is based on static priorities. This allows RM scheduling to be used in standard OS, such as Windows NT. No idle capacity is needed if "i: p i+1 =F p i : i.e. if the period of each task is a multiple of the period of the next higher priority task, schedulability is then also guaranteed if µ 1. A huge number of variations of RM scheduling exists. In the context of RM scheduling, many formal proofs exist. ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.132 Θεοχαρίδης, ΗΜΥ, 2017

133 EDF q EDF can also be applied to periodic scheduling. q EDF optimal for every period F F Optimal for periodic scheduling EDF must be able to schedule the example in which RMS failed. ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.133 Θεοχαρίδης, ΗΜΥ, 2017

134 Comparison EDF/RMS q RMS : EDF: T2 not preempted, due to its earlier deadline. EDF-animation ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.134 Θεοχαρίδης, ΗΜΥ, 2017

135 EDF: Properties q EDF requires dynamic priorities q F EDF cannot be used with a standard operating system just providing static priorities. q However, a recent paper (by Margull and Slomka) at DATE 2008 demonstrates how an OS with static priorities can be extended with a plug-in providing EDF scheduling (key idea: delay tasks becoming ready if they shouldn t be executed under EDF scheduling. ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.135 Θεοχαρίδης, ΗΜΥ, 2017

136 Comparison RMS/EDF RMS EDF Priorities Static Dynamic Works with std. OS with fixed priorities Uses full computational power of processor Possible to exploit full computational power of processor without provisioning for slack Yes No, just up till µ=n(2 1/n -1) No No* Yes Yes * Unless the plug-in by Slomka et al. is added. ΗΜΥ653 Δ7-8 Mapping Applications, Optimization and Scheduling.136 Θεοχαρίδης, ΗΜΥ, 2017