Introduction The Models ASAP/ALAP ILP Formulation Experimental Evaluation Case Study Conclusion Bibliography Optimal Scheduling of Precedence-constrained Task Graphs on Heterogeneous Distributed Systems with Shared Buses Sanjit Kumar Roy 1 , Sayani Sinha 2 , Kankana Maji 2 , Rajesh Devaraj 1 , and Arnab Sarkar 1 IEEE ISORC 2019 May 9, 2019 1 Indian Institute of Technology Guwahati 2 Jadavpur University
Introduction The Models ASAP/ALAP ILP Formulation Experimental Evaluation Case Study Conclusion Bibliography Table of Contents Introduction 1 The Models 2 ASAP/ALAP 3 ILP Formulation 4 ILP1 ILP2 Experimental Evaluation 5 Experiment-1 Experiment-2 Experiment-3 Case Study 6 Conclusion 7 Bibliography 8
Introduction The Models ASAP/ALAP ILP Formulation Experimental Evaluation Case Study Conclusion Bibliography Introduction Applications in many time-critical cyber-physical systems are often represented as Precedence-constrained Task Graphs (PTGs) There is an increasing trend towards their implementation on distributed heterogeneous platforms – consisting of heterogeneous processing elements – shared buses (CAN, LIN, FlexRay etc.) [1] On a distributed platform consisting of heterogeneous processing and communication resources, – execution of a task may require different amounts of time on different processing elements. – transmission of a message may require different amounts of time on different communication resources
Introduction The Models ASAP/ALAP ILP Formulation Experimental Evaluation Case Study Conclusion Bibliography Introduction Contd. Given a PTG representing a real-time application and a heterogeneous platform, successful execution/transmission of the task/message nodes while satisfying all timing, precedence and resource related specifications, is ultimately a scheduling problem Scheduler design schemes for PTGs can be broadly classified as static (offline) and dynamic (online) [2] In safety-critical systems such as automotive/avionic systems [3], it is often advisable that all timing requirements be guaranteed off-line, before putting the system in operation [4] Hence, static off-line scheduling schemes are preferred in such systems to provide a high degree of timing predictability [5]
Introduction The Models ASAP/ALAP ILP Formulation Experimental Evaluation Case Study Conclusion Bibliography Introduction Contd. Most existing real-time static scheduling approaches for PTGs are list scheduling based heuristic schemes [2, 6, 7] A majority of them attempt to minimize the overall schedule length ( makespan minimization) Such an objective allows maximization of the spare computation bandwidth in the system, which may be used to perform other useful activities Many of them assume that the underlying execution platform consists of a fully connected system of processing elements There exists a significant class of cyber-physical systems with bus based shared communication links among processors
Introduction The Models ASAP/ALAP ILP Formulation Experimental Evaluation Case Study Conclusion Bibliography Introduction: Heuristic vs Optimal Heuristic schedules – typically based on the satisfaction of a set of sufficiency conditions – cannot take into consideration all necessary schedulability requirements – schedules are sub-optimal in nature Optimal solutions – can make a fundamental difference in resource-constrained time-critical systems with respect to performance, reliability and other non-functional metrics like cost, power, space etc – Optimal schedules can act as benchmarks allowing accurate comparison and evaluation of heuristic solutions [8]
Introduction The Models ASAP/ALAP ILP Formulation Experimental Evaluation Case Study Conclusion Bibliography We design an Integer Linear Programming (ILP) based static optimal real-time scheduling strategy for PTGs executing on a distributed platform consisting of heterogeneous processing nodes and inter-connected through a set of heterogeneous shared buses
Introduction The Models ASAP/ALAP ILP Formulation Experimental Evaluation Case Study Conclusion Bibliography Platform Model R 1 = Processing R 2 = Processing R p = Processing Element P 1 Element P 2 Element P p R p+1 = Bus B 1 R p+2 = Bus B 2 R p+b = Bus B b Figure: Platform Model A set of resources { R 1 , R 2 , . . . , R p + b } among which, { R 1 , R 2 , . . . , R p } denote a set P = { P 1 , P 2 , . . . , P p } of p heterogeneous processing elements { R p + 1 , R p + 2 , . . . , R p + b } denote a set B = { B 1 , B 2 , . . . , B b } of b heterogeneous shared buses Each processing node P i is connected to all b buses
Introduction The Models ASAP/ALAP ILP Formulation Experimental Evaluation Case Study Conclusion Bibliography Computation Model R 1 = Processing R 2 = Processing V 1 = T 1 Node P 1 Node P 2 V 7 = M 1 V 8 = M 2 V 9 = M 3 R 3 = Bus B 1 R 4 = Bus B 2 V 2 = T 2 V 3 = T 3 V 4 = T 4 (b) D ID B 1 B 2 V 10 = M 4 V 11 = M 5 V 12 = M 6 ID P 1 P 2 M 1 8 4 M 2 T 1 12 7 4 5 V 5 = T 5 T 2 6 11 M 3 3 3 M 4 T 3 6 9 8 5 V 13 = M 7 M 5 4 3 T 4 14 10 M 6 12 14 9 7 T 5 V 6 = T 6 M 7 T 6 6 15 7 10 (a) (c) (d) Figure: (a) PTG G , (b) Platform Model ρ , (c) Computation-time Matrix ( CT ) and (d) Communication-time Matrix ( CM ).
Introduction The Models ASAP/ALAP ILP Formulation Experimental Evaluation Case Study Conclusion Bibliography Computation Model Contd. A Precedence-constrained Task Graph (PTG) G is described by a quadruple G = ( V , E , CT , CM ) where, V = { V 1 , V 2 , . . . , V n + m } represents a set of nodes { V 1 , V 2 , . . . , V n } represent a set T = { T 1 , T 2 , . . . , T n } of n task nodes { V n + 1 , V n + 2 , . . . , V n + m } denote a set M = { M 1 , M 2 , . . . , M m } of m message nodes E ⊆ V × V is a set of edges that describe the precedence-constraints among nodes in V . CT is a n × p computation-time matrix CM is a m × b communication-time matrix
Introduction The Models ASAP/ALAP ILP Formulation Experimental Evaluation Case Study Conclusion Bibliography Assumptions Single source node T 1 Single sink node T n T 1 Both source ( T 1 ) and sink ( T n ) nodes are tasks. Each task node T i is preceded/succeeded by one M 1 M 2 or more message nodes. T 2 T 3 Each message node M k is preceded/succeeded by M 3 M 4 a single task node. The communication time for M k is negligible if T 4 both preceding and succeeding task nodes are mapped to same processing element.
Introduction The Models ASAP/ALAP ILP Formulation Experimental Evaluation Case Study Conclusion Bibliography Problem Formulation Given a PTG G = ( V , E , CT , CM ) with end-to-end deadline D , p processing elements and b buses, find: A task node assignment V i �→ R j ; 1 ≤ i ≤ n and 1 ≤ j ≤ p A message node assignment V i �→ R j ; n + 1 ≤ i ≤ n + m and p + 1 ≤ j ≤ p + b – If both the preceding and succeeding task nodes of message node M i are mapped to the same processing element then, V i → ∅ A start time for each task node and message node, such that – length of the total schedule is minimized and – meets the deadline D
Introduction The Models ASAP/ALAP ILP Formulation Experimental Evaluation Case Study Conclusion Bibliography Earliest/Latest Start Times for PTG Nodes Let, t s i and t l i be the ASAP and ALAP time of node V i , respectively T 1 M 1 M 2 T 2 T 3 M 3 M 4 ASAP time computation of task nodes: T 4 – Ignore message nodes in the PTG – Set ASAP time of the source task node, t s 1 = 1 Figure: PTG with – Compute ASAP times of the remaining task message nodes nodes recursively (downward) as follows: T 1 t s T j ∈ pred ( T i ) ( t s i = max j + min r ∈ [ 1 , p ] CT jr ) T 2 T 3 where, pred ( T i ) is the set of immediate predecessors of task node T i T 4 Figure: PTG without message nodes
Introduction The Models ASAP/ALAP ILP Formulation Experimental Evaluation Case Study Conclusion Bibliography Earliest/Latest Start Times for PTG Nodes T 1 M 1 M 2 ALAP time computation of task nodes: T 2 T 3 – Ignore message nodes in the PTG M 3 M 4 – Set ALAP time for the sink task node as, T 4 t l n = D − min r ∈ [ 1 , p ] CT nr Figure: PTG with message nodes – Compute ALAP times of the remaining task T 1 nodes recursively (upward) as follows: t l T j ∈ succ ( T i ) ( t l i = min j − min r ∈ [ 1 , p ] CT ir ) T 2 T 3 where, succ ( T i ) is the set of immediate T 4 successors of task node T i Figure: PTG without message nodes
Introduction The Models ASAP/ALAP ILP Formulation Experimental Evaluation Case Study Conclusion Bibliography Earliest/Latest Start Times for PTG Nodes ASAP/ALAP computation procedure for message nodes: T 1 – ASAP time of a message node M k is, M 1 M 2 t s n + k = t s i + min r ∈ [ 1 , p ] CT ir T 2 T 3 M 3 M 4 where, T i is the predecessor task node of M k – ALAP time of a message node M k is, T 4 Figure: PTG with t l n + k = t l j − min r ∈ [ 1 , b ] CM kr message nodes where, T j is the successor task node of M k
Introduction The Models ASAP/ALAP ILP Formulation Experimental Evaluation Case Study Conclusion Bibliography ILP Formulation: ILP1 We define binary decision variable, 1 if node i starts its execution/transmission on r th resource at time step t X irt = 0 Otherwise where, i = 1 , 2 , . . . , n + m ; r = 1 , 2 , . . . , p + b ; t = 1 , 2 , . . . , D
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