On the Tractability of Digraph-Based Task Models Martin Stigge - PowerPoint PPT Presentation

On the Tractability of Digraph-Based Task Models Martin Stigge Uppsala University, Sweden Joint work with Pontus Ekberg, Nan Guan and Wang Yi Martin Stigge Tractability of Digraph-Based Models 1

Analysis of Abstract Models Model hard real-time systems ◮ Analysis: Guarantee deadlines ◮ Expressiveness of models? ◮ Efficiency of analysis? � 2 � 2 � Question C 9 � 2 � 2 � 2 How expressive can a model be? 2 B 3 5 � 1 � 2 � E � 4 � 5 � � 6 2 ... with a tractable feasibility test? 7 2 D � 1 � 2 � Martin Stigge Tractability of Digraph-Based Models 2

Context: Real-Time Task Models System is composed of tasks , releasing jobs J = ( r , e , d ) ◮ Release time r ◮ Worst-case execution time e ◮ Deadline d Scheduling window e t r d Feasibility: Can we schedule s.t. all jobs meet their deadlines? In this work: ◮ Preemptive schedules ◮ On uniprocessors ◮ Independent jobs � In this setting: EDF is optimal. Feasible ⇔ Schedulable with EDF Martin Stigge Tractability of Digraph-Based Models 3

Context: Real-Time Task Models System is composed of tasks , releasing jobs J = ( r , e , d ) ◮ Release time r ◮ Worst-case execution time e ◮ Deadline d Scheduling window e t r d Feasibility: Can we schedule s.t. all jobs meet their deadlines? In this work: ◮ Preemptive schedules ◮ On uniprocessors ◮ Independent jobs � In this setting: EDF is optimal. Feasible ⇔ Schedulable with EDF Martin Stigge Tractability of Digraph-Based Models 3

The Liu and Layland (L&L) Task Model (Liu and Layland, 1973) Tasks are periodic ◮ Job WCET e ◮ Minimum inter-release delay p (implicit deadline) ... ➋ ❀ e e ( e , p ) t p p Advantages: Well-known model; efficient schedulability test Disadvantage: Very limited expressiveness Martin Stigge Tractability of Digraph-Based Models 4

Hierarchy of Models high difficult Feasibility test Expressiveness branching, loops, . . . arbitrary graph DRT [S. et al., 2011] branching DAG RRT [Baruah, 2003] cycle graph different job types GMF [Mok et al., 1999] two integers periodic L&L efficient low [Liu et al., 1973] Martin Stigge Tractability of Digraph-Based Models 5

Hierarchy of Models high difficult Strongly (co)NP-hard Pseudo-Polynomial Feasibility test Expressiveness branching, loops, . . . arbitrary graph DRT [S. et al., 2011] branching DAG RRT [Baruah, 2003] cycle graph different job types GMF [Mok et al., 1999] two integers periodic L&L efficient low [Liu et al., 1973] Martin Stigge Tractability of Digraph-Based Models 5

Hierarchy of Models high difficult Question: How close to this border? Strongly (co)NP-hard Pseudo-Polynomial Feasibility test Expressiveness branching, loops, . . . arbitrary graph DRT [S. et al., 2011] branching DAG RRT [Baruah, 2003] cycle graph different job types GMF [Mok et al., 1999] two integers periodic L&L efficient low [Liu et al., 1973] Martin Stigge Tractability of Digraph-Based Models 5

Hierarchy of Models high difficult Question: How close to this border? EDRT Strongly (co)NP-hard k -EDRT Pseudo-Polynomial Feasibility test Expressiveness branching, loops, . . . arbitrary graph DRT [S. et al., 2011] branching DAG RRT [Baruah, 2003] cycle graph different job types GMF [Mok et al., 1999] two integers periodic L&L efficient low [Liu et al., 1973] Martin Stigge Tractability of Digraph-Based Models 5

The Digraph Real-Time (DRT) Task Model Branching, cycles (loops), ... Directed graph for each task ◮ Vertices J : jobs to be released (with WCET and deadline) ◮ Edges ( J i , J j ): minimum inter-release delays p ( J i , J j ) J 3 11 15 20 J 2 J 4 10 20 J 1 10 20 J 5 Theorem (S. et al., RTAS 2011) For DRT task systems τ with a utilization bounded by any c < 1 , feasibility can be decided in pseudo-polynomial time. Martin Stigge Tractability of Digraph-Based Models 6

The Digraph Real-Time (DRT) Task Model Branching, cycles (loops), ... Directed graph for each task ◮ Vertices J : jobs to be released (with WCET and deadline) ◮ Edges ( J i , J j ): minimum inter-release delays p ( J i , J j ) J 3 11 15 20 J 2 J 4 10 20 J 1 10 20 J 5 Theorem (S. et al., RTAS 2011) For DRT task systems τ with a utilization bounded by any c < 1 , feasibility can be decided in pseudo-polynomial time. Martin Stigge Tractability of Digraph-Based Models 6

Hierarchy of Models high difficult EDRT Strongly (co)NP-hard k -EDRT Pseudo-Polynomial Feasibility test Expressiveness branching, loops, . . . arbitrary graph DRT [S. et al., 2011] branching DAG RRT [Baruah, 2003] cycle graph different job types GMF [Mok et al., 1999] two integers periodic L&L efficient low [Liu et al., 1973] Martin Stigge Tractability of Digraph-Based Models 7

Hierarchy of Models high difficult EDRT Strongly (co)NP-hard k -EDRT Pseudo-Polynomial Feasibility test Expressiveness branching, loops, . . . arbitrary graph DRT [S. et al., 2011] branching DAG RRT [Baruah, 2003] cycle graph different job types GMF [Mok et al., 1999] two integers periodic L&L efficient low [Liu et al., 1973] Martin Stigge Tractability of Digraph-Based Models 7

Extending DRT: Global Timing Constraints – This Work Delays in DRT only for adjacent jobs What about adding global delay constraints ? J 3 9 2 5 J 2 3 5 2 J 5 J 1 J 2 J 3 J 1 6 2 2 2 7 2 J 4 Motivation: ◮ Mode sub-structures ◮ Burstiness ◮ ... Martin Stigge Tractability of Digraph-Based Models 8

Extended DRT (EDRT) – This Work Extends DRT with global delay constraints Directed graph for each task ◮ Vertices J : jobs to be released (with WCET and deadline) ◮ Edges ( J i , J j ): minimum inter-release delays p ( J i , J j ) ◮ k global constraints ( J i , J j , γ ) 2 J 3 9 J 2 5 3 2 J 1 6 J 5 2 7 2 J 4 Theorem (Our technical result) For k-EDRT task systems with bounded utilization, feasibility is decidable in pseudo-polynomial time if k is constant, and 1 strongly coNP-hard in general. 2 Martin Stigge Tractability of Digraph-Based Models 9

Extended DRT (EDRT) – This Work Extends DRT with global delay constraints Directed graph for each task ◮ Vertices J : jobs to be released (with WCET and deadline) ◮ Edges ( J i , J j ): minimum inter-release delays p ( J i , J j ) ◮ k global constraints ( J i , J j , γ ) 2 J 3 9 J 2 5 3 2 J 1 6 J 5 2 7 2 J 4 Theorem (Our technical result) For k-EDRT task systems with bounded utilization, feasibility is decidable in pseudo-polynomial time if k is constant, and 1 strongly coNP-hard in general. 2 Martin Stigge Tractability of Digraph-Based Models 9

Hierarchy of Models high difficult EDRT Strongly (co)NP-hard k -EDRT Pseudo-Polynomial Feasibility test Expressiveness branching, loops, . . . arbitrary graph DRT [S. et al., 2011] branching DAG RRT [Baruah, 2003] cycle graph different job types GMF [Mok et al., 1999] two integers periodic L&L efficient low [Liu et al., 1973] Martin Stigge Tractability of Digraph-Based Models 10

Fahrplan Martin Stigge Tractability of Digraph-Based Models 11

Fahrplan Martin Stigge Tractability of Digraph-Based Models 11

Hardness for EDRT Number of constraints now not constant Reduction from Hamiltonian Path Problem (strongly NP-hard) T 1 J ′ � 1 , 6 � 1 T 2 12 � 1 , 1 � J 3 1 � 1 , 1 � 1 J 4 � 1 , 1 � 1 ❀ 12 J 1 1 � 1 , 1 � J 5 12 1 1 1 1 12 � 1 , 1 � J 6 � 1 , 1 � J 2 1 12 12 Martin Stigge Tractability of Digraph-Based Models 12

Hardness for EDRT Number of constraints now not constant Reduction from Hamiltonian Path Problem (strongly NP-hard) T 1 J ′ � 1 , 6 � 1 T 2 12 � 1 , 1 � J 3 1 � 1 , 1 � 1 J 4 � 1 , 1 � 1 ❀ 12 J 1 1 � 1 , 1 � J 5 12 1 1 1 1 12 � 1 , 1 � J 6 � 1 , 1 � J 2 1 12 12 Martin Stigge Tractability of Digraph-Based Models 12

Hardness for EDRT Number of constraints now not constant Reduction from Hamiltonian Path Problem (strongly NP-hard) T 1 J ′ � 1 , 6 � 1 T 2 12 � 1 , 1 � J 3 1 � 1 , 1 � 1 J 4 � 1 , 1 � 1 ❀ 12 J 1 1 � 1 , 1 � J 5 12 1 1 1 1 12 � 1 , 1 � J 6 � 1 , 1 � J 2 1 12 12 Martin Stigge Tractability of Digraph-Based Models 12

Analysing k -EDRT: The Problem Detour: Analyzing DRT ◮ Using demand bound functions ◮ Compute exec. demand and deadline for all paths in G ( T ) � 2 , 2 � 2 J 3 J 2 5 3 2 J 1 � 1 , 2 � J 5 2 7 2 J 4 � 1 , 2 � Problem: Constraints ignored during path exploration ◮ � 2 , 4 � is lacking constraint information ◮ ... about the active constraint Martin Stigge Tractability of Digraph-Based Models 13

Analysing k -EDRT: The Problem Detour: Analyzing DRT ◮ Using demand bound functions ◮ Compute exec. demand and deadline for all paths in G ( T ) � 2 , 2 � 2 J 3 Demand pair: � 2 , 4 � J 2 5 3 2 J 1 � 1 , 2 � J 5 2 7 2 J 4 � 1 , 2 � Problem: Constraints ignored during path exploration ◮ � 2 , 4 � is lacking constraint information ◮ ... about the active constraint Martin Stigge Tractability of Digraph-Based Models 13