IntroductionFormal models

Scheduling problemPerformance evaluation

Scheduling with timed automata

for temporal properties veri�cation via

model-checking

Antonio Filieri1

1Politecnico di Milano and University of Illinois at Chicago

720463 antonio.�lieri@mail.polimi.it

Tutors: Marco D. Santambrogio, Paola Spoletini

HPPS PrjDAY - Jun 19, Milan

Antonio Filieri Scheduling with Timed Automata - 1/24

IntroductionFormal models

Scheduling problemPerformance evaluation

Outline

1 IntroductionWhy this work?What are we trying to do?

2 Formal modelsHow to describe a process?What is a Timed Automata?

3 Scheduling problemWhat do we mean with scheduling?How can we use Timed Automata?How can this stu� schedule?

4 Performance evaluationIs the method e�ective?

Antonio Filieri Scheduling with Timed Automata - 2/24

IntroductionFormal models

Scheduling problemPerformance evaluation

Problem statement and motivationsObjectives

Outline

1 IntroductionWhy this work?What are we trying to do?

2 Formal modelsHow to describe a process?What is a Timed Automata?

3 Scheduling problemWhat do we mean with scheduling?How can we use Timed Automata?How can this stu� schedule?

4 Performance evaluationIs the method e�ective?

Antonio Filieri Scheduling with Timed Automata - 3/24

IntroductionFormal models

Scheduling problemPerformance evaluation

Problem statement and motivationsObjectives

Scheduling problem

Several tasks with dependencies one to each other, have to beexecuted on a limited, typically not su�cient amount ofresources.

Common approach: existence of certain known results drivesthe choice of problem formulation.

Optimal very expensive schedules Xor Sub-optimal faster ones.

Lot of ad-hoc solutions in each �eld.

Antonio Filieri Scheduling with Timed Automata - 4/24

IntroductionFormal models

Scheduling problemPerformance evaluation

Problem statement and motivationsObjectives

Outline

1 IntroductionWhy this work?What are we trying to do?

2 Formal modelsHow to describe a process?What is a Timed Automata?

3 Scheduling problemWhat do we mean with scheduling?How can we use Timed Automata?How can this stu� schedule?

4 Performance evaluationIs the method e�ective?

Antonio Filieri Scheduling with Timed Automata - 5/24

IntroductionFormal models

Scheduling problemPerformance evaluation

Problem statement and motivationsObjectives

Objectives of the project

1 Analyze scheduling algorithms based on Timed Automata interms of expressiveness and reusability of the model,complexity vs solution quality trade-o�, performance in simpleFPGA scheduling problem.

2 Complexity considerations, possible improvements and main

drawbacks.

Antonio Filieri Scheduling with Timed Automata - 6/24

IntroductionFormal models

Scheduling problemPerformance evaluation

Process ModelTimed automata

Outline

1 IntroductionWhy this work?What are we trying to do?

2 Formal modelsHow to describe a process?What is a Timed Automata?

3 Scheduling problemWhat do we mean with scheduling?How can we use Timed Automata?How can this stu� schedule?

4 Performance evaluationIs the method e�ective?

Antonio Filieri Scheduling with Timed Automata - 7/24

IntroductionFormal models

Scheduling problemPerformance evaluation

Process ModelTimed automata

Process de�nition: static viewpoint

A process is de�ned as a set of tasks.

Basic description by extended DAGs (G =< V ,E ,W ,R >)based on DFGs.

Simplifying assumptions:

A task has associated a functional unit that can execute it.

A task cannot be preempted.

For each task con�guration and execution time are known apriori.

Antonio Filieri Scheduling with Timed Automata - 8/24

IntroductionFormal models

Scheduling problemPerformance evaluation

Process ModelTimed automata

Outline

1 IntroductionWhy this work?What are we trying to do?

2 Formal modelsHow to describe a process?What is a Timed Automata?

3 Scheduling problemWhat do we mean with scheduling?How can we use Timed Automata?How can this stu� schedule?

4 Performance evaluationIs the method e�ective?

Antonio Filieri Scheduling with Timed Automata - 9/24

IntroductionFormal models

Scheduling problemPerformance evaluation

Process ModelTimed automata

Timed Automata, UPPAAL extension

Timed automata concepts:

It is a �nite-state automaton with a �nite collection ofreal-valued clocks.Boolean conditions as well as clock evaluation.Keep in a state may require a staying condition holds.Transitions taken if �re condition satis�ed.Transition �ring can reset clocks.

Some UPPAAL enhancement:

Guard condition on transition must hold in order to �re.Synchronization channels point2point and broadcast.Finite domain integer variables.

Antonio Filieri Scheduling with Timed Automata - 10/24

IntroductionFormal models

Scheduling problemPerformance evaluation

FormalizationCase of studyVeri�cation

Outline

1 IntroductionWhy this work?What are we trying to do?

2 Formal modelsHow to describe a process?What is a Timed Automata?

3 Scheduling problemWhat do we mean with scheduling?How can we use Timed Automata?How can this stu� schedule?

4 Performance evaluationIs the method e�ective?

Antonio Filieri Scheduling with Timed Automata - 11/24

IntroductionFormal models

Scheduling problemPerformance evaluation

FormalizationCase of studyVeri�cation

Scheduling formalization

A schedule for a problem J = (P,≺,M,µ,d) is determined bythe function st : P→R+ indicating the start time of each task.

Deterministic setting =⇒ en(p) = st(p)+d(p).

Feasible schedule if:1 Precedence: for every p,p′ ∈ P,p ≺ p′⇒ en(p)≤ st(p′).2 Mutual exclusion: for every two tasks p,p′ such that

µ(p) = µ(p′), [st(p),en(p)]∩ [st(p′),en(p′)] = /0.

The duration of a schedule is max{en(p) : p ∈ P}.An optimal schedule is one whose length is minimal.

Antonio Filieri Scheduling with Timed Automata - 12/24

IntroductionFormal models

Scheduling problemPerformance evaluation

FormalizationCase of studyVeri�cation

Outline

1 IntroductionWhy this work?What are we trying to do?

2 Formal modelsHow to describe a process?What is a Timed Automata?

3 Scheduling problemWhat do we mean with scheduling?How can we use Timed Automata?How can this stu� schedule?

4 Performance evaluationIs the method e�ective?

Antonio Filieri Scheduling with Timed Automata - 13/24

IntroductionFormal models

Scheduling problemPerformance evaluation

FormalizationCase of studyVeri�cation

1D FPGA scheduling, con�icts and assumptions

Con�guration prefetch: allow FU to be placed sometimesbefore execution.

FPGA uniformly recon�gurable: no reserved areas.

Partially Runtime-Recon�gurable (PRTR) FPGA (e.g. Virtexseries).

No speci�c policy with respect to placement: anywhere, free=⇒ usable.

Time to recon�gure proportional to required area.

Antonio Filieri Scheduling with Timed Automata - 14/24

IntroductionFormal models

Scheduling problemPerformance evaluation

FormalizationCase of studyVeri�cation

Task modeling in UPPAALDynamic task model.

Task model

Done

Executingc<=E[tid]

Reconfigured

Reconfiguringc<=R[tid]

Init

c==E[tid]

RemoveTask(tid),done[tid]=true

PredAllDone(tid)

go?

c:=0

c==R[tid]

inuseR:=false

col:int[1,nC-1]

!inuseR && CanProceed(tid,col)

AllocateTask(tid,col),inuseR:=true,c:=0

Variables and constants

number of columns on the FPGA:int nC

number of tasks: int nTcolumns busy: bool inuse[nC +1]Recon�guration controller busy:bool inuseR

Task recon�guration times:int R[nT +1]Task execution times: int E [nT +1]Task required area: int W [nT +1]Task's leftmost column:int position[nT +1]

Antonio Filieri Scheduling with Timed Automata - 15/24

IntroductionFormal models

Scheduling problemPerformance evaluation

FormalizationCase of studyVeri�cation

Outline

1 IntroductionWhy this work?What are we trying to do?

2 Formal modelsHow to describe a process?What is a Timed Automata?

3 Scheduling problemWhat do we mean with scheduling?How can we use Timed Automata?How can this stu� schedule?

4 Performance evaluationIs the method e�ective?

Antonio Filieri Scheduling with Timed Automata - 16/24

IntroductionFormal models

Scheduling problemPerformance evaluation

FormalizationCase of studyVeri�cation

Scheduling by model-checking

Operational model and model-checking:

Possible behaviors of a system: correct sequences of states.

A model-checker search paths in a �nite state space from aninitial state.

Temporal logics (mainly LTL & CTL) can express a path'sproperties.

Properties of a schedule:

Reachability property: E<> P1.Done and P2.Done and ...

A feasible schedule is a path satisfying the reachabilityproperty.

An optimal schedule is one of the "fastest" feasible schedules(UPPAAL can �nd it).

Antonio Filieri Scheduling with Timed Automata - 17/24

IntroductionFormal models

Scheduling problemPerformance evaluation

FormalizationCase of studyVeri�cation

Example of results

ProcessFeasible Sch. Optimal Sch.

Antonio Filieri Scheduling with Timed Automata - 18/24

IntroductionFormal models

Scheduling problemPerformance evaluation

Computation TimeConclusionsQuestions

Outline

1 IntroductionWhy this work?What are we trying to do?

2 Formal modelsHow to describe a process?What is a Timed Automata?

3 Scheduling problemWhat do we mean with scheduling?How can we use Timed Automata?How can this stu� schedule?

4 Performance evaluationIs the method e�ective?

Antonio Filieri Scheduling with Timed Automata - 19/24

IntroductionFormal models

Scheduling problemPerformance evaluation

Computation TimeConclusionsQuestions

Performance environment

Tests results coming RTAS07 proceedings, our own not yetcompleted.

Task graph randomly generated with TGFF utility.

1 to 3 start nodes, max input 2, max output 2.

Recon�guration time = width.

Computation time ∈ [4,12] width ∈ [3,9].

Model checking on Linux workstation, 4x AMD Opteron844(1.8GHz) 1MB E-Cache, 8GB RAM.

UPPAL command −t2 option.

Heuristic from [Banerjee, Bozorgzadeh 2005].

Antonio Filieri Scheduling with Timed Automata - 20/24

IntroductionFormal models

Scheduling problemPerformance evaluation

Computation TimeConclusionsQuestions

Memory and time costs

Memory cost Time cost

100 test cases from 8 to 20 tasks per set.

Antonio Filieri Scheduling with Timed Automata - 21/24

IntroductionFormal models

Scheduling problemPerformance evaluation

Computation TimeConclusionsQuestions

Quality of results vs heuristic

Remark: Optimal DAG scheduling is, in general, NP-complete.

15 columns 20 columns

100 test cases per each task number.

Antonio Filieri Scheduling with Timed Automata - 22/24

IntroductionFormal models

Scheduling problemPerformance evaluation

Computation TimeConclusionsQuestions

Quality of results vs heuristic

Timed automata are very high level descriptions.

Identifying an already computed subnet, its possible to reuseresults for sub-optimal fast scheduling.

Performance really depends on num of tasks, timing, andshape.

Worst-case exponential, best case (sequential tasks) linearwith respect to number of tasks.

Not so bad until 18 tasks because of optimality (really needit?).

Possible to exploit well-known shortest path algorithm to buildspecial purpose checkers.

Possible to express complex scheduling request (e.g. E<>T<20 and T1.Done and (T2.Done xor T3.Done) ).

Antonio Filieri Scheduling with Timed Automata - 23/24

IntroductionFormal models

Scheduling problemPerformance evaluation

Computation TimeConclusionsQuestions

Thanks for listening

Any question?

Any suggestion?

Any comment?

Antonio Filieri Scheduling with Timed Automata - 24/24