ARRIVAL CURVES FOR REAL-TIME CALCULUS : THE CAUSALITY

PROBLEM AND ITS SOLUTIONS

- Matthieu Moy and Karine Altisen

Vasvi Kakkad

School of Information Technologies

University of Sydney

OUTLINE

Real Time Calculus

Arrival Curves

Causality Problem

Computing Causality Closure

Algorithm

Conclusion2

REAL TIME CALCULUS Modeling & Analysis Techniques for Real time

systems Computational Analytical

Computational Approach Simulation, testing and verification Very precise result Only for one simulation and one instance of system

Analytical Approach Based on mathematical equations Gives strict worst case execution times Fast Only for theoretical cases

3

Real T

ime C

alcu

lus

REAL TIME CALCULUS

RTC – Framework to model and analyse heterogeneous system

Models timing property of event streams with curves – Arrival Curves

Component – works with input streams and gives output stream as a function of input stream

Can not handle the notion of state in System modeling 4

Real T

ime C

alcu

lus

ARRIVAL CURVES

5

ARRIVAL CURVES Arrival Curves –

Function of relative time that constraints the no of events that can occur in an interval of time

For Sliding Window ∆ & Arrival Curves (u, l) l (∆) lower bounds and u(∆) upper bounds on no

of events

6

Arriv

al C

urv

es

ARRIVAL CURVES - NOTATIONS

R+ - Set of non-negative Reals

- R+ U {+ ∞}

N – Set of Naturals

- N U {+ ∞}

T – Time – R+ or N

- Event Count - R+ or N

- Event Count - or 7

Arriv

al C

urv

es

ARRIVAL CURVE – FORMAL DEFINITION

Pair of functions (u, l) in F X Ffinite, s.t. l ≤ u

(u, l) is satisfiable if cumulative curve R satisfies it - R ╞ (u, l)

8

Arriv

al C

urv

es

ARRIVAL CURVES - FUNCTIONS

Wide-sense increasing function f : T

For f, g : T

9

Arriv

al C

urv

es

ARRIVAL CURVES – SUB/SUPER ADDITIVITY & SUB/SUPER ADDITIVE CLOSURE

Sub additive function f – For all s, t T. f(t + s) ≥ f(t) + f(s)

Sub additive Closure of f –

Super additive function f – For all s, t T. f(t + s) ≤ f(t) + f(s)

Super additive Closure of f –10

Arriv

al C

urv

es

IMPLICIT CONSTRAINTS ON ARRIVAL CURVES

Some Constraints cause problems – E.g. Deadlock with Generator of Events Spurious counter-example with Formal

verification

Two types Unreachable Regions – region between curves

and sub/super additive closure Forbidden Regions

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Arriv

al C

urv

es

CAUSALITY PROBLEM

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CAUSALITY PROBLEM

Causal – Pair of curves for which beginning of execution

never prevents continuation Having no forbidden regions

Causality Problem Curve having forbidden regions

Paper describes – algorithm to transform pair of arrival curves to equivalent causal representation

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Causa

lity P

roble

m

CAUSAL ARRIVAL CURVES

Pair of Arrival Curves (u, l) is causal iff “any cumulative curve R satisfies (u, l) up to

T can be extended indefinitely into a cumulative curve R’ that also satisfies (u,

l).”

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Causa

lity P

roble

m

CAUSALITY CHARACTERISATION

Forbidden region – area between u and u

l & area between l and l u

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Causa

lity P

roble

m

EXAMPLE

0 1 2 3 4 5 6 7 8 9 100

87

654321

# events

αu

αl

αl SA Curve

time

Forbidden

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Causa

lity P

roble

m

COMPUTING CAUSALITY CURVE

C Operator – removes forbidden region from pair of curves.

Removing forbidden regions on l will introduce new ones on u and vice-versa.

Remove it until fix-point

17

Com

putin

g C

ausa

lity C

urv

e

C OPERATOR

For arrival curve (αu, αl),

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Com

putin

g C

ausa

lity C

urv

e

DISCRETE FINITE CURVE

Restriction of infinite curves on a finite interval

(u |T, l |T) – restriction of (u, l) to [0,T] :

SA-SA on an interval

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Discre

te Fin

ite C

urv

e

ALGORITHM & EXAMPLE

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ALGORITHM

A A0Repeat

A SA-SA-closure(A) /* not mandatory, speeds up convergence and ensures SA-SA */A’ AA C(A)

Until A ≠ ┴AC or A’ = A

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Alg

orith

m &

Exam

ple

EXAMPLE

22

Exam

ple

EXAMPLE – CONTD..

23

Exam

ple

EXAMPLE – CONTD..

24

Exam

ple

EXAMPLE – CONTD..

25

Exam

ple

CONCLUSION

RTC – framework to analyse heterogeneous systems

Arrival Curves - (u, l) : Time no of Events

Causal Curve and Causality Problem

C operator – Remove forbidden region Causal curve

Algorithm to Compute Causality Closure for finite, discrete graph 26

THANK YOU...