Performance Analysis of ARQ Protocols using a Theorem Proverohasan.seecs.nust.edu.pk/talks/ispass_2008.pdf · O. Hasan Performance Analysis of ARQ Protocols using Theorem Proving

Performance Analysis of ARQ Protocols using a Theorem Prover

Osman Hasan Sofiene Tahar

Hardware Verification Group Concordia University

Montreal, Canada

ISPASS 2008

O. Hasan 2 Performance Analysis of ARQ Protocols using Theorem Proving

Objectives

n Probabilistic Theorem Proving “A robust and precise probabilistic analysis technique”

n  What is it?

n  Why do we need it? n  How can we apply it for the performance analysis of

ARQ Protocols?


Outline

n  Introduction

n Theorem Proving based Performance Analysis

n Performance Analysis of ARQ Protocols

n Conclusions


Motivation

n Performance Analysis

Environmental Conditions

Aging Phenomena

Probabilistic Choice

Unpredictable Inputs

Noise

n  Simulation n  State-of-the-art n  Inaccurate results

n  Theorem Proving n  Proposed Solution


Performance Analysis

Hardware Software

System Model

Property Satisfied?

Random Components

Probabilistic and Statistical Properties

Computer Based Analysis Framework

R andom Variables(Discrete/

C ontinuous)


Probabilistic Analysis Approaches

Simulation Formal Methods

Model Checking Theorem Proving

Random Components

Probabilistic State Machine

good

Analysis

Accuracy

Expressiveness

No CPU Time Issue

Automation

Approximate random variable

functions

Observing some test cases

û

ü

û

ü

Probabilistic State Machine

Exhaustive Verification

ü

û

û

ü

Precise random variable

functions

Mathematical Reasoning

ü

ü

ü

û

Simulation Formal Methods

Model Checking Theorem Proving


Theorem Prover

n  A notation (syntax) n  A small set of fundamental axioms (facts)

n  A Boolean variable can be True or False: ∀ a.(a = T) ∨ (a =F) n  A small set of inference (deduction) rules

n  Equality is transitive: ∀ a b c. (a = b) ∧ (b = c) ⇒ (a = c)

n  Soundness n  Every new theorem must be created from

n  Basic axioms and primitive inference rules n  Already proved theorems or inference rules

n  Theory (collection of verified theorems in a file) n  Can be reloaded in theorem provers n  Facilitates the instant utilization of already verified theorems


Theorem Proving – Example

n  Check if y>x for the given system (x is a natural number)

1 y>x Problem statement

2 (x+1)2>x Implementation

3 (x+1).(x+1)>x Definition of Square

4 (x+1).x+(x+1).1>x Distributivity

5 x.x+1.x+x.1+1.1>x Distributivity

6 x.x+x+x+1>x Multiplicative Identity

7 x.x+x+1+x>x Additive Commutivity

8 x.x+x+1>0 Addition Cancellation

9 True Natural numbers > 0

2)1( +xx y


Outline

n  Introduction



n Conclusions


HOL Theorem Prover

n Higher-order logic theorem prover n  University of Cambridge, UK

n  5 axioms n  8 primitive inference rules

n Numerous proof assistants are available

n  Inbuilt mathematical theories of Boolean, list, set, integers, real analysis, measure, and probability theory


Theorem Proving Based Performance Analysis

System Description

Sys

tem

Pro

pert

ies

(Dis

cret

e R

ando

m V

aria

bles

)

Sys

tem

Pro

pert

ies

(Con

tinuo

us R

ando

m V

aria

bles

)

System Model

Probabilistic Analysis

Theorems

Discrete Random Variables

Continuous Random Variables

Random Components

Probabilistic Properties

Statistical Properties

Probabilistic Properties

Statistical Properties

Theorem Prover

Formal Proofs of Properties

System Properties


Formal Verification of Random Variables

n Measure Theory n Probability space of Infinite Boolean sequence (B

∞)

B∞: positive integers → Boolean

n A random variable that n  Accepts : α n  Returns: β

can be modeled in HOL as a function

f : α → B∞ → (β x B

∞ )

0 1 2 3 4 5 6 7 T/F T/F T/F T/F T/F T/F T/F T/F


Random Variables in HOL Example

n  Coin Flip (Head, Tail)

B∞ → (flip_outcome x B∞

)

n  Algorithm

flip s = (if (top element of s) then

Head else Tail, remaining portion of s)

n  Probabilistic Properties

P {s | flip s = Head} = ½


Discrete Random Variables in HOL

Theorems: Discrete Random Variables

Random variable

HOL Funtions PMF (Pr (X = n))

Uniform(m) unif_rv

Bernoulli(p) bern_rv

Geometric(p) geom_rv

m1

p

npp )1( −


Continuous Random Variables in HOL

Theorems: Continuous Random Variables

Random Variable HOL Functions CDF (Pr (X ≤ x)

Exponential(l)

exp_rv

Uniform(a,b) uniform_rv

Rayleigh(l) rayleigh_rv

⎭⎬⎫

⎩⎨⎧

<

≤

x0 ,exp-10 x ,0

lx-

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

<

≤<

≤

xb 1,

bxa ,a-ba-x

a x,0

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

<

≤

x0 ,exp-1

0 x ,0

2

2

2x-l


Verification of Statistical Properties

Theorem: Expectation Properties

[ ]∑∑==

=⎥⎦

⎤⎢⎣

⎡ n

ii

n

ii XExXEx

11

Definition: Expectation for Discrete Random Variables

∑∞

=

==1

)Pr(][i

iXiXEx

ccEx =][


Verification of Statistical Properties

Theorems: Discrete Random Variables

Random variable

HOL Function

Expectation Variance

Uniform(m)

unif_rv

Bernoulli(p) bern_rv

Geometric(p) geom_rv

2m

121)1( 2 −+m

p )1( pp −

p1

2

1pp−


Probabilistic Theorem Proving – Case Studies

n Very few examples n Roundoff error analysis of a Digital Processer

n  Verification of a couple of probabilistic properties

n Probabilistic Analysis of Algorithms n  Miller Rabin Test

n  Coupon-Collector’s Problem


Outline

n  Introduction



n Conclusions


Automatic Repeat Request (ARQ)

n  Reliable communication between computers

n  Transmitter n  Repeats transmission of a data frame until it receives an ACK

n  Receiver n  Discards erroneous data frames

n  Sends Acknowledgment (ACK) for Error-free data frames

n  Applications n  Transmission Control Protocol (TCP)

n  High-level Data Link Control (HDLC) Standard


ARQ Protocols

n  Implementation variants of ARQ principle n  Stop-and-Wait n  Go-Back-N n  Selective Repeat

n Performance Analysis Metric n  Message Delay

n Both simulation and state-based formal techniques fail to produce reasonable results n  A subtle interaction of a number of distributed

components


Stop-and-Wait Protocol

n Delay (Unsuccessful Transmission Trial)

n Delay (Successful Transmission Trial)

outfu ttT +=

)(2 procpropafs ttttT +++=


Go-Back-N Protocol



outfu ttT +=

fs tT =


Selective Repeat Protocol



fu tT =

fs tT =


Average Message Delay of ARQ Protocols

n p: Bit-error probability of the channel

n Average (Message Delay) = ?

n Step 1: Message Delay (Tu,Ts,p) n  Geometric Random Variable

§  Delay = (G-1)Tu + Ts

n Step 2: Average of the above random variable


Step 1: Message Delay in HOL

n Geometric random variable function (geom_rv) n Success probability = ?

n  Error behaviour of single bit: bern_rv(p)

⊢∀ n p. f_err 0 p = false ∧ f_err (n + 1) p = bern_rv(p) ∨ (f_err n p)

Definition: Frame Error

⊢∀ nf na p. suc_p_arq nf na p = P { (f_err nf p) ∨ (f_err na p) = false }

Definition: Probability of Successful Transmission


Step 1: Message Delay in HOL

n Proof n  Boolean Logic, Positive Integers, Real Numbers, Set,

Probability

⊢∀ nf na p. 0 ≤ p ∧ p ≤ 1 ⇒ suc_p_arq nf na p = (1-p) (nf + na)

Theorem: Probability of Successful Transmission

⊢∀ nf na p Tu Ts. arq_del = Tu (geom_rv ((1-p) (nf + na)) – 1) + Ts

Definition: ARQ Message Delay


Step 2: Average Message Delay

n Proof n  Already verified Expectation properties

n  Boolean Logic, Positive Integers, Real Numbers, Set, Probability

Theorem: Linearity of Expectation

bXaEbaXEx +=+ ][][

[ ]∑∑==

=⎥⎦

⎤⎢⎣

⎡ n

ii

n

ii XExXEx

11

ccEx =][


Average Message Delay in HOL Stop-and-Wait Protocol

n  Proof: n  n  Expectation of Geometric random variable

⊢∀ nf na p tout tprop tproc tf ta. sw_del nf na p tout tprop tproc tf ta = (tf + tout) (geom_rv ((1-p) (nf + na)) – 1) + tf + ta + 2(tproc + tprop)

Definition: Stop-and-Wait Message Delay

⊢∀ nf na p tout tprop tproc tf ta. (0 ≤ p) ∧ (p < 1) ⇒ expec (sw_del nf na p tout tprop tproc tf ta) = (tf + tout) (1 - (1-p) (nf + na))/((1-p) (nf + na)) + tf + ta + 2(tproc + tprop)

Theorem: Average Stop-and-Wait Message Delay

bXaEbaXEx +=+ ][][


Average Message Delay in HOL Go-Back-N Protocol


⊢∀ nf na p tout tf. gbn_del nf na p tout tf = (tf + tout) (geom_rv ((1-p) (nf + na)) – 1) + tf

Definition: Go-Back-N Message Delay

⊢∀ nf na p tout tf. (0 ≤ p) ∧ (p < 1) ⇒ expec (gbn_del nf na p tout tf) = (tf + tout) (1 - (1-p) (nf + na))/((1-p) (nf + na)) +tf

Theorem: Average Go-Back-N Message Delay

bXaEbaXEx +=+ ][][


Average Message Delay in HOL Selective Repeat Protocol


⊢∀ nf na p tf. sr_del nf na p tf = (tf) (geom_rv ((1-p) (nf + na)) – 1) + tf

Definition: Stop-and-Wait Message Delay

⊢∀ nf na p tf. (0 ≤ p) ∧ (p < 1) ⇒ expec (sr_del nf na p tf) = (tf)/((1-p) (nf + na))

Theorem: Average Stop-and-Wait Message Delay

bXaEbaXEx +=+ ][][


Outline

n  Introduction



n Conclusions


Conclusions

n Probabilistic Theorem Proving n  Model randomness in systems with higher-order-logic random

variables n  Verify probabilistic and statistical properties in a theorem prover n  Exact Answers

n  Useful for the analysis of Safety critical application

n  Performance Analysis of ARQ Protocols n  Delay Characteristic → Higher-order-logic random variable n  Verification of Linearity of Expectation Property in HOL

n  Results exactly match the paper-and-pencil based analysis methods §  100% precise


Conclusions

n  Probabilistic Theorem Proving is not a “golden solution” to all performance analysis problems n  Interactive and tedious nature

n  Less critical sections of the system n  Simulation

n  Critical sections of the system that can be expressed as a Markov Chain n  Model Checking

n  Critical sections of the system that cannot be handled by Model Checking n  Thereom Proving


Thank you

For more information: http://hvg.ece.concordia.ca

Contact: [email protected]


Additional Slides


Performance Analysis Basics – Random Variables

n Discrete Random Variables n  Attain a countable number of values

n  Examples n  Uniform (countable values in an interval [a,b])

n  Bernoulli (True, False)

n Continuous Random Variables n  Attain an uncountable (infinite) number of values

n  Examples n  Uniform (all real values in an interval [a,b])

n  Exponential (The time between independent events)


Performance Analysis Basics – Properties of Random Variables

n Used to characterize system’s behaviour n Probabilistic properties

n  Probability (Multiplier delay = x)

n Statistical properties n  Average message delay of a telecommunication

protocol

n  Major decision making criteria in performance analysis

Performance Analysis of ARQ Protocols using a Theorem Proverohasan.seecs.nust.edu.pk/talks/ispass_2008.pdf · O. Hasan Performance Analysis of ARQ Protocols using Theorem Proving

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