deploying analytical redundancy for system fault tolerance n.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Deploying Analytical Redundancy for System Fault Tolerance PowerPoint Presentation
Download Presentation
Deploying Analytical Redundancy for System Fault Tolerance

Loading in 2 Seconds...

play fullscreen
1 / 29

Deploying Analytical Redundancy for System Fault Tolerance - PowerPoint PPT Presentation


  • 385 Views
  • Uploaded on

FY2001 University Software Initiative for the NASA IV&V Facility - Fairmont WV. Deploying Analytical Redundancy for System Fault Tolerance. V. Cortellessa, D. Del Gobbo, A. Mili, M. Shereshevsky, and Z. Zhuang CSEE Dept. West Virginia University - Morgantown. Outline.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Deploying Analytical Redundancy for System Fault Tolerance' - paul2


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
deploying analytical redundancy for system fault tolerance

FY2001 University Software Initiative for the NASA IV&V Facility - Fairmont WV

Deploying Analytical Redundancyfor System Fault Tolerance

V. Cortellessa, D. Del Gobbo, A. Mili, M. Shereshevsky, and Z. Zhuang

CSEE Dept. West Virginia University - Morgantown

outline
Outline
  • Characterizing Redundancy
  • Quantifying Redundancy
  • Qualifying Redundancy
objectives
Objectives
  • To develop a classification of redundancy by identifying the orthogonal dimensions in redundancy
  • To analyze physical and analytical redundancy on the basis of the obtained classification
  • To answer general questions about redundancy:
    • What is redundancy?
    • Can we talk about redundancy outside the context of fault tolerance?
    • Can we distinguish between intrinsic redundancy and redundancy-by-design?
    • Is redundancy a representation issue or a design issue?
    • Is physical redundancy an extreme case of redundancy?
definition of redundancy
Definition of Redundancy
  • From IEEE Dictionary
    • duplication of elements for the purpose of enhancing system reliability
    • presence of auxiliary components in a system for the purpose of preventing or recovering from failures
    • the existence of more than one means for performing a given function
    • pertaining to characters that do not contribute to the information content
    • Log (# symbols) - average information content per symbol
definition of redundancy functional vs state redundancy
Definition of RedundancyFunctional vs. State Redundancy
  • State redundancy
    • system state [x0, x1, … xn](implementation dependent)
  • Functional redundancy
    • System level requirements R={(u,y)| …}
    • Subsystem/component level requirements R={(xi, xj)|…}(implementation dependent)
content redundancy english language sentence shannon
Content Redundancy English language sentence (Shannon)
  • No redundancy
    • symbols are independent and equiprobable
  • First-level redundancy
    • symbols are independent but with frequency of English text
    • digram structure as in English text
    • trigram structure as in English text
  • Word redundancy
    • words are independent but with frequency of English text
    • word transition probability is that of English text
content redundancy physical system
Content Redundancy Physical system
  • Rigid body in free fall ( p, v, a, F, M)
  • No redundancy
    • quantities are independent and each uniformly distributed
  • Local redundancy (quantities are still independent)
    • each quantity is assigned a probability distribution
    • relationship among each quantity at different time instants
  • System redundancy
    • instantaneous dependency between different quantities
    • temporal dependency between different quantities
representation redundancy parity bit
Representation RedundancyParity-bit
  • Information in order to be processed needs to be represented in some suitable manner
  • The parity-bit in serial communication allows detecting non-admissible strings of bits.
  • Admissibility of the string of bits is independent of the information content
temporal sequential redundancy
Temporal/Sequential Redundancy
  • Some applications are characterized by a sequential introduction of data
  • Shannon’s example
    • first-order redundancy is a single-step redundancy
    • following orders of redundancy are multiple-step
  • Physical system example
    • F(ti) = M(ti)a(ti) is single-step (instantaneous) redundancy
    • v(t2) = [p(t2)-p(t1)]/(t2-t1) is multiple-step (temporal) redundancy
analytical redundancy
Analytical Redundancy
  • System/Subsystem/component level functional redundancy
  • State redundancy
  • Content redundancy
  • Representation redundancy
  • Single/multiple-step redundancy
physical redundancy
Physical Redundancy
  • Component level functional redundancy
  • State redundancy
  • Content redundancy
  • Representation redundancy
  • Single-step redundancy (deterministic asset)
objectives1
Objectives
  • To quantify the amount of redundancy by means of a numeric function
  • To characterize analytical vs physical redundancy by means of this function
  • To characterize Fault Tolerance Capabilities (e.g., detection, identification, etc.) by means of this function
  • Use this function to support decision making in redundancy vs Fault Tolerant Capability tradeoffs
redundancy as the ability to choose among representations
Redundancy as the ability to choose among representations

X : system state

P : set of all the “possible” system states

C : set of all the “correct” system states

Prob ( X  C | X  P )

The corresponding conditional entropy is a suitable metric of “how fully the potential domain is being exploited” (or, conversely, how sparsely populated it is), i.e. how much redundancy the system shows in terms of unused possible states

redundancy as logical relation among state variables
Redundancy as logical relation among state variables
  • State made up of two (aggregate of) variables, say X and Y
  • P(X|Y) : to what extent the value of Y determines the values of X
  • H(X|Y) : Amount of uncertainty that remains about X if we know Y
        • H(X|Y) = H(X,Y) – H(Y)
a simple example
A simple example

a: system variable

 : vector of readings of a

SYSTEM

Hypothesis: there is redundancy only if  uniquely determines a

H(a | ) = 0 ( = H(a , ) – H() )

f

a

 a : P(f -1(a)) = P(a)

slide18

This property holds:

H(a)  H()

and the distance depends on the injectivity of f

(e.g., one-to-one mapping gives H(a) = H() )

Again we may consider, as a measure of redundancy:

 () = H() - H(a) ( = H( | a) )

i.e., how fully the potential domain of values is being exploited.

slide19

 () = H() - H(a)

We voluntarily omit a as a parameter of  because:

  • P(a) comes from the intrinsic system operational profile (there is no control on it)
  • while
  • P() is the result of design choices and fault hypotheses (its value can be controlled by design)
objectives2
Objectives
  • Whereas the previous section quantifies redundancy, this section qualifies it. The same amount of redundancy may or may not be useful, depending on functional properties
  • Whereas in quantifying redundancy we need to distinguish between correct and representable (possible) states, in this section we will distinguish between:
    • Correct states
    • Maskable states
    • Recoverable states
    • Representable states
notation
Notation
  • s0 : system initial state
  • milestone: breaking point between past and future behavior of the system
  •  : relation that describes the past behavior
  •  : relation that describes the future behavior
  •  : system requirements
slide23

s0

s is a correct state:

(s0,s)  

milestone

   (s0)

(s0)

slide24

s0

s is a maskable state:

(s0,s)  K (, )

milestone

maskable

   (s0)

(s0)

slide25

s0

s is a recoverable state:

’

r : ’ r K (, )

r

milestone

maskable

   (s0)

(s0)

question
Question

r : ’ r K (, )

For what ’ and K this equation has a solution?

Analogy: for what a,b does the equation ax=b have a solution?

Answer: a0

answer conditions for existence of r
Answer: conditions for existence of r

- C1 - K L  ’ L

- C2 - (K L ’)^ K must be a total relation

In practice, we look for the smallest’ s.t. C1 and C2 hold

(i.e., the relation that maps initial to recoverable states only)

- C1 - K L = ’ L

- C2 -’ K must be a total relation

a sufficient condition for c2
A sufficient condition for C2

If the domain partition determined by K is preserved by ’

then condition C2 holds

’ ’ K K

’ K is a total relation

A simple example

K = { (s,s’) | s’ = s mod 6}

’1 = { (s,s’) | s’ = s mod 12}

Only produces

recoverable states

recovery: s’ = s mod 6

’2 = { (s,s’) | s’ = (s+5) mod 18}

Only produces

recoverable states

recovery: s’ = (s+1) mod 6

’3 = { (s,s’) | s’ = s mod 10}

It does not produce

recoverable states

conclusions and future work
Conclusions and Future Work
  • We have developed a framework for reasoning about redundancy
  • It includes: Classification/Quantification/Qualification
  • Future work
    • Refining/reorganizing classification
    • Evaluate quantification
    • Validate qualification