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

2. Outline Characterizing Redundancy Quantifying Redundancy Qualifying Redundancy

3. CHARACTERIZING REDUNDANCY

4. 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?

5. 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

6. Definition of Redundancy Functional vs. State Redundancy State redundancy –system state [x 0, x 1, … x n ] (implementation dependent) Functional redundancy –System level requirements R={(u,y)| …} –Subsystem/component level requirements R={(x i, x j )|…} (implementation dependent)

7. 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

8. 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

9. Representation Redundancy Parity-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

10. 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(t i ) = M(t i )a(t i ) is single-step (instantaneous) redundancy –v(t 2 ) = [p(t 2 )-p(t 1 )]/(t 2 -t 1 ) is multiple-step (temporal) redundancy

11. Analytical Redundancy System/Subsystem/component level functional redundancy State redundancy Content redundancy Representation redundancy Single/multiple-step redundancy

12. Physical Redundancy Component level functional redundancy State redundancy Content redundancy Representation redundancy Single-step redundancy (deterministic asset)

13. QUANTIFYING REDUNDANCY

14. 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

15. 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

16. 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)

17. A simple example a : system variable SYSTEM  : vector of readings of a Hypothesis: there is redundancy only if  uniquely determines a H( a |  ) = 0 ( = H( a,  ) – H(  ) ) a  f  a : P(f -1 ( a )) = P( a )

18. 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.

19.  (  ) = 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)

20. QUALIFYING REDUNDANCY

21. 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

22. Notation s 0 : 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

23. s0s0    (s0)(s0) milestone s is a correct state: (s 0,s)      (s 0 )

24. s0s0    (s0)(s0) maskable milestone (s 0,s)  K ( ,  ) s is a maskable state:    (s 0 )

25.    (s0)(s0) maskable ’’ r milestone s is a recoverable state: s0s0  r :  ’ r K ( ,  )    (s 0 )

26. Question 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  r :  ’ r K ( ,  )

27. 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

28. 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}  ’ 2 = { (s,s’) | s’ = (s+5) mod 18} Only produces recoverable states recovery: s’ = s mod 6 Only produces recoverable states recovery: s’ = (s+1) mod 6  ’ 3 = { (s,s’) | s’ = s mod 10} It does not produce recoverable states

29. 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