1. ECE 753: FAULT-TOLERANT COMPUTING Kewal K.Saluja Department of Electrical and Computer Engineering System Diagnosis

2. ECE 753 Fault Tolerant Computing2 Overview Introduction System Model Diagnosis Problem - PMC model Other Models and Comments Sequential Diagnosability Other Formulations, Algorithms, and ProblemsOther Formulations, Algorithms, and Problems Summary

3. ECE 753 Fault Tolerant Computing3 Introduction Reference [prad:96] Chapter 8, Original paper in IEEETC (Dec 1967) Diagnosis: an important part of recovery, maintenance and reconfigurationDiagnosis: an important part of recovery, maintenance and reconfiguration What is system level diagnosis: diagnose failed components in a large, possibly multiprocessor, systemWhat is system level diagnosis: diagnose failed components in a large, possibly multiprocessor, system Underlying needs: failures inevitable, units are smart/intelligent to test other units, hence need a different model and corresponding theoryUnderlying needs: failures inevitable, units are smart/intelligent to test other units, hence need a different model and corresponding theory

4. ECE 753 Fault Tolerant Computing4 System Model Model and Assumptions –Graph modelGraph model Processors/processes expressed as nodes Interconnects as links between nodes –Each processor is sufficiently powerful to test other processors comprehensivelyEach processor is sufficiently powerful to test other processors comprehensively –An example model with four nodesAn example model with four nodes –Test model: node V i tests V j then draw a directed link from V i to V jTest model: node V i tests V j then draw a directed link from V i to V j

5. ECE 753 Fault Tolerant Computing5 Diagnosis - PMC model (contd.) Example – Test Model v4v4 v3v3 v2v2 v1v1

6. ECE 753 Fault Tolerant Computing6 Diagnosis - PMC model (contd.) Assumptions –System with n unitsSystem with n units –Tests are comprehensiveTests are comprehensive –Test results are binary: good (0) /faulty (1)Test results are binary: good (0) /faulty (1) –Faulty units can not be trusted for their test outcomes (denote x – means can be 0 or 1)Faulty units can not be trusted for their test outcomes (denote x – means can be 0 or 1) –Total number of faulty units in the system is upper-bounded to tTotal number of faulty units in the system is upper-bounded to t –Example: system with four nodes and one faultExample: system with four nodes and one fault

7. ECE 753 Fault Tolerant Computing7 Diagnosis - PMC model (contd.) Example – Test outcomes Assume V 2 is faultyAssume V 2 is faulty v4v4 v3v3 v2v2 v1v1 1 0 0 0 x x

8. ECE 753 Fault Tolerant Computing8 Diagnosis - PMC model (contd.) One-step diagnosis –Analysis problem – give a system with n units, all the interconnects, and the test outcomes, identify the faulty units subject to the constraint that no more than t units in the system are faulty.Analysis problem – give a system with n units, all the interconnects, and the test outcomes, identify the faulty units subject to the constraint that no more than t units in the system are faulty. –Design problem – design a system using fewest possible test links such that all the faulty units can be correctly identified in one-step knowing the outcomes of the tests.Design problem – design a system using fewest possible test links such that all the faulty units can be correctly identified in one-step knowing the outcomes of the tests.

9. ECE 753 Fault Tolerant Computing9 Diagnosis - PMC model (contd.) One-step diagnosis - Example –Consider all possible outcomes -Consider all possible outcomes - fault a 12 a 23 a 24 a 31 a 41 a 43 none 0 0 0 0 0 0 V 1 faulty x 0 0 1 1 0 V 2 faulty 1 x x 0 0 0 V 3 faulty 0 1 0 x 0 1 V 4 faulty 0 0 1 0 x x each row is called Syndrome of the fault

10. ECE 753 Fault Tolerant Computing10 Diagnosis - PMC model (contd.) Observations 1. Two possible syndromes associated with the fault V 1 and these are: 0 0 0 1 1 0 and 1 0 0 1 1 0 2. No two faults have overlapping syndromes Hence: we can correctly identify (diagnose) the faulty unit

11. ECE 753 Fault Tolerant Computing11 Diagnosis - PMC model (contd.) Consider two faulty units – say V 1 and V 2Consider two faulty units – say V 1 and V 2 possible syndrome x x x 1 1 0 implies 0 0 0 1 1 0 a possible outcome Therefore we can not determine if V 1 alone or both V 1 and V 2 are faulty. Thus two faults in this system can not be diagnosed in one- step.

12. ECE 753 Fault Tolerant Computing12 Diagnosis - PMC model (contd.) Result: A system is one-step t-fault diagnosable provided syndrome for each fault ( 0-fault, 1-fault, 2-faults, …, t-faults) are all distinct (non overlappling/non intersecting)Result: A system is one-step t-fault diagnosable provided syndrome for each fault ( 0-fault, 1-fault, 2-faults, …, t-faults) are all distinct (non overlappling/non intersecting) More results: - but first one more assumption – no two units test each other

13. ECE 753 Fault Tolerant Computing13 Diagnosis - PMC model (contd.) Result 1: For a system to be one-step t-fault diagnosable n ≧ 2t + 1 Result 2: For a system to be one-step t-fault diagnosable each unit must be tested by at least t other units Theorem: A system of n units in which no two units test each other is one step t-fault diagnosable if and only if each unit is tested by t other units.

14. ECE 753 Fault Tolerant Computing14 6 0 1 5 43 2 Diagnosis - PMC model (contd.) Design Problem – one-step t-fault diagnosable systemDesign Problem – one-step t-fault diagnosable system Example – n = 7, t = 3

15. ECE 753 Fault Tolerant Computing15 Diagnosis - PMC model (contd.) Design Problem: Algorithm for a simple one- step t-fault diagnosable with n ≧ 2t + 1Design Problem: Algorithm for a simple one- step t-fault diagnosable with n ≧ 2t + 1 1. Number the nodes from 0 to n-1 2. draw a link from node i to i+1 (mod n), i+2 (mod n), …, i+t (mod n). 3. System so designed is t-fault one-step diagnosable.

16. ECE 753 Fault Tolerant Computing16 Diagnosis - PMC model (contd.) Systems in which some units test each otherSystems in which some units test each other One-step t-fault diagnosability conditions are some what complex – See [prad:96]One-step t-fault diagnosability conditions are some what complex – See [prad:96] How does one check if a given system is one-step t-fault diagnosable –How does one check if a given system is one-step t-fault diagnosable – –Simple if no two units test each otherSimple if no two units test each other –Some what complex if units test each otherSome what complex if units test each other –There is a body of literature dealing with diagnosis algorithemsThere is a body of literature dealing with diagnosis algorithems

17. ECE 753 Fault Tolerant Computing17 Other Models and Comments Consider possible test outcomes when a unit V i tests unit V j – see the listing below V i V j outcomes G G 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 G F 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 F G 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 F F 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

18. ECE 753 Fault Tolerant Computing18 Other Models/Comments(contd.) –4,5,6,7 PMC model4,5,6,7 PMC model –8,9,10,11 PMC with complement encoding8,9,10,11 PMC with complement encoding –0,15 of little value0,15 of little value –etc.etc. –Some subset of PMC are more interesting – for example 5,7 – this implies that a unit being tested is always correctly identified, if faulty, independent of the status of the testing unit. Many such variations have been studied.Some subset of PMC are more interesting – for example 5,7 – this implies that a unit being tested is always correctly identified, if faulty, independent of the status of the testing unit. Many such variations have been studied.

19. ECE 753 Fault Tolerant Computing19 Other Models/Comments(contd.) –Comparison based testing and diagnosisComparison based testing and diagnosis A paper is in the IEEE Transactions of Computers - February 2009 IssueA paper is in the IEEE Transactions of Computers - February 2009 Issue –Basically the model is built on PMC modelBasically the model is built on PMC model

20. ECE 753 Fault Tolerant Computing20 Sequential Diagnosability Consider the following repair strategy identify one or more faulty units repair them test system again and continue till we know that there are no more faulty units –This is called sequential diagnosis

21. ECE 753 Fault Tolerant Computing21 Sequential Diagnosability (contd.) Assumptions –Same as before:Same as before: System with n units Tests are comprehensive Test results are binary: good (0) /faulty (1) Faulty units can not be trusted for their test outcomes (denote x – means can be 0 or 1)Faulty units can not be trusted for their test outcomes (denote x – means can be 0 or 1) Total number of faulty units in the system is upper-bounded to tTotal number of faulty units in the system is upper-bounded to t

22. ECE 753 Fault Tolerant Computing22 Sequential Diagnosability (contd.) Result 1: For a system to be sequntially t-fault diagnosable n ≧ 2t + 1 It is not necessary for every unit to be tested by t units

23. ECE 753 Fault Tolerant Computing23 0 Sequential Diagnosability (contd.) Example – n = 7, t = 3 6 1 5 43 2

24. ECE 753 Fault Tolerant Computing24 Sequential Diagnosability (contd.) It is easy to show that the example system is sequentially 3-fault diagnosableIt is easy to show that the example system is sequentially 3-fault diagnosable Above construction will require n+2t–1 linksAbove construction will require n+2t–1 links A better solution: A system with n+2t-2 links can be designed that is sequentially t-fault diagnosableA better solution: A system with n+2t-2 links can be designed that is sequentially t-fault diagnosable

25. ECE 753 Fault Tolerant Computing25 Sequential Diagnosability (contd.) Proof: –First construct the system – n nodes form a single loop, thus containing n linksFirst construct the system – n nodes form a single loop, thus containing n links –Next choose some 2t-2 units and let these units test V 0 unitNext choose some 2t-2 units and let these units test V 0 unit –Now show that this system is sequentially t-fault diagnosable using the following three cases. Let n 1 indicate the number of units which find V 0 faulty. Similarly n 0 indicate the units that find V 0 not faulty. Clearly n 1 + n 0 = 2t-1Now show that this system is sequentially t-fault diagnosable using the following three cases. Let n 1 indicate the number of units which find V 0 faulty. Similarly n 0 indicate the units that find V 0 not faulty. Clearly n 1 + n 0 = 2t-1

26. ECE 753 Fault Tolerant Computing26 Sequential Diagnosability (contd.) Proof: –Case 1: n 1 > t ---- V 0 is faultyCase 1: n 1 > t ---- V 0 is faulty –Case 1: n 1 < t ---- V 0 is not faultyCase 1: n 1 < t ---- V 0 is not faulty –Case 1: n 1 = t ---- a fault free unit exists that is not involved in testing V 0Case 1: n 1 = t ---- a fault free unit exists that is not involved in testing V 0

27. ECE 753 Fault Tolerant Computing27 Sequential Diagnosability (contd.) Sequential diagnosis – single loop system –Example single loop system with n=5Example single loop system with n=5 –This is sequentially 2-fault diagnosable and can be demonstrated by constructing syndromes for different fault conditions. However, a system with n=9 is NOT sequentially 4-fault diagnosableThis is sequentially 2-fault diagnosable and can be demonstrated by constructing syndromes for different fault conditions. However, a system with n=9 is NOT sequentially 4-fault diagnosable –General result: A single loop system is sequentially t-fault diagnosable if and only ifGeneral result: A single loop system is sequentially t-fault diagnosable if and only if n  t + t 2 /4 + 2 for even t n  t + [(t-1)(t+1)/4] + 2 for odd t

28. ECE 753 Fault Tolerant Computing28 Other Formulations, Algorithms, and Problems Generalization of sequential diagnosability –Diagnose s faulty units at a time thus making a system t/s-sequentially diagnosableDiagnose s faulty units at a time thus making a system t/s-sequentially diagnosable Allow replacing up to t units – but not all units there are replaced are faulty. In other words non faulty units can be replaced as long as all the faulty units are within the replaced units (t/t fault diagnosability )Allow replacing up to t units – but not all units there are replaced are faulty. In other words non faulty units can be replaced as long as all the faulty units are within the replaced units (t/t fault diagnosability ) –An example in [prad:96] shows a system with 13 units, each unit is tested by 3 other units. Clearly such a system is only one-step 3-fault diagnosable. But it is shown to be 5/5 diagnosable.An example in [prad:96] shows a system with 13 units, each unit is tested by 3 other units. Clearly such a system is only one-step 3-fault diagnosable. But it is shown to be 5/5 diagnosable. Even additional formulations exist

29. ECE 753 Fault Tolerant Computing29 Other Formulations, Algorithms, and Problems Diagnosis algorithms – Given a syndrome and knowing that the system is t diagnosable, determine the set of faulty unitsDiagnosis algorithms – Given a syndrome and knowing that the system is t diagnosable, determine the set of faulty units –Possible solutionsPossible solutions Dictionary approach – some what impractical for large systemsDictionary approach – some what impractical for large systems Algorithmic approach – based on graph models and using solution to maximum matching problemAlgorithmic approach – based on graph models and using solution to maximum matching problem –Central v/s distributed algorithmsCentral v/s distributed algorithms Diagnosis and reconfiguration in homogenous and heterogeneous multicore systemsDiagnosis and reconfiguration in homogenous and heterogeneous multicore systems

30. ECE 753 Fault Tolerant Computing30 Summary System diagnosis model One-step t-fault diagnosis Sequential diagnosis