ece 753: fault-tolerant computing kewal k.saluja department of electrical and computer engineering...

ECE 753: FAULT-TOLERANT COMPUTING

Kewal K.SalujaDepartment of Electrical and Computer

Engineering

System Diagnosis

ECE 753 Fault Tolerant Computing 2

Overview• Introduction• System Model• Diagnosis Problem - PMC model• Other Models and Comments• Sequential Diagnosability• Other Formulations, Algorithms, and

Problems• Summary


Introduction• Reference

• [prad:96] Chapter 8, Original paper in IEEETC (Dec 1967)

• Diagnosis: an important part of recovery, maintenance and reconfiguration

• What 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 theory


System Model• Model and Assumptions

– Graph model• Processors/processes expressed as nodes• Interconnects as links between nodes

– Each processor is sufficiently powerful to test other processors comprehensively

– An example model with four nodes

– Test model: node Vi tests Vj then draw a

directed link from Vi to Vj


Diagnosis - PMC model (contd.)• Example – Test Model

v4 v3

v2v1


Diagnosis - PMC model (contd.)• Assumptions

– 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)– Total number of faulty units in the system is

upper-bounded to t– Example: system with four nodes and one

fault


Diagnosis - PMC model (contd.)

• Example – Test outcomes

• Assume V2 is faulty

v4 v3

v2v11

0

0

0

xx


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.

– 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.



• One-step diagnosis - Example– Consider all possible outcomes -

fault a12 a23 a24 a31 a41 a43

none 0 0 0 0 0 0

V1 faulty x 0 0 1 1 0

V2 faulty 1 x x 0 0 0

V3 faulty 0 1 0 x 0 1

V4 faulty 0 0 1 0 x x

each row is called Syndrome of the fault



• Observations1. Two possible syndromes associated with the

fault V1 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


Diagnosis - PMC model (contd.)• Consider two faulty units – say V1 and V2

possible syndrome

x x x 1 1 0

implies

0 0 0 1 1 0 a possible outcome

Therefore we can not determine if V1 alone or both V1 and V2 are faulty. Thus two faults in this system can not be diagnosed in one-step.



• 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


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.


6

0

1

5

4 3

2


• Design Problem – one-step t-fault diagnosable system

• Example – n = 7, t = 3


Diagnosis - PMC model (contd.)• Design 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.


Diagnosis - PMC model (contd.)• Systems in which some units test each

other• 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 – – Simple if no two units test each other– Some what complex if units test each other– There is a body of literature dealing with diagnosis

algorithems


Other Models and CommentsConsider possible test outcomes when a unit

Vi tests unit Vj – see the listing below

Vi Vj 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


Other Models/Comments(contd.)

– 4,5,6,7 PMC model– 8,9,10,11 PMC with complement encoding– 0,15 of little value– 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.


Other Models/Comments(contd.)

– Comparison based testing and diagnosis• A paper is in the IEEE Transactions of

Computers - February 2009 Issue

– Basically the model is built on PMC model


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


Sequential Diagnosability (contd.)

• Assumptions– 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)• Total number of faulty units in the system is

upper-bounded to t



• 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


0


• Example – n = 7, t = 3

6 1

5

4 3

2



• It is easy to show that the example system is sequentially 3-fault diagnosable

• Above 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 diagnosable



• Proof:– First construct the system – n nodes form a

single loop, thus containing n links– Next choose some 2t-2 units and let these units

test V0 unit– Now show that this system is sequentially t-fault

diagnosable using the following three cases. Let n1 indicate the number of units which find V0 faulty. Similarly n0 indicate the units that find V0 not faulty. Clearly n1+ n0 = 2t-1



• Proof:– Case 1: n1 > t ---- V0 is faulty

– Case 1: n1 < t ---- V0 is not faulty

– Case 1: n1 = t ---- a fault free unit exists that is not involved in testing V0


Sequential Diagnosability (contd.)• Sequential diagnosis – single loop system

– Example 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 diagnosable

– General result: A single loop system is sequentially t-fault diagnosable if and only if

n t + t2/4 + 2 for even t

n t + [(t-1)(t+1)/4] + 2 for odd t


Other Formulations, Algorithms, and Problems

• Generalization of sequential diagnosability– Diagnose 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)– 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


Other Formulations, Algorithms, and Problems

• Diagnosis algorithms – Given a syndrome and knowing that the system is t diagnosable, determine the set of faulty units– Possible solutions

• Dictionary approach – some what impractical for large systems

• Algorithmic approach – based on graph models and using solution to maximum matching problem

– Central v/s distributed algorithms• Diagnosis and reconfiguration in homogenous

and heterogeneous multicore systems


Summary

• System diagnosis model

• One-step t-fault diagnosis

• Sequential diagnosis

ece 753: fault-tolerant computing kewal k.saluja department of electrical and computer engineering...

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