1. ECE 753: FAULT-TOLERANT COMPUTING
5/15/2018
ECE 753: FAULT-TOLERANT COMPUTING
Kewal K.Saluja
Department of Electrical and Computer Engineering
Reliability Modeling and Analysis
Lectures 8-10

2. ECE 753 Fault Tolerant Computing
5/15/2018
Overview
Recap
Introduction
Reliability Modeling
reliability block diagram
combinatorial model
Markov model
Other Parameters and analysis
General remarks and Summary
Do not discuss much about topics here.
Under computer system
overall implies what is a compute system - its architecture and
components
Then focus on hardware and software components
ECE 753 Fault Tolerant Computing

3. ECE 753 Fault Tolerant Computing
5/15/2018
Recap
Course introduction
Fundamental principles - Four types of redundancy
FEF and breaking FEF chain
Fault modeling
models at different levels, error models, process failure models
Testing and Test Generation
test generation, fault simulation, DFT and BIST concepts
Simple concepts in fault-tolerance
hardware redundancy, information redundancy, time redundancy, and software redundancy methods
Be sure everyone has a conduct sheet.
Re GROUND RULES:
1. In terms of allowed collaboration vs. individual work, ask if you are not sure.
2. Deactivate all cell phones or pagers during class unless you are on-call during your job.
3. No tape-recording permitted. Take notes.
ECE 753 Fault Tolerant Computing

4. ECE 753 Fault Tolerant Computing
5/15/2018
Introduction
References
[prad:96]
[john:89]
[triv:82]
These three books contain sufficient material covering this part of the course
Recap of definitions
Importance of analysis and analytical model
Mathematical formulation for quantitative analysis
Be sure everyone has a conduct sheet.
Re GROUND RULES:
1. In terms of allowed collaboration vs. individual work, ask if you are not sure.
2. Deactivate all cell phones or pagers during class unless you are on-call during your job.
3. No tape-recording permitted. Take notes.
ECE 753 Fault Tolerant Computing

5. ECE 753 Fault Tolerant Computing
5/15/2018
Introduction (contd.)
Recap of definitions
Reliability R(t)
Availability A(t)
Performability and Dependability
Importance of analysis and analytical model
to evaluate a design
a metric to compare different designs
to provide feedback to the designer during early design stages
use a model for performance analysis
used for quantitative and qualitative analysis
Be sure everyone has a conduct sheet.
Re GROUND RULES:
1. In terms of allowed collaboration vs. individual work, ask if you are not sure.
2. Deactivate all cell phones or pagers during class unless you are on-call during your job.
3. No tape-recording permitted. Take notes.
ECE 753 Fault Tolerant Computing

6. ECE 753 Fault Tolerant Computing
5/15/2018
Introduction (contd.)
Mathematical formulation for quantitative analysis
consider a large experiment with N systems
observation at time t
N0(t) - number of correctly operating systems
Nf(t) - number of failed systems
Hence
Reliability R(t) = N0(t)/N(t) = 1 - Nf(t)/N
Unreliability Q(t) = 1 - R(t)
Derivative of reliability: dR/dt = -(1/N)(dNf(t)/dt)
dNf(t)/dt is called instantaneous failure rate of the component
Be sure everyone has a conduct sheet.
Re GROUND RULES:
1. In terms of allowed collaboration vs. individual work, ask if you are not sure.
2. Deactivate all cell phones or pagers during class unless you are on-call during your job.
3. No tape-recording permitted. Take notes.
ECE 753 Fault Tolerant Computing

7. ECE 753 Fault Tolerant Computing
5/15/2018
Introduction (contd.)
Mathematical formulation (contd.)
Also
failure rate at time t
(instantaneous failure rate at time t) / N0(t)
(1/No(t))(dNf(t)/dt) - called z(t)
this and the previous expressions together reduce to
z(t) = -(1/R(t))(dR(t)/dt)
Z(t) is called failure rate, hazard function or hazard rate
We can solve the above for R(t) provided we know instantaneous failure rate
Bath tub curve for failure rate
implies constant failure rate during useful life
infant mortality and wear out periods have variable failure rates
Be sure everyone has a conduct sheet.
Re GROUND RULES:
1. In terms of allowed collaboration vs. individual work, ask if you are not sure.
2. Deactivate all cell phones or pagers during class unless you are on-call during your job.
3. No tape-recording permitted. Take notes.
ECE 753 Fault Tolerant Computing

8. ECE 753 Fault Tolerant Computing
5/15/2018
Introduction (contd.)
Mathematical formulation (contd.)
Reliability computation - constant failure rate
solve the equations - exponential function for reliability and for unreliability, R(t) = 1- Q(t) = exp(-λt)
Reliability computation - time varying failure rate
Waibull distribution z(t) = αλ(λt)**(α-1)
solve the equations - exponential function for reliability and for unreliability
Failure rate computation - military standard
function of - learning factor, quality factor, temperature factor, environmental factor, and # of pins on IC
Be sure everyone has a conduct sheet.
Re GROUND RULES:
1. In terms of allowed collaboration vs. individual work, ask if you are not sure.
2. Deactivate all cell phones or pagers during class unless you are on-call during your job.
3. No tape-recording permitted. Take notes.
ECE 753 Fault Tolerant Computing

9. ECE 753 Fault Tolerant Computing
5/15/2018
Introduction (contd.)
Mathematical formulation (contd.)
Reliability computation - mean time to failure (MTTF)
Definition: expected time that a system will operate before the first failure occurs
Probability measure: S-sample space, E-event space
for A in E P(A) >= 0
P(S) = 1
P(AB) = P(A) + P(B), when A and B are non-intersecting
Random Variable (RV) - X maps events of S to real-numbers
Probability distribution function of a RV
Probability density function (pdf) - derivative of the distribution function
Be sure everyone has a conduct sheet.
Re GROUND RULES:
1. In terms of allowed collaboration vs. individual work, ask if you are not sure.
2. Deactivate all cell phones or pagers during class unless you are on-call during your job.
3. No tape-recording permitted. Take notes.
ECE 753 Fault Tolerant Computing

10. ECE 753 Fault Tolerant Computing
5/15/2018
Introduction (contd.)
Mathematical formulation (contd.)
Reliability computation - mean time to failure
Probability density function - properties
always >= 0
integrates to 1 (between limits)
Expectation
Integrate xf(x)
Σ xi p(xi) in discrete case
Application in our case
unreliability Q(t) is a probability distribution function of failure - in fact it is cumulative probability that system fails in time [0,t]
Be sure everyone has a conduct sheet.
Re GROUND RULES:
1. In terms of allowed collaboration vs. individual work, ask if you are not sure.
2. Deactivate all cell phones or pagers during class unless you are on-call during your job.
3. No tape-recording permitted. Take notes.
ECE 753 Fault Tolerant Computing

11. ECE 753 Fault Tolerant Computing
5/15/2018
Introduction (contd.)
Mathematical formulation (contd.)
Reliability computation - MTTF and MTTR
Application in our case (contd.)
derivative of Q(t) , written as f(t), is pdf of failure - or failure density function
Expected value can be computed using integration and is Mean Time To Failure (MTTF)
constant failure rate
MTTF = 1/λ
Mean time to repair - MTTR
assume constant repair rate (μ) and arguments similar to those used for failure analysis and conclude MTTR = 1/ μ
Be sure everyone has a conduct sheet.
Re GROUND RULES:
1. In terms of allowed collaboration vs. individual work, ask if you are not sure.
2. Deactivate all cell phones or pagers during class unless you are on-call during your job.
3. No tape-recording permitted. Take notes.
ECE 753 Fault Tolerant Computing

12. ECE 753 Fault Tolerant Computing
5/15/2018
Introduction (contd.)
Mathematical formulation (contd.)
Reliability computation - mean time between failure (MTBF)
Mean time between failure - MTBF
use heuristic arguments to conclude
MTBF = (total time T)/(average number of failures)
can also argue MTBF = MTTF + MTTR
Note: often λ << μ and hence MTTF >> MTTR , therefore the words MTTF and MTBF are used interchangeably by some practioners
Be sure everyone has a conduct sheet.
Re GROUND RULES:
1. In terms of allowed collaboration vs. individual work, ask if you are not sure.
2. Deactivate all cell phones or pagers during class unless you are on-call during your job.
3. No tape-recording permitted. Take notes.
ECE 753 Fault Tolerant Computing

13. ECE 753 Fault Tolerant Computing
5/15/2018
Reliability Modeling
Application of the previous analysis to system models
Assumptions
system consists of modules
each module assigned a probability of working R(t), a function of time
once a module fails it is assumed to yield incorrect results
module failures are independent
Be sure everyone has a conduct sheet.
Re GROUND RULES:
1. In terms of allowed collaboration vs. individual work, ask if you are not sure.
2. Deactivate all cell phones or pagers during class unless you are on-call during your job.
3. No tape-recording permitted. Take notes.
ECE 753 Fault Tolerant Computing

14. ECE 753 Fault Tolerant Computing
5/15/2018
Reliability Modeling
Application of the previous analysis to system models
Reliability block diagrams
consider a system - microP, controller, mem, bus, …
the system will fail if any of the components fails
Rsys = P(all subsystems work correctly)
= P(bus correct).P(mem correct)…. Etc.
(follows from the assumption that component
failures are independent)
Rsys = Rbus.Rmem.Rmicro.Rcont
Be sure everyone has a conduct sheet.
Re GROUND RULES:
1. In terms of allowed collaboration vs. individual work, ask if you are not sure.
2. Deactivate all cell phones or pagers during class unless you are on-call during your job.
3. No tape-recording permitted. Take notes.
ECE 753 Fault Tolerant Computing

15. ECE 753 Fault Tolerant Computing
5/15/2018
Reliability Modeling
Reliability block diagrams - Series Systems
Assume system has n components
All components should survive for system to operate
Reliability of system
R sys = Pi Ri (t)
For exponential distributions of each component
R sys = Pi e - l i t = e - (l1 + l ln)t =exp(- Slit)
Effect is that the system failure rate is the summation of failure rates of components
Note these are nonredundant systems
Be sure everyone has a conduct sheet.
Re GROUND RULES:
1. In terms of allowed collaboration vs. individual work, ask if you are not sure.
2. Deactivate all cell phones or pagers during class unless you are on-call during your job.
3. No tape-recording permitted. Take notes. R1 R2 Rn
ECE 753 Fault Tolerant Computing

16. ECE 753 Fault Tolerant Computing
5/15/2018
Reliability Modeling
Reliability block diagrams - Parallel Systems
Assume system with spares
faulty component is replaced by a spare as fault occurs
only one component needs to survive for the system to operate
Model is to represent all components connected in parallel
P(sys fail) = P(M1 fails).P(M2 fails). .. .P(Mn fails)
Rsys = 1 - P(sys fail) = 1- (1-R1)(1-R2) …(1-Rn)
Be sure everyone has a conduct sheet.
Re GROUND RULES:
1. In terms of allowed collaboration vs. individual work, ask if you are not sure.
2. Deactivate all cell phones or pagers during class unless you are on-call during your job.
3. No tape-recording permitted. Take notes.
ECE 753 Fault Tolerant Computing

17. ECE 753 Fault Tolerant Computing
5/15/2018
Reliability Modeling
Reliability block diagrams - Series-Parallel Systems
straight forward
Reliability block diagrams - MTTF of system
1/(system failure rate)
Series systems - 1/(sum of individual falure rates)
Parallel systems and series parallel systems – work out by integration from the reliability or unreliability equations
Be sure everyone has a conduct sheet.
Re GROUND RULES:
1. In terms of allowed collaboration vs. individual work, ask if you are not sure.
2. Deactivate all cell phones or pagers during class unless you are on-call during your job.
3. No tape-recording permitted. Take notes.
ECE 753 Fault Tolerant Computing

18. ECE 753 Fault Tolerant Computing
5/15/2018
Reliability Modeling
Reliability block diagrams -Non series parallel systems
Bayes rule: consider a sample space S. Partitions this into space B andB (complement of B). Now consider an event that falls partly in B and partly inB. We can write:
A = (AB)(AB)
P(A) = P[(AB)(AB)]
= P[(AB)] + P[(AB)]
= P(A/B)P(B) + P(A/B)P(B)
In general the set S can be partitioned into (B1, B2, … ,Bn)
P(A) = Σ P(A/Bi)P(Bi)
This can be viewed graphically also (draw a tree)
Be sure everyone has a conduct sheet.
Re GROUND RULES:
1. In terms of allowed collaboration vs. individual work, ask if you are not sure.
2. Deactivate all cell phones or pagers during class unless you are on-call during your job.
3. No tape-recording permitted. Take notes.
ECE 753 Fault Tolerant Computing

19. ECE 753 Fault Tolerant Computing
5/15/2018
Reliability Modeling
Reliability block diagrams -Non series parallel systems
Example - consider the following non series parallel system
list all paths for system to survive, namely c1c4, c2c4, c2c5, c3c5
These paths are not disjoint, sum of reliabilities of all path gives an upper bound on the system reliability
Exact computation is possible using Bayes rule – complete in class C5 C4 C3 C2 C1
Be sure everyone has a conduct sheet.
Re GROUND RULES:
1. In terms of allowed collaboration vs. individual work, ask if you are not sure.
2. Deactivate all cell phones or pagers during class unless you are on-call during your job.
3. No tape-recording permitted. Take notes.
ECE 753 Fault Tolerant Computing

20. ECE 753 Fault Tolerant Computing
5/15/2018
Reliability Modeling
Combinatorial model
Consider an NMR system
Assume voter reliability to be 1
Divide all events for success to disjointed events
Compute probability of each event and add them
Example – TMR system
Can be used to compute MTTF
Can also analyze other systems such as an m-of-n system
Be sure everyone has a conduct sheet.
Re GROUND RULES:
1. In terms of allowed collaboration vs. individual work, ask if you are not sure.
2. Deactivate all cell phones or pagers during class unless you are on-call during your job.
3. No tape-recording permitted. Take notes.
ECE 753 Fault Tolerant Computing

21. ECE 753 Fault Tolerant Computing
5/15/2018
Reliability Modeling
Markov model
Difficulty with the previous models
incorporating repairs in the model and analysis
Incorporation of coverage factor – such as in duplicates system we may be less than 100% certain that only faulty unit will be eliminated when system is re-configured
Markov modeling - basic
Define the concept of state using TMR system example (8 states)
Transitions between states occur with certain probabilities
Markov model – assumption
Probability of transition from a state si to sj is independent of the method of arrival into state si
Example – develop a Markov model for a TMR in class
Be sure everyone has a conduct sheet.
Re GROUND RULES:
1. In terms of allowed collaboration vs. individual work, ask if you are not sure.
2. Deactivate all cell phones or pagers during class unless you are on-call during your job.
3. No tape-recording permitted. Take notes.
ECE 753 Fault Tolerant Computing

22. ECE 753 Fault Tolerant Computing
5/15/2018
Reliability Modeling
Markov model
Markov model for a TMR – all details not shown
011
001
λΔt
1-3λΔt
000
111
Be sure everyone has a conduct sheet.
Re GROUND RULES:
1. In terms of allowed collaboration vs. individual work, ask if you are not sure.
2. Deactivate all cell phones or pagers during class unless you are on-call during your job.
3. No tape-recording permitted. Take notes.
101
010
λΔt
λΔt
100
110
ECE 753 Fault Tolerant Computing

23. ECE 753 Fault Tolerant Computing
5/15/2018
Reliability Modeling
Markov model- Reduced
Reduced Markov model for a TMR system
Previous eight state model can be reduced to a three state model by merging states and re-computing the transition probabilities
Markov model- accounting for repairs
We can include links between states knowing the repair rates of components
Be sure everyone has a conduct sheet.
Re GROUND RULES:
1. In terms of allowed collaboration vs. individual work, ask if you are not sure.
2. Deactivate all cell phones or pagers during class unless you are on-call during your job.
3. No tape-recording permitted. Take notes.
ECE 753 Fault Tolerant Computing

24. ECE 753 Fault Tolerant Computing
5/15/2018
Reliability Modeling
Markov model- analyzing systems
Consider a duplicate compare system – no repairs
Develop Markov model with 3 states
Develop a difference equation for computing probabilities for being in different states of the system
Develop a differential equation model
Solution methods
Numerical approach
Solving differential equation
direct approach
Using Laplace transforms
Be sure everyone has a conduct sheet.
Re GROUND RULES:
1. In terms of allowed collaboration vs. individual work, ask if you are not sure.
2. Deactivate all cell phones or pagers during class unless you are on-call during your job.
3. No tape-recording permitted. Take notes.
ECE 753 Fault Tolerant Computing

25. ECE 753 Fault Tolerant Computing
5/15/2018
Reliability Modeling
Markov model- analyzing systems
Consider a duplicate compare system – with repairs
Develop Markov model with 3 states
Develop a differential equation model
Solve using Laplace transforms
Yet one more example
duplicate compare system – with imperfect coverage
Develop Markov model with 5 states
Reduce model for different scenarios
Be sure everyone has a conduct sheet.
Re GROUND RULES:
1. In terms of allowed collaboration vs. individual work, ask if you are not sure.
2. Deactivate all cell phones or pagers during class unless you are on-call during your job.
3. No tape-recording permitted. Take notes.
ECE 753 Fault Tolerant Computing

26. Other Parameters and analysis
5/15/2018
Other Parameters and analysis
Markov model- Can use other parameters
Safety –
Availability
Consider a simplex system
Develop Markov model with 2 states
Solve the system for probability of system being in available state
Define and compute steady state availability
Provide a intuitive explanation of the computed value of steady state availability and its relation of MTTF and MTTR
Maintainability
Be sure everyone has a conduct sheet.
Re GROUND RULES:
1. In terms of allowed collaboration vs. individual work, ask if you are not sure.
2. Deactivate all cell phones or pagers during class unless you are on-call during your job.
3. No tape-recording permitted. Take notes.
ECE 753 Fault Tolerant Computing

27. ECE 753 Fault Tolerant Computing
5/15/2018
General remarks
Voter reliability issue
Performance and states with degraded performance
Mission time improvement
Redundancy Ratio
Law of diminishing return
Be sure everyone has a conduct sheet.
Re GROUND RULES:
1. In terms of allowed collaboration vs. individual work, ask if you are not sure.
2. Deactivate all cell phones or pagers during class unless you are on-call during your job.
3. No tape-recording permitted. Take notes.
ECE 753 Fault Tolerant Computing

28. ECE 753 Fault Tolerant Computing
5/15/2018
Summary
Introduction of mathematical models
Solving models to carry out analysis
Example systems
Duplicate
Duplicate with repair
Simplex with repair for avialability
Be sure everyone has a conduct sheet.
Re GROUND RULES:
1. In terms of allowed collaboration vs. individual work, ask if you are not sure.
2. Deactivate all cell phones or pagers during class unless you are on-call during your job.
3. No tape-recording permitted. Take notes.
ECE 753 Fault Tolerant Computing