ECE 753: FAULTTOLERANT COMPUTING

1
ECE 753 FAULT-TOLERANT COMPUTING

• Kewal K.Saluja
• Department of Electrical and Computer Engineering
• Reliability Modeling and Analysis

2
Overview

• Introduction
• Reliability Modeling
• reliability block diagram
• combinatorial model
• Markov model
• Other Parameters and analysis
• General remarks and Summary

3
Introduction

• References
• prad96, swew99, shooman02
• triv82 and triv01
• Books in the first line (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

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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

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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

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Introduction (contd.)

• Mathematical formulation (contd.)
• Also
• failure rate at time t
• (instantaneous failure rate at time t) / N0(t)
• (1/N0(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

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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(-?t)
• Reliability computation - time varying failure
  rate
• Waibull distribution z(t) a?(?t)(a-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

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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) gt 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

9
Introduction (contd.)

• Mathematical formulation (contd.)
• Reliability computation - mean time to failure
• Probability density function - properties
• always gt 0
• integrates to 1 (between limits)
• Expectation
• Integrate xf(x)
• S 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

10
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/?
• Mean time to repair - MTTR
• assume constant repair rate (µ) and arguments
  similar to those used for failure analysis and
  conclude MTTR 1/ µ

11
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 ? ltlt µ and hence MTTF gtgt MTTR ,
  therefore the words MTTF and MTBF are used
  interchangeably by some practioners

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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

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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

14
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 l2 . . .
  ln)t exp(- Slit)
• Effect is that the system failure rate is the
  summation of failure rates of components
• Note these are nonredundant systems

R1
R2
Rn
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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)

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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

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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) S P(A/Bi)P(Bi)
• This can be viewed graphically also (draw a
  tree)

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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

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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

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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

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Reliability Modeling

• Markov model
• Markov model for a TMR all details not shown

011
001
??t
1-3??t
000
111
101
010
??t
??t
100
110
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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

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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

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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

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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

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General remarks

• Voter reliability issue
• Performance and states with degraded performance
• Mission time improvement
• Redundancy Ratio
• Law of diminishing return

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Summary

• Introduction of mathematical models
• Solving models to carry out analysis
• Example systems
• Duplicate
• Duplicate with repair
• Simplex with repair for avialability