1. Dependability & Maintainability Theory and Methods Part 1: Introduction and definitions
Andrea Bobbio
Dipartimento di Informatica
Università del Piemonte Orientale, “A. Avogadro”
15100 Alessandria (Italy)
IFOA, Reggio Emilia, June 17-18, 2003
A. Bobbio
Reggio Emilia, June 17-18, 2003

2. Dependability: Definition
Dependability is the property of a system to be dependable in time, i.e. such that reliance can justifiably be placed on the service it delivers.
Dependability extends the interest on the system from the design and construction phase to the operational phase (life cycle).
A. Bobbio
Reggio Emilia, June 17-18, 2003

3. What dependability theory and practice wants to avoid A. Bobbio
Reggio Emilia, June 17-18, 2003

4. Dependability: Taxonomy
reliability
availability
maintainability
safety
security
measures
means
fault forecasting
fault tolerance
fault removal
fault prevention
dependability
threats
faults
errors
failures
A. Bobbio
Reggio Emilia, June 17-18, 2003

5. Quantitative analysis
The quantitative analysis aims at numerically evaluating measures to characterize the dependability of an item:
Risk assessment and safety
Design specifications
Technical assistance and maintenance
Life cycle cost
Market competition
A. Bobbio
Reggio Emilia, June 17-18, 2003

6. Risk assessment and safety
The risk associated to an activity is given proportional to the probability of occurrence of the activity and to the magnitute of the consequences.
R = P  M
A safety critical system is a system whose incorrect behavior may cause a risk to occur, causing undesirable consequences to the item, to the operators, to the population, to the environment.
A. Bobbio
Reggio Emilia, June 17-18, 2003

7. Design specifications
Technological items must be dependable.
Some times, dependability requirements (both qualitative and quantitative) are part of the design specifications:
Mean time between failures
Total down time
A. Bobbio
Reggio Emilia, June 17-18, 2003

8. Technical assistance and maintenance
The planning of all the activity related to the technical assistance and maintenance is linked to the system dependability (expected number of failure in time).
planning spare parts and maintenance crews;
cost of the technical assistance (warranty period);
preventive vs reactive maintenance.
A. Bobbio
Reggio Emilia, June 17-18, 2003

9. Market competition
The choice of the consumers is strongly influenced by the perceived dependability.
advertisement messages stress the dependability;
the image of a product or of a brand may depend on the dependability.
A. Bobbio
Reggio Emilia, June 17-18, 2003

10. Purpose of evaluation Observation Operational environment Reasoning
Understanding a system
Observation
Operational environment
Reasoning
Predicting the behavior of a system
Need a model
A model is a convenient abstraction
Accuracy based on degree of extrapolation
A. Bobbio
Reggio Emilia, June 17-18, 2003

11. Methods of evaluation Measurement-Based
Most believable, most expensive
Not always possible or cost effective during system design
Model-Based
Less believable, Less expensive
Analytic vs Discrete-Event Simulation
Combinatorial vs State-Space Methods
A. Bobbio
Reggio Emilia, June 17-18, 2003

12. Measurement-Based Most believable, most expensive;
Data are obtained observing the behavior of physical objects.
field observations;
measurements on prototypes;
measurements on components (accelerated tests).
A. Bobbio
Reggio Emilia, June 17-18, 2003

13. Models Closed-form Answers Numerical Solution Analytic Simulation
All models are wrong; some models are useful
A. Bobbio
Reggio Emilia, June 17-18, 2003

14. Methods of evaluation Measurements + Models data bank A. Bobbio
Reggio Emilia, June 17-18, 2003

15. The probabilistic approach
The mechanisms that lead to failure a technological object are very complex and depend on many physical, chemical, technical, human, environmental … factors.
The time to failure cannot be expressed by a determin-istic law.
We are forced to assume the time to failure as a random variable.
The quantitative dependability analysis is based on a probabilistic approach.
A. Bobbio
Reggio Emilia, June 17-18, 2003

16. Reliability
The reliability is a measurable attribute of the dependability and it is defined as:
The reliability R(t) of an item at time t is the probability that the item performs the required function in the interval (0 – t) given the stress and environmental conditions in which it operates.
A. Bobbio
Reggio Emilia, June 17-18, 2003

17. Basic Definitions: cdf
Let X be the random variable representing the time to failure of an item.
The cumulative distribution function (cdf) F(t) of the r.v. X is given by:
F(t) = Pr { X  t }
F(t) represents the probability that the item is already failed at time t (unreliability) .
A. Bobbio
Reggio Emilia, June 17-18, 2003

18. Basic Definitions: cdf
Equivalent terminoloy for F(t) :
CDF (cumulative distribution function)
Probability distribution function
Distribution function
A. Bobbio
Reggio Emilia, June 17-18, 2003

19. Basic Definitions: cdf
F(t) 1
F(b)
F(a) a b t
F(0) = 0
lim F(t) = 1
t
F(t) = non-decreasing
A. Bobbio
Reggio Emilia, June 17-18, 2003

20. Basic Definitions: Reliability
Let X be the random variable representing the time to failure of an item.
The survivor function (sf) R(t) of the r.v. X is given by:
R (t) = Pr { X > t } = 1 - F(t)
R(t) represents the probability that the item is correctly working at time t and gives the reliability function .
A. Bobbio
Reggio Emilia, June 17-18, 2003

21. Basic Definitions Equivalent terminology for R(t) = 1 -F(t) :
Reliability
Complementary distribution function
Survivor function
A. Bobbio
Reggio Emilia, June 17-18, 2003

22. Basic Definitions: Reliability
R(t) 1
R(a) a b t
R(0) = 1
lim R(t) = 0
t
R(t) = non-increasing
A. Bobbio
Reggio Emilia, June 17-18, 2003

23. Basic Definitions: density
Let X be the random variable representing the time to failure of an item and let F(t) be a derivable cdf:
The density function f(t) is defined as:
d F(t)
f (t) = ——— dt
f (t) dt = Pr { t  X < t + dt }
A. Bobbio
Reggio Emilia, June 17-18, 2003

24. Basic Definitions: Density
f (t) t a b b
 f(x) dx = Pr { a < X  b } = F(b) – F(a) a
A. Bobbio
Reggio Emilia, June 17-18, 2003

25. Basic Definitions: Density
f (t) 1 t
A. Bobbio
Reggio Emilia, June 17-18, 2003

26. Basic Definitions Equivalent terminology: pdf
probability density function
density function
density
f(t) =
For a non-negative
random variable
A. Bobbio
Reggio Emilia, June 17-18, 2003

27. Quiz 1: The higher the MTTF is, the higher the item reliability is.
Correct
Wrong
The correct answer is wrong !!!
A. Bobbio
Reggio Emilia, June 17-18, 2003

28. Hazard (failure) rate h(t) t = Conditional Prob. system will fail in
(t, t + t) given that it is survived until time t
f(t) t = Unconditional Prob. System will fail in
(t, t + t)
A. Bobbio
Reggio Emilia, June 17-18, 2003

29. The Failure Rate of a Distribution
is the conditional probability that the unit will fail in the interval given that it is functioning at time t.
is the unconditional probability that the unit will fail in the interval
Difference between the two sentences:
probability that someone will die between 90 and 91, given that he lives to 90
probability that someone will die between 90 and 91
A. Bobbio
Reggio Emilia, June 17-18, 2003

30. (infant mortality – burn in)
Bathtub curve
h(t)
(infant mortality – burn in)
(wear-out-phase)
CFR
Constant fail. rate
(useful life)
DFR
IFR t
Decreasing failure rate
Increasing fail. rate
A. Bobbio
Reggio Emilia, June 17-18, 2003

31. Infant mortality (dfr)
Also called infant mortality phase or reliability growth phase. The failure rate decreases with time.
Caused by undetected hardware/software defects;
Can cause significant prediction errors if steady-state failure rates are used;
Weibull Model can be used;
A. Bobbio
Reggio Emilia, June 17-18, 2003

32. Useful life (cfr)
The failure rate remains constant in time (age independent) .
Failure rate much lower than in early-life period.
Failure caused by random effects (as environmental shocks).
A. Bobbio
Reggio Emilia, June 17-18, 2003

33. Wear-out phase (ifr) The failure rate increases with age.
It is characteristic of irreversible aging phenomena (deterioration, wear-out, fatigue, corrosion etc…)
Applicable for mechanical and other systems.
(Properly qualified electronic parts do not exhibit wear-out failure during its intended service life)
Weibull Failure Model can be used
A. Bobbio
Reggio Emilia, June 17-18, 2003

34. Exponential Distribution
Failure rate is age-independent (constant).
Cumul. distribution function:
Reliability :
Density Function :
Failure Rate (CFR):
Mean Time to Failure:
A. Bobbio
Reggio Emilia, June 17-18, 2003

35. The Cumulative Distribution Function of an Exponentially Distributed Random Variable With Parameter  = 1
F(t)
1.0
F(t) = 1 - e
-  t
0.5
1.25
2.50
3.75
5.00 t
A. Bobbio
Reggio Emilia, June 17-18, 2003

36. The Reliability Function of an Exponentially Distributed Random Variable With Parameter  = 1
R(t)
1.0
R(t) = e
-  t
0.5
1.25
2.50
3.75
5.00 t
A. Bobbio
Reggio Emilia, June 17-18, 2003

37. Exponential Density Function (pdf)
f(t)
MTTF = 1/ 
A. Bobbio
Reggio Emilia, June 17-18, 2003

38. Memoryless Property of the Exponential Distribution
Assume X > t. We have observed that the component has not failed until time t
Let Y = X - t , the remaining (residual) lifetime
A. Bobbio
Reggio Emilia, June 17-18, 2003

39. Memoryless Property of the Exponential Distribution (cont.)
Thus Gt(y) is independent of t and is identical to the original exponential distribution of X
The distribution of the remaining life does not depend on how long the component has been operating
An observed failure is the result of some suddenly appearing failure, not due to gradual deterioration
A. Bobbio
Reggio Emilia, June 17-18, 2003

40. Quiz 3: If two components (say, A and B) have independent identical exponentially distributed times to failure, by the “memoryless” property, which of the following is true?
1. They will always fail at the same time
2. They have the same probability of failing at time ‘t’ during operation
3. When these two components are operating simultaneously, the component which has been operational for a shorter duration of time will survive longer
A. Bobbio
Reggio Emilia, June 17-18, 2003

41. Weibull Distribution Distribution Function: Density Function:
Reliability:
A. Bobbio
Reggio Emilia, June 17-18, 2003

42. Weibull Distribution Dfr Cfr Ifr  : shape parameter;
 : scale parameter.
Failure Rate:
Dfr
Cfr
Ifr
A. Bobbio
Reggio Emilia, June 17-18, 2003

43. Failure Rate of the Weibull Distribution with Various Values of 
A. Bobbio
Reggio Emilia, June 17-18, 2003

44. Weibull Distribution for Various Values of 
Cdf
density
A. Bobbio
Reggio Emilia, June 17-18, 2003

45. Failure-Rate Multiplier
Failure Rate Models
We use a truncated Weibull Model
Infant mortality phase modeled by DFR Weibull and the steady-state phase by the exponential
Figure 2.34 Weibull Failure-Rate Model 7 6 5 4 3 2 1
Failure-Rate Multiplier
2,190
4,380
6,570
8,760
10,950
13,140
15,330
17,520
Operating Times (hrs)
A. Bobbio
Reggio Emilia, June 17-18, 2003

46. Failure Rate Models (cont.)
This model has the form:
where:
steady-state failure rate
is Weibull shape parameter
Failure rate multiplier =
A. Bobbio
Reggio Emilia, June 17-18, 2003

47. Failure Rate Models (cont.)
There are several ways to incorporate time dependent failure rates in availability models
The easiest way is to approximate a continuous function by a piecewise constant step function 7 6 5 4 3 2 1
Discrete Failure-Rate Model
Failure-Rate Multiplier
2,190
4,380
6,570
8,760
10,950
13,140
15,330
17,520
Operating Times (hrs)
A. Bobbio
Reggio Emilia, June 17-18, 2003

48. Failure Rate Models (cont.)
Here the discrete failure-rate model is defined by:
A. Bobbio
Reggio Emilia, June 17-18, 2003

49. A lifetime experiment
X 1 1
X 2 2
X 3 3
X 4 4
X N N
t = 0
N i.i.d components are put in a life test experiment.
A. Bobbio
Reggio Emilia, June 17-18, 2003

50. A lifetime experiment 1 X 1 2 X 2 3 X 3 4 X 4 X N N A. Bobbio
Reggio Emilia, June 17-18, 2003