Oct. 2007 Terminology, Models, and Measures Slide 1
Fault-Tolerant Computing
Basic Concepts
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Terminology, Models, and Measures for Fault-Tolerant Computing - Prof. B. Parhami, Assignments of Electrical and Electronics Engineering
This presentation covers the terminology, models, and measures related to dependability in fault-tolerant computing. It includes concepts such as impairments to dependability, the fault-error-failure cycle, the four-universe model, and multilevel models. The document also discusses the importance of dependability and various types of dependable computer systems.
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Oct. 2007
Terminology, Models, and Measures
Slide 1
Fault-Tolerant Computing^ Basic Conceptsand Tools
Oct. 2007
Terminology, Models, and Measures
Slide 2
About This Presentation
Edition
Released
Revised
Revised
First
Oct. 2006
Oct. 2007
This presentation has been prepared for the graduatecourse ECE 257A (Fault-Tolerant Computing) byBehrooz Parhami, Professor of Electrical and ComputerEngineering at University of California, Santa Barbara.The material contained herein can be used freely inclassroom teaching or any other educational setting.Unauthorized uses are prohibited. © Behrooz Parhami
Oct. 2007
Terminology, Models, and Measures
Slide 4
Oct. 2007
Terminology, Models, and Measures
Slide 5
Impairments to Dependability
ERROR
Malfunction
Degradation
Failure
Fault
Intrusion
Hazard
Defect
Flaw
Bug
Crash
Oct. 2007
Terminology, Models, and Measures
Slide 7
The Four-Universe Model
Cause-effect diagram for Avižienis’ four-universemodel of impairments to dependability.
Universe
Impairment
Physical
Failure
Logical
Fault
Informational
Error
External
Crash
Oct. 2007
Terminology, Models, and Measures
Slide 8
Unrolling the Fault-Error-Failure Cycle
Cause-effect diagram for an extended six-levelview of impairments to dependability.
Abstraction
Impairment
Component
Defect
⇓
⇓
Logic
Fault
⇓
⇓
Information
Error
⇓
⇓
System
Malfunction
⇓
⇓
Service
Degradation
⇓
⇓
Result
Failure
Low-Level Mid-Level High-Level
FirstCycle SecondCycle
Failure
Aspect
Impairment
Structure
Fault
⇓^
⇓
State
Error
⇓^
⇓
Behavior
Oct. 2007
Terminology, Models, and Measures
Slide 10
Analogy for the Multilevel Model
An analogy for ourmulti-level model ofdependable computing.Defects, faults, errors,malfunctions,degradations, andfailures are represented by pouringwater from above.Valves representavoidance andtolerance techniques.The goal is to avoidoverflow.
Wall heights represent
inter-level latencies
Drain valves representtolerance techniques
Concentric reservoirs areanalogs of the six model levels,with defect being innermost
I I I I I
I^ I
I^
I^
I^
I^
I
Inlet valves representavoidance techniques
Oct. 2007
Terminology, Models, and Measures
Slide 11
Why Our Concern with Dependability?
Reliability of
n
-transistor system, each having failure rate
λ
R
( t
e
- n
λ t
There are only 3 ways of making systems more reliable Reduce
Reduce
n
1.0 0.8 0.6 0.4 0.2 0.
–ne
tλ
.
.
.
.
.
10 10
8 10
6 10
4 10
nt
Reduce
t
Alternative: Change the reliabilityformula by introducingredundancy in system
Oct. 2007
Terminology, Models, and Measures
Slide 13
Aspects of Dependability
RELIABILITY
Maintainability
Availability
Performability
Security Integrity
Serviceabilit
y
Testability
Safety
Robustness
Resilience
Reliability, MTTF = MTFF
Risk, conseque
nce
Controllability,
observability
Performability, MCBF
Pointwise av., Interval av., MTBF, MTTR
Oct. 2007
Terminology, Models, and Measures
Slide 14
Concepts from Probability Theory
Cumulative distribution function: CDF F
( t
) = prob[
x ≤
t
] =
∫
0
f (
x )^ dx
t
Probability density function: pdf f (
t ) = prob[
t^
≤
x
≤
t
dt
] /
dt
=
dF
( t
) /
dt
Time
0
10
20
30
40
50
Time
0
10
20
30
40
50
Time
0
10
20
30
40
50
1.00.80.60.40.20.
CDF pdf
0.050.040.030.020.010.
F(t)
f(t)
Expected value of
x
E
x^
=
∫
−∞
x f
( x
)^ dx
=
∑
k^
x
fk ( x
) k
+∞
Covariance of
x
and
y
ψ
x , y^
=
E
[(
x
E
)( x
y
E
)] y
=
E
[ x y
] –
E
x^ E
y
Variance of
x
σ
x^
=
∫
−∞
( x
E
) x 2 f (
x )^ dx
=
∑
k^
( x
k^
E
) x 2
f (
x
) k
+∞
2
Lifetimes of 20 identical systems
Oct. 2007
Terminology, Models, and Measures
Slide 16
Reliability and MTTF
Reliability:
R
( t
)
Probability that system remains in the“Good” state through the interval [0,
t
]
Two-statenonrepairablesystem
R
( t
dt
) =
R
( t
) [1 –
z
( t
)^ dt
]
Hazard function
Constant hazard function
z
( t
) =
λ ⇒
R
( t
) =
e
- λ
t
R (system failure rate is independent of its age)
( t
) = 1 –
F
( t
)^
CDF of the system lifetime, or its unreliability
Exponentialreliability law
Mean time to failure: MTTF MTTF =
∫
0
t f
( t
)^ dt
=
∫
0
R
( t
)^ dt
+∞
+∞
Expected value of lifetime
Area under the reliability curve(easily provable)
Startstate
Failure
Up
Down
Oct. 2007
Terminology, Models, and Measures
Slide 17
Failure Distributions of Interest
Exponential:
z
( t
) =
λ
R
( t
) =
e
- λ
t^
MTTF = 1/
λ
Weibull:
z
( t
) =
αλ
( λ
t )
α
-
R
( t
) =
e
(−λ
α t )
MTTF = (1/
λ
)^
Γ
(1 + 1/
α
)
Erlang:
MTTF =
k
/λ
Rayleigh: Gamma: Erlang and exponential are special cases Normal: Reliability and MTTF formulas are complicated
z
( t
) = 2
λ
( λ
t )
R
( t
) =
e
(−λ
(^2) t )
MTTF = (1/
λ
)^
√π / 2
Discrete versions Geometric^ R Discrete Weibull Binomial
( k
) =
q
k
Oct. 2007
Terminology, Models, and Measures
Slide 19
Availability, MTTR, and MTBF
(Interval) Availability:
A
( t
)
Fraction of time that system is in the“Up” state during the interval [0,
t
]
Two-staterepairablesystem
Availability = Reliability, when there is no repair Availability is a function not only of how rarely a system fails (reliability)but also of how quickly it can be repaired (time to repair)
MTTF
MTTF
μ
MTTF + MTTR
MTBF
λ
μ
Pointwise availability:
a
( t
)
Probability that system available at time
t
A
( t
) = (1/
t )
∫
0
a
( x
)^ dx
t
Steady-state availability:
A =
lim
t →∞
A
( t
)
A
=
=
=
Repair rate1/
μ
= MTTR (Will justify thisequation later)
In general,
μ
λ
, leading to
A
≅
1
Repair
Startstate
Failure
Up
Down
Oct. 2007
Terminology, Models, and Measures
Slide 20
System Up and Down Times
Time
Up Down
0
t
Time to first failure
Time between failures
Repair time t^1
t^2
t'^1
t'^2
Short repair time impliesgood
maintainability
(
serviceability
)
Repair
Startstate
Failure
Up
Down