1. Fault-Tolerant Computing Systems #4 Reliability and Availability
Pattara Leelaprute
Computer Engineering Department
Kasetsart University

2. Reliability and Availability
The probability that a system survives till time t (it has not fail till t)
Availability
The probability that a system works properly at time t

3. Preliminaries of Probability
Discrete sample space:
Tossing a coin
{head, tail} sample space
Continuous sample space:
How long the pc stays up after reboot
{t | t>0} sample space
Random variable
A function mapping each element of sample space to a real number
Ex. heads=1, tails=0

4. Preliminaries Random variable CDF (Cumulative distributed function)
A function mapping each element of sample space to a real number
CDF (Cumulative distributed function)
FX (t) = Pr [X ≤ t]
Pr : probability that the system has gone down by time t
Pdf (Probability density function)
f(t) = dF(t) / dx
Expected Value, Mean
E[X] = 0 t f(t)dt (X≥0)
Average outcome of the random experiment expect value, mean of a random variable

5. Exponential Distribution
The most commonly used distribute function in reliability modeling.
CDF
F(t) = 1 – e-lt
pdf
f(t) = l e-lt
Mean
1/l
Memoryless property
Y = X – t
Gt(y) = Pr [Y ≤ y | X > t ] = 1 – e-ly
Distribute of remaining life of a component does not depend on how long it has been working.
The component does not AGE !
(remaining life of X does not depend on the time that has passed)
f(t) = 2e-2t
F(t) = 1 – e-2t

6. Reliability Reliability
The probability that a system survives till time t
R(t) = Pr [X > t]
= 1 – F(t)
X : Random probability variable X which represents a time to failure of the system (the life of the system)
R(t): represents probability that the system survives till time t
F(t) = exponential Distribution
F(t) = 1 – e-2t
R(t) = e-2t t
time 0 X
time t

7. Reliability Reliability R(t) = Pr [X > t] = 1 – F(t) t time 0
The system is initially working
R() = 0
No system has infinite lifetime
F(t) = exponential Distribution
R(t) = reliability
F(t) = 1 – e-2t
R(t) = e-2t t
time 0 X
time t

8. Failure Rate
Probability that fault will occur in an interval time [t, t+Dt] =
f(t)Dt
Probability that fault will occur in time [t, t+Dt]
f(t)Dt / R(t)
Probability of occurrence of fault at time [t, t+Dt], when the system is working properly at t
Failure Rate
f(t) / R(t)
f(t) = probability of fault
F(t) = exponential Distribution
R(t) = reliability
f(t) = 2e-2t
R(t) = e-2t
F(t) = 1 – e-2t
[t, t+Dt]

9. Bathtub Curve Failure Rate Bathtub Curve
f(t) / R(t)
Bathtub Curve
General Failure Rate observed from the empirical data collected from mechanical and electronic component
When lifetime of a system F(t) is exponential distribution，it has a constant Failure Rate (see previous slide)
2.constant failure rate
1.Initial stage:
Inherit defects
faulty design
3.last stage:
faults caused by age

10. MTTF (Mean Time To Failure)
E[X] = 0 t f(t)dt = 0 R(t)dt
X: the Expected value of the probability variable which represents time till fault occurs in the system
When R(t) = e-lt (X is exponential distribution)
Failure Rate = l
MTTF = 1 / l
time 0
expected value

11. Availability The probability that a system works properly at time t
Availability is a measure that is frequently used for describing the behavior of the system
*If the system has no repair or replacement, availability is equal to reliability R(t)
R(t): the probability that no failures have occurred during the whole period (0,t)
fails
repairs
fails
repairs
Operational
Under repair
Operational t Xi
Xi+1
Xi+2 Ui
Ui+1

12. Availability Instantaneous availability (ทันทีทันใด)
A(t) = Pr [probability that the component is functioning correctly at t ]
Steady-State Availability (general meaning)
A = limt→∞ A(t)
fails
repairs
fails
repairs t Xi
Xi+1
Xi+2 Ui
Ui+1

13. Availability When Xi, Ui is exponential distribution
FXi(t) = 1 – e-lt, FUi(t) = 1 – e-mt
Instantaneous Availability
A(t) = (m + le-（l + m)t ) /(m + l)
Steady-State Availability
A = limt→∞ A(t) = m /(m + l) t Xi
Xi+1
Xi+2 Ui
Ui+1

14. MTTR (Mean Time To Repair)
MTTR = E [ Ui ]
Ui : the random variable that represents the downtime for i th repair or replacement
E[Ui] : the Expected value of Ui
MTTF (mean time to failure)
MTTF = E [ Xi ]
Xi : the random variable that represents the duration of the i th function period.
E[Xi] : the Expected value of Xi
Steady-State Availability
A = MTTF / (MTTF+MTTR)
= m /(m + l) 　
(Xi,Ui is the exponential distribution of parameter l,m) t Xi
Xi+1
Xi+2 Ui
Ui+1