FV3102 Probabilistic Risk Analysis
Part A (55 Marks) (Learning Outcome 4)
A.1 Test is performed to estimate the failure rate of a new designed component. Ten
identical components were tested until they either fail or reach 1000 hours. The test
data are shown in the following table.
a) Based on the empirical method, calculate the reliability 𝑅(𝑡ⅈ), failure probability
), failure density 𝑓(𝑡𝑖
) and hazard rate ℎ(𝑡𝑖
) in each 100-hour period;
b) Plot the ‘bath-tub’ diagram using the hazard rate and estimate the duration of burnin period, useful life period and time at which burn-out period starts.
A.2 A system contains two subsystems in series. System 1 consists of four identical
components and for this particular group it is necessary that two out of the four
components functions satisfactorily for system success. System 2 has four identical
components in parallel and system success requires that at least three of these
components must function.
a) Present a Reliability Block Diagram to represent the system.
b) Assume each component has the same reliability in terms of 𝑅, derive a general
expression for the unreliability of the system (i.e. Failure probability of the
c) Assume each component has an identical failure rate of λ = 0.0002 per hour.
Estimate the reliability of the system at 2000-hours operation time.
A.3 Components Following please find three designed systems which consisted of six
Assume the identical components are independently and exponential distributed with
a constant failure rate 0.02 per hours. For each system, derive the average failure rate
and mean time to failure (MTTF) of the system.
A.4 A system is defined to have three states: (a) working; (b) under repair; (c) waiting for
a new task.
a) Suppose that if the system was working yesterday, today the probability to break is
0.1 and the probability to go to waiting is 0.2; if the system was under repair
yesterday, then today the system will get repaired and become waiting state with the
probability of 0.1. A broken system will never be brought directly to work in one
step. If the system was waiting yesterday, there will be 0.9 probability to get into
working. A waiting system will never break directly. Describe the system as a
Markov process and find out the probability of the system at working status after a
b) Consider the same system in continuous time. Assume the state transition takes an
exponential amount of time with constant rate. Suppose the failure rate (i.e. rate from
working to repair) is 0.02 per day, the rate from waiting to operating is 1 per day, the
rate from operating to waiting is 0.1 per day and the repair rate (i.e. rate from repair to
waiting) is denoted as r. Describe the system as a Markov Chain and find out the repair
rate r so that the system in steady state will be in the repair state less than 5% of the
A.5 Consider the pumping system shown in the figure below. The purpose of the system
is to pump water from point A to point B. The time to failure of all the valves and
pump can be represented by the exponential distributions with failure rate 𝜆𝑣 and 𝜆𝑝
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