The Parametric Models for Common Cause Failure Analysis
Common Cause Failures (CCF) occur when multiple (usually
identical) components fail due to shared causes. Typical
examples of shared causes include impact, vibration, temperature,
contaminants, miscalibration and improper maintenance. In
the September 2010 issues of HotWire, the
Reliability Basics article explained how to implement
the Beta factor model using the duty cycle in
BlockSim. In this article, we will show how to
apply these models via mirror blocks in BlockSim.
The five parametric models for CCF are listed in Table
1. The Basic Parameter Model (BPM) is used to estimate the
basic event probabilities directly. The other four models
are each a reparameterization of the BPM. The main characteristics
of the models are summarized in the table. For more details,
please refer to [1].
Table 1  Main characteristics
of five popular parametric models for CFF
Model

Parameters

General Form for Multiple Component
Failure Frequency

Basic Parameter Model (BPM)

where is the probability of
a basic event involving k specific components in
a common cause component group of size m.


Beta Factor

Q_{t }, β
where:
 Q_{t } is the total probability of each component
failing due to all independent and common cause
events.
 β is a constant fraction of the component
failure probability that can be associated with common
cause events shared by other components in a
common cause component group.


Multiple Greek Letters (MGL)

Q_{t }, β, γ, ...
where:
 Q_{t } is the total probability of each component
failing due to all independent and common cause
events.
 β is the conditional
probability that the cause of a component
failure will be shared by one or more
additional components, given that a specific
component has failed.
 γ is the conditional probability that the
cause of a component failure that is shared
by one or more components will be shared by
two or more components, given that two specific
components have failed.


Alpha Factor

Q_{t }, α_{1}, α_{2}, α_{3}, ..., α_{m}
where:
 Q_{t } is the total probability of each component
failing due to all independent and common cause
events.
 α_{k} is the probability that when a common
cause basic event occurs in a common cause group
of size m, it involves the failure of k
components.

Nonstaggered testing (all components
tested simultaneously):
Staggered testing (components
tested sequentially):
where:

Binomial Failure Rate (BFR)

Q_{t }, μ, ρ, ω
where:
 Q_{t } is the independent failure frequency
for each component.
 μ is the frequency of occurrence of nonlethal
shocks.
 ρ is the conditional probability of failure
of each component, given a nonlethal shock.
 ω is the frequency of occurrence of lethal
shocks.


From Table 1, we can see that the only difference between
these five models is the way in which they calculate the
common factor failure probability,
.
Once we have , it is easy to model the
system reliability via mirror blocks in BlockSim.
Mirror Blocks
Mirror blocks allow you to place duplicates of an original
block in multiple locations within a reliability block diagram
(RBD) or fault tree. This can be useful for many purposes,
such as modeling
bidirectional paths or common cause failures. The duplicate
block behaves in the same way as the original block. In
BlockSim, a gray box will appear in the upper left
corner of the block to indicate that the block is a mirror
block, as shown next.
Example
Let’s consider a system with three identical components
A, B and C which form a common cause component group of
size 3 (m = 3). The success criterion we will consider
is that one of the three components must function. Thus,
for the system to fail, all three components must fail.
We will initially use the BPM and assume we can obtain
the different failure probabilities
directly.
We will then illustrate how the Alpha Factor parameterization
of the BPM would be used. The other methods can be used
in a similar way.
Using the BPM, we define the total failure of component
A with the following equation:
where:
 A_{t }is the total failure of component
A from any cause.
 A_{I }is the failure of component A from
independent causes.
 S_{AB }is the failure of A and B from
common causes.
 S_{AC }is the failure of A and C from
common causes.
 S_{ABC }is the failure of A, B and C
from common causes.
Similarly, we define total failure of components B and
C by the following equations:
The terms in these equations are defined analogously
to the ones for component A.
Table 2 lists the failure probabilities and the CCF parameters
for the Alpha Factor model.
Table 2  Parameters
By using the staggered testing equations for the Alpha
Factor model in Table 1, we can calculate
, and
as follows:
Table 3 lists the basic event failure probabilities and
reliabilities.
Table 3  Basic event probabilities
The fault tree in BlockSim is shown in Figure
1. Note that event S_AB under "B_Fails" is a duplicate of
S_AB under "A Fails." Also, the events named S_ABC under
"B Fails" and "C Fails" are duplicates of S_ABC under "A
Fails." Events S_AC and S_BC follow a similar logic.
Figure 1  Fault tree for a system with three components
Now we can obtain the reliability of the system using
the Quick Calculation Pad, as shown in Figure 2. The system
reliability is 0.9995.
Figure 2  QCP results
Conclusion
This article briefly reviewed five parametric models
for common cause failure analysis and illustrated how one
of these models can be applied in BlockSim. After
parameterization of the BPM using any one of the other four
models, it is very easy to model the common cause failure
via mirror blocks in BlockSim. Note that while these
models make simplifying assumptions (for example, that all
components are identical), using mirror blocks does not
always require these assumptions.[2]
References
[1] A. Mosleh, K. Fleming, G. Parry,
H. Paula, D. Worledge, and D. Rasmuson, "Procedures for
treating common cause failure," Safety and Reliability
Studies, NUREG/CR4780 EPRI NP5613, vol. 1, Jan. 1988.
[2] ReliaSoft Corporation. (2005,
Aug.). "Treating common cause failure in fault trees,"
Reliability HotWire. 2005. Available:
http://www.weibull.com/hotwire/issue54/relbasics54.htm
