Determining Reliability for Complex Systems
Part 2 - Simulation
In
the previous issue, we discussed methods
to analytically determine the
reliability of a complex system. While the analytical method has a number
of advantages, such as being able to determine the pdf or the
failure rate for the entire system, there are also some drawbacks. A major
disadvantage of analytical analysis of complex systems is the complexity
of the solutions. Calculating the analytical reliability solution for a
sizable complex system may tax the resources of even the most powerful PC.
In situations such as this, it may be more advantageous to use simulation
to determine the complex system's reliability. This article discusses the
methodology used by ReliaSoft's BlockSim to simulate system reliability. (NOTE:
you may want to download a free evaluation version of
BlockSim in order to perform some of the following examples.)
A complex system is one that cannot
be broken down into groups of series and parallel components. In many cases it is not easy to recognize which components
are in series and which are in parallel in a complex system. The
following network is a good example of such a complex system:
As the figure illustrates, this system cannot be broken down into a
group of series and parallel systems. If
the system can be broken down into series/parallel configurations, it is a
relatively simple matter to determine the mathematical or analytical
formula that describes the system's reliability. However, for a
complex system, determination of the system reliability becomes more
involved.
In this article, we will look at some of the
techniques that can be employed to determine a system's reliability via
simulation. It is assumed that the reliability
values for the components have been determined using standard (or
accelerated) life data analysis techniques, so that the reliability
function for each component is known. With this component-level
reliability information available, simulation can then be performed to
determine the reliability of the entire system.
Monte Carlo
Simulation
Simulation in
system reliability analysis is based on the Monte
Carlo simulation method that generates random failure times from each
component's failure distribution. The overall system reliability is then
obtained by simulating system operation and empirically calculating the
reliability values for a series of time values. Through the use of
computers, simulation has become a very popular analysis tool. Simulation is
simple to apply and it can produce results that can be rather difficult to solve
analytically. On the other hand,
simulation methods also have certain drawbacks, not the least of which is
that the results depend on the
number of simulations, which results in a lack of repeatability. Other drawbacks
are that systems with static components (i.e., components in which
the reliability does not change with time) cannot be simulated, and that most of the reliability optimization and allocation techniques cannot be
applied.
To
illustrate how Monte Carlo data points are generated, we will demonstrate
how to generate
times to failure based on a two-parameter Weibull distribution with beta
equal to two ( =2)
and eta equal to 100 ( =100).
The reliability equation for the two-parameter Weibull distribution is
given by:
where
0 < R(T) < 1. If we assume that the values of R(T) are
uniformly distributed over the interval between 0 and 1, then we can let U,
a uniformly distributed random number in the same interval, represent R(T).
Substituting U for R(T), beta (),
eta (),
and solving for T yields:
This
equation is valid for any uniform random number U, 0 < U <
1. The procedure is then repeated using newly generated random
numbers, U, until the desired number of simulated failure times, T,
are reached.
The
same methodology, using different equations, is used for other
distributions.
System
Simulation Methodology
The system simulation methodology process is based on the Monte Carlo
simulation method which was described in the previous section. This is
different from the analytical methodology discussed in last month's issue.
While one can perform a Monte Carlo simulation based on the results of the
analytical system reliability solution, this should not be confused with
the methodology described below, which uses Monte Carlo simulation of the individual
components to estimate the overall system reliability.
In BlockSim,
the reliability simulation option requires a number of inputs. The first
input is the end
time at which the reliability is to be estimated. The
second input is the number of increments. The end time is divided into the
number of increments specified. When the simulation is performed, a table
of reliabilities and instantaneous failure rates is generated for each
incremental time up to the end time. However, only the instantaneous
failure rate estimation is affected by the number of increments. The Use
Seed option allows the user to choose the seed value for the generation of
random numbers. Use of the same seed value will result in identical
simulation results, provided the other inputs remain the same.
The next two inputs for the simulation,
the number of inner loops and the number of outer loops, can be found on the Setup
page of the Reliability/Maintainability Simulation window.
The product of the two values will determine the total number of simulations to be performed.
The number of inner loops indicates the number of simulation points to be generated for each component.
The number of outer loops indicates the number of repetitions of the inner loops.
If, for example, 1000 inner loops and 10 outer loops are to be performed, this means that first 1000 simulation points
will be generated and the reliability of the system at the end of each of the 1000 runs will be calculated.
This will then be repeated 10 times, each time with a new stream of random numbers for the simulation points.
This will yield 10 different system reliability values each obtained from 1000 runs.
The average of these 10 reliability values will be the returned system reliability at the specified time.
In summary, the simulation procedure
consists of the following steps:
-
Step
1 - Decide on the number of points to generate (Inner Loops).
-
Step
2 - For each run, generate a random number between 0 and 1.
-
Step
3 - Obtain a failure time for each component based on this random
number.
-
Step
4 - Keep the smallest time-to-failure with the corresponding component
(i.e., time-to-failure with a value less than the desired mission
time).
-
Step
5 - Check which components or combination of components cause system
failure.
-
Step
6 - The unreliability of the system is the number of times the system
was found to have failed divided by the total number of runs. The
reliability of the system is 100% minus the unreliability.
-
Step
7 - Return to Step 2 and repeat the procedure for the desired number
of cycles (Outer Loops).
-
Step
8 - The reliability of the system is the summation of the
reliabilities of the Outer Loops divided by the number of Outer Loops
(i.e., the average reliability).
BlockSim
System Simulation Example
In order to
illustrate these principles, consider the following complex system:
Given
that components A through E are identical, with a two-parameter Weibull
failure distribution with a beta value of 1.2 (
=1.2)
and an eta value of 1230 ( =1230),
determine the reliability of the system at 1500 hours. Note that the Start
and End blocks cannot fail. The
Reliability/Maintainability Simulation utility in BlockSim is used for
this example. Since
we are not solving for system reliability using analytical techniques, the
reliability equation for the system cannot be obtained.
However, a table of reliability vs. time can be generated. First, open the
Reliability/Maintainability Simulation window. On the Reliability page, enter an
End Time of
1500 hrs, 15 Increments, and a Seed Value of 1. When you
perform the simulation, these settings will generate a table of 15 reliability values
with the corresponding times and failure rates.
On
the Setup page of the Reliability/Maintainability Simulation window, specify 5 Outer Loops and 10,000
Inner Loops.
This
means that 10,000 random times-to-failure will be generated for each
component. This failure time will be compared to
the simulation time increment. If the failure time is less than the
time increment, a failure will be counted against the system. The system
reliability is the ratio of the number of successes to the number of
trials (in this case, there are 10,000 trials). The process is
repeated 5 times, and the results averaged to get a system reliability
value at each time increment. When the simulation is complete, the Results Panel window will appear with the corresponding results.
As
you can see from the preceding table, the reliability of the system at
1500 hours is 0.1738, or 17.38%. This gives a simple demonstration of how
system reliability simulation works. While the technique is rather simple,
it also requires many repetitions in order to develop a realistic
solution, thus making the use of a computer necessary to be able to perform
the analysis in a timely fashion.
In
future issues of the Reliability HotWire, we will look at how simulation
can be used to determine a system's availability as well as its
reliability.
|
