
Failure Rate of Redundant Configurations in Lambda Predict
Standards based reliability prediction methods typically assume that the failure rates for the components in a system are constant (i.e., the components are modeled by exponential distributions) and that the system fails if any component in the system fails (i.e., the system is arranged reliabilitywise in series). When these two conditions are met, the system failure rate is constant. ReliaSoft's reliability prediction software, Lambda Predict, allows the user to improve upon the traditional standards based reliability method through the use of redundant reliability configurations at the block and system levels. This article addresses how redundancy affects the failure rates of systems composed of constant failure rate components.
Lambda Predict can be configured to show two failure rate results. The Failure Rate(t=INF) column is shown by default and is used to indicate the steadystate failure rate. The Failure Rate(t) column is hidden by default and is used to indicate the system failure rate at a given mission time (defined at the system level).
When all the components in a system are arranged in a series configuration, the system failure rate is constant so the system failure rate at any mission time is equal to the steadystate failure rate. When redundancy is present in the system, the system failure rate is not constant, so the system failure rate at a given mission time, t, is less than the steadystate failure rate. (See the derivations of the system failure rates for series and parallel configurations.)
Consider the system shown in Figure 1. It shows a single block that contains three components with constant failure rates, arranged in a series configuration. Since there is no redundancy in this system, the system failure rate is equal to the sum of the component failure rates. Furthermore, since the components all have constant failure rates, the system failure rate is constant. Therefore, the Failure Rate (t) column and the Failure Rate (t=INF) column in Lambda Predict display identical information.
Figure 1 – Failure Rate at 1,000,000 hours and SteadyState Failure Rate for a Series System
To examine the effect of redundancy on the system failure rate, the block properties are altered so that Redundancy is set to True and Quantity is set to 4. In addition, at the system level the Mission Time is set to 1,000,000 hours. Varying the input for the Quantity Required field from 1 to 4 gives the results shown in Table 1.
For Quantity Required values less than Quantity values, the system failure rate at 1,000,000 hours is significantly less than the steadystate system failure rate. Therefore, even after 1,000,000 hours of operation, the system failure rate has not reached a steadystate value and is still increasing. For a Quantity Required value equal to the Quantity value, the system failure rate at 1,000,000 hours is identical to the steadystate system failure rate. This result is expected since an noutofn configuration is equivalent to a series configuration with n blocks.
Quantity Required  Failure Rate at 1,000,000 hours (FPMH)  Failure Rate at Time = Infinity (FPMH) 
1  0.0061  36.0123 
2  0.5032  72.0246 
3  13.8193  108.0368 
4  144.0491  144.0491 
Figure 2 shows a plot of the failure rate for each of the configurations considered in Table 1. This plot was created by publishing the capacitor, inductor and resistor models, using those published models in BlockSim to create a separate analytical RBD for each row of Table 1, and then creating an overlay plot of the RBD analyses. The plot indicates that for the systems with redundancy (i.e., the Quantity Required is less than the Quantity), the failure rates start at zero and increase to the steadystate failure rates shown in Table 1 by about 0.15 billion hours. In addition, it can be seen that for the system without redundancy (i.e., the Quantity Required is equal to the Quantity), the failure rate is constant.
Figure 2  Failure Rate Plot for Systems with Various Amounts of Redundancy
Conclusion
This article used Lambda Predict and BlockSim software to show that the system failure rate for constant failure rate components arranged in a redundant configuration is nonconstant (i.e., timedependent). In addition, the use of the Failure Rate(t) and Failure Rate(t=INF) columns in the Lambda Predict software was explained.