Figure 4.5: Simple parallel system.
In a simple parallel system, as shown in Figure 4.5, at least one of the units must succeed for the system to succeed. Units in parallel are also referred to as redundant units. Redundancy is a very important aspect of system design and reliability in that adding redundancy is one of several methods of improving system reliability. It is widely used in the aerospace industry and generally used in mission critical systems. Other example applications include the RAID computer hard drive systems, brake systems and support cables in bridges.
The probability of failure, or unreliability, for a system with n statistically independent parallel components is the probability that unit 1 fails and unit 2 fails and all of the other units in the system fail. So in a parallel system, all n units must fail for the system to fail. Put another way, if unit 1 succeeds or unit 2 succeeds or any of the n units succeeds, then the system succeeds. The unreliability of the system is then given by:
(3)
Where:
Qs = unreliability of the system.
Xi = event of failure of unit i.
P(Xi) = probability of failure of unit i.
In the case where the failure of a component affects the failure rates of other components, then the conditional probabilities in Eqn. (3) must be considered.
However, in the case of independent components, Eqn. (3) becomes:
Or:
Or, in terms of component unreliability:
(4)
Observe the contrast with the series system, in which the system reliability was the product of the component reliabilities; whereas the parallel system has the overall system unreliability as the product of the component unreliabilities.
The reliability of the parallel system is then given by:
(5)
(Note: This is similar to determining the probability of rolling a six on either die, when rolling two dice. The first block is the probability of rolling a six from the first die (1/6) and the second block is the probability of rolling a six from the second die (1/6). Since either must happen, then they are in a simple parallel configuration with a probability of 11/36 (almost 1/3) of rolling at least one six from the two dice.)
Consider a system consisting of three subsystems arranged reliability-wise in parallel. Subsystem 1 has a reliability of 99.5%, Subsystem 2 has a reliability of 98.7% and Subsystem 3 has a reliability of 97.3% for a mission of 100 hours. What is the overall reliability of the system for a 100 hour mission?
Since the reliabilities of the subsystems are specified for 100 hours, the reliability of the system for a 100 hour mission is:
When we examined a system of components in series, we found that the least reliable component has the biggest effect on the reliability of the system. However, the component with the highest reliability in a parallel configuration has the biggest effect on the system's reliability, since the most reliable component is the one that will most likely fail last. This is a very important property of the parallel configuration, specifically in the design and improvement of systems.
Consider three components arranged reliability-wise in parallel with R1 = 60%, R2 = 70% and R3 = 80% (for a given time). The corresponding reliability for the system is Rs = 97.6%. In Table 3, we can examine the effect of each component's reliability on the overall system reliability. The first row of the table shows the given reliabilities for each component and the corresponding system reliability for these values. In the second row, the reliability of Component 1 is increased by a value of 10% while keeping the reliability of the other two components constant. Similarly, by increasing the reliabilities of Components 2 and 3 in the third and fourth rows by a value of 10% while keeping the reliabilities of the other components at the given values, we can observe the effect of each component's reliability on the overall system reliability. It is clear that the highest value for the system's reliability was achieved when the reliability of Component 3, which is the most reliable component, was increased. Once again, this is the opposite of what was encountered with a pure series system.
This conclusion can also be illustrated graphically, as shown in Figure 4.6.
Figure 4.6: Rate of change of parallel system reliability when increasing the reliability of each component.
In the case of the parallel configuration, the number of components has the opposite effect of the one observed for the series configuration. For a parallel configuration, as the number of components/subsystems increases, the system's reliability increases.
Figure 4.7: Reliability of a system with n statistically independent and identical components arranged reliability-wise in parallel.
Figure 4.7 illustrates that a high system reliability can be achieved with low-reliability components, provided that there are a sufficient number of components in parallel. Note that Figure 4.7 is the mirror image of Figure 4.3, which presents the effect of the number of components in a series configuration.
Consider a system that consists of a single component. The reliability of the component is 60%, thus the reliability of the system is 60%. What would the reliability of the system be if the system were composed of two, four or six such components in parallel?
Figure 4.8: Effect of the number of components in a parallel configuration.
Clearly, the reliability of a system can be improved by adding redundancy. However, it must be noted that doing so is usually costly in terms of additional components, additional weight, volume, etc.
Reliability optimization and costs are covered in detail in later sections.
See Also:
RBDs and Analytical System Reliability
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