## Series Systems

In a series configuration, a failure of any component results in failure for the entire system. In most cases when considering complete systems at their basic subsystem level, it is found that these are arranged reliability-wise in a series configuration. For example, a personal computer may consist of four basic subsystems: the motherboard, the hard drive, the power supply and the processor. These are reliability-wise in series and a failure of any of these subsystems will cause a system failure. In other words, all of the units in a series system must succeed for the system to succeed.

The reliability of the system is the probability that unit 1 succeeds and unit 2 succeeds and all of the other units in the system succeed. So, all n units must succeed for the system to succeed. The reliability of the system is then given by:

(1)

Where:

• = reliability of the system.

• = event of unit i being operational.

• P() = probability that unit i is operational.

In the case where the failure of a component affects the failure rates of other components (i.e. the life distribution characteristics of the other components change when one fails), then the conditional probabilities in Eqn. (1) must be considered.

However, in the case of independent components, Eqn. (1) becomes:

Or:

Or, in terms of individual component reliability:

In other words, for a pure series system, the system reliability is equal to the product of the reliabilities of its constituent components. (Note: This is similar to determining the probability of rolling double sixes. 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 both must happen, they are in a series configuration with a probability of 1/36.)

### RBDs and Analytical System Reliability Example 1

Three subsystems are reliability-wise in series and make up a system. 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?

#### Solution to RBDs and Analytical System Reliability Example 1

Since the reliabilities of the subsystems are specified for 100 hours, the reliability of the system for a 100 hour mission is simply:

### Effect of Component Reliability in a Series System

In a series configuration, the component with the smallest reliability has the biggest effect on the system's reliability. There is a saying that "a chain is only as strong as its weakest link." This is a good example of the effect of a component in a series system. In a chain, all the rings are in series and if any of the rings break, the system fails. In addition, the weakest link in the chain is the one that will break first. The weakest link dictates the strength of the chain in the same way that the weakest component/subsystem dictates the reliability of a series system. As a result, the reliability of a series system is always less than the reliability of the least reliable component.

### RBDs and Analytical System Reliability Example 2

Consider three components arranged reliability-wise in series, where = 70%, = 80% and = 90% (for a given time).

In Table 1, 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 reliabilities of the other two components constant. Similarly, by increasing the reliabilities of Components 2 and 3 in rows 3 and 4 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 1, which is the least reliable component, was increased by a value of 10%.

This conclusion can also be illustrated graphically, as shown in Figure 4.1.

Figure 4.1: Effect of component reliability on the overall system reliability.

Note the slight difference in the slopes of the three lines in Figure 4.1. The difference in these slopes represents the difference in the effect of each of the components on the overall system reliability. In other words, the system reliability's rate of change with respect to the each component's change in reliability is different. This observation will be explored further when the importance measures of components are considered in later chapters. The rate of change of the system's reliability with respect to each of the components is plotted in Figure 4.2. It can be seen in Figure 4.2 that Component 1 has the steepest slope, which indicates that an increase in the reliability of Component 1 will result in a higher increase in the reliability of the system. In other words, Component 1 has a higher reliability importance.

Figure 4.2: Rate of change of series system reliability when increasing the reliability of each component.

### Effect of Number of Components in a Series System

The number of components is another concern in systems with components reliability-wise in series. As the number of components in series increases, the system's reliability decreases.

Figure 4.3: Reliability of a system with n statistically independent and identical components arranged reliability-wise in series.

Figure 4.3 illustrates the effect of the number of components arranged reliability-wise in series on the system's reliability for different component reliability values. This figure also demonstrates the dramatic effect that the number of components has on the system's reliability, particularly when the component reliability is low. In other words, in order to achieve a high system reliability, the component reliability must be high also, especially for systems with many components arranged reliability-wise in series.

### RBDs and Analytical System Reliability Example 3

Consider a system that consists of a single component. The reliability of the component is 95%, thus the reliability of the system is 95%. What would the reliability of the system be if there were more than one component (with the same individual reliability) in series? Table 2 shows the effect on the system's reliability by adding consecutive components (with the same reliability) in series. Figure 4.4 illustrates the same concept graphically for components with 90% and 95% reliability.

Figure 4.4: Effect of the number of components in a series configuration for two different cases.