# Time-Dependent System Reliability for Components in Series

*This example appears in the System Analysis Reference book*.

**Time-Dependent System Reliability for Components with Constant Failure Rates in Series**

Consider a system consisting of 3 exponential units, connected in series, with the following failure rates (in failures per hour): , and .

- Obtain the reliability equation for the system.

- What is the reliability of the system after 150 hours of operation?

- Obtain the system's
*pdf*.

- Obtain the system's

- Obtain the system's failure rate equation.

- What is the MTTF for the system?

- What should the warranty period be for a 90% reliability?

**Solution**

The analytical expression for the reliability of the system is given by:

At 150 hours of operation, the reliability of the system is:

In order to obtain the system's *pdf*, the derivative of the reliability equation given in the first equation above is taken with respect to time, or:

The system's failure rate can now be obtained simply by dividing the system's *pdf* given in the equation above by the system's reliability function given in the first equation above, and:

Combining and the first equation above, the system's MTTF can be obtained:

Solving the first equation above with respect to time will yield the corresponding warranty period for a 90% reliability. In this case, the system reliability equation is simple and a closed form solution exists. The warranty time can now be found by solving:

Thus, the warranty period should be 132 hours.