Reliability HotWire

Issue 108, February 2010

Reliability Basics

Failure Rate of a Series System Using Weibull++

In many reliability prediction standards, systems are assumed to have components described by exponential distributions (i.e. constant failure rates) arranged in series. The goal of these standards is to determine the system failure rate, which is computed by summation of the component failure rates. However, many reliability engineers do not realize that for components arranged reliability-wise in series, the system failure rate at a given time is always equal to the sum of the component failure rates at that time, regardless of the distributions used to describe the components. This article will provide the mathematical justification of the preceding statement and show an example of two methods of plotting the system failure rate function using Weibull++.
Failure Rate of a Series System
The failure rate function, λ(t), is defined as:


where f(t) is the probability density function, R(t) is the reliability function, and t is time. This equation is valid at the system, subsystem or component level. It can be rewritten using the relationship between the probability density function and the reliability function as:


For a system of N components arranged reliability-wise in series, the system reliability, RS, is given by:


where Ri is the reliability of the ith component.

Taking the derivative of both sides with respect to t yields:


Using the product rule to differentiate the right hand side, we obtain:


Dividing both sides by -RS gives:


Rewriting in terms of the failure rate function, the system failure rate, λS, is given by:


Since we did not have to assume a distribution for the components to derive this formula, we can say that for any system of components arranged in series, the system failure rate at time t is always equal to the sum of the failure rates of the components at time t.

Example in Weibull++
A system is made up of seven components arranged in series. Test data are collected (in hours) for each component. In a Weibull++ Standard Folio, a separate Data Sheet is created for each component and a distribution is fitted to each data set. The resulting distributions and parameters are listed in Table 1.

Table 1: Distributions and Parameters for Each Component

Component Distribution Parameter 1 Parameter 2
Dorothy Weibull β = 1.9883 η = 400.7599
Toto Exponential λ = 6.9357x10-3  
Scarecrow Weibull β = 2.8303 η = 237.5418
Lion Lognormal μ = 5.3518 σ = 0.6695
Tin Man Exponential λ = 7.8934x10-3  
Witch Lognormal μ = 4.1516 σ = 0.7787
Aunt Em Weibull β = 5.6797 η = 279.9178


Suppose that your boss wants to see a plot of the failure rate for the first 200 hours of system operation. There are two ways to construct this plot using Weibull++. The first is to create a reliability block diagram and plot the system failure rate curve; the second is to use a General Spreadsheet to compute the component and system failure rates at discrete points in time and then create a graph showing the component and system failure rates.

For the first method, use the following steps to create the reliability block diagram representing the system.

  • Create a new Diagram by choosing Project > Add Diagram. Each component (i.e. each Data Sheet) appears as a block in the Template Panel at the bottom of the Diagram.
  • Drag each block from the template to the Diagram Sheet.
  • Choose Diagram > Join Blocks to enable the relationship line tool. When you point to a block, the cursor will change to a cross hair.
  • To draw a relationship line between two blocks, click the source block and drag the relationship line to the destination block.
  • Continue to draw relationship lines between blocks until all blocks are joined in a series configuration, as shown in Figure 1.
  • Reliability Block Diagram
    Figure 1: Reliability Block Diagram

  • Once all relationship lines are in place, click the Plot icon to create a plot.

Plot Icon

  • In the Control Panel, choose Failure Rate vs. Time from the drop-down list to create the system failure rate function plot shown in Figure 2.

Failure Rate vs. Time Plot
Figure 2: System Failure Rate Function

For the second method, the table shown in Figure 3 is constructed in a General Spreadsheet as follows:

  • In the Standard Folio containing the times-to-failure data, choose Folio > Insert General Spreadsheet.
  • In the General Spreadsheet, enter 0 in cell A3.
  • Enter =A3+10 in cell A4.
  • Select cell A4 and point to the black box at the lower right corner of the cell. Drag the box down to cell A23, thereby populating each cell with a function that adds 10 to the previous cell. This will create a list of times ranging from 0 to 200 in increments of 10.
  • Select cell B3. Use the Function Wizard to obtain the failure rate for the first component at time 0, as shown in Figure 4. Drag this formula to row 23, as in the previous step. Repeat for all components.
  • Enter =sum(B3:H3) in cell I3. Drag this formula to row 23.

Component and System Failure Rates
Figure 3: Component and System Failure Rates for Times from 0 to 200 Hours 

Function Wizard
Figure 4: Using the Function Wizard to Obtain the Component Failure Rate

To create the failure rate plot, follow the steps in this month's Hot Topics article to create a plot of the component and system failure rates versus time. Figure 5 shows the resulting graph.

Scatter Plot
Figure 5: Component and System Failure Rate Functions

[1] ReliaSoft Corporation, Life Data Analysis Reference, Tucson, AZ: ReliaSoft Publishing, 2005.

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