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 reliabilitywise 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
reliabilitywise in series, the system reliability, R_{S}, is given
by:
where R_{i} is the reliability of the
i^{th} 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 R_{S} gives:
Rewriting in terms of the failure rate function, the system
failure rate, λ_{S}, 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 =
6.9357x103 

Scarecrow 
Weibull 
β = 2.8303 
η =
237.5418 
Lion 
Lognormal 
μ = 5.3518 
σ = 0.6695 
Tin Man 
Exponential 
λ =
7.8934x103 

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.
 In the Control Panel, choose Failure Rate vs.
Time from the dropdown list to create the system failure rate
function plot shown in
Figure 2.
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 timestofailure 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.
Figure 3: Component and System Failure Rates for Times from
0 to 200 Hours
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.
Figure
5: Component and System Failure Rate Functions
References [1] ReliaSoft Corporation,
Life Data Analysis Reference, Tucson, AZ: ReliaSoft
Publishing, 2005. 