### APPENDIX 8C. LIFETIME DISTRIBUTIONS

TABLE OF CONTENTS

8C.1 INTRODUCTION ...................................................................................................8C-1

8C.2 DERIVATION OF WEIBULL DISTRIBUTION PARAMETERS .......................8C-1

8C.3 COOKING PRODUCTS – COOKTOP AND OVEN LIFETIME

DISTRIBUTIONS ...................................................................................................8C-2

8C.4 MICROWAVE OVEN LIFETIME DISTRIBUTIONS..........................................8C-5

8C.5 COMMERCIAL CLOTHES WASHER LIFETIME DISTRIBUTIONS ...............8C-6

LIST OF TABLES

Table 8C.3.1 Cooktops and Ovens ....................................................................................8C-2

Table 8C.4.1 Microwave Ovens ........................................................................................8C-5

Table 8C.5.1 Commercial Clothes Washers......................................................................8C-6

LIST OF FIGURE

Figure 8C.3.1 Fraction of the Electric Cooktops and Ovens Failing..................................8C-3

Figure 8C.3.2 Cumulative Lifetime Length of Electric Cooktops and Ovens....................8C-3

Figure 8C.3.3 Fraction of the Gas Cooktops and Ovens Failing ........................................8C-4

Figure 8C.3.4 Cumulative Lifetime Length of Gas Cooktops and Ovens..........................8C-4

Figure 8C.4.1 Fraction of the Microwave Ovens Failing ...................................................8C-5

Figure 8C.4.2 Cumulative Lifetime Length of Microwave Ovens .....................................8C-6

Figure 8C.5.1 Fraction of the Commercial Clothes Washers Failing, Multi-Family .........8C-7

Figure 8C.5.2 Cumulative Lifetime Length of Commercial Clothes Washers, Multi-

Family ..........................................................................................................8C-7

Figure 8C.5.3 Fraction of the Commercial Clothes Washers Failing, Laundromat ...........8C-8

Figure 8C.5.4 Cumulative Lifetime Length of Commercial Clothes Washers,

Laundromat..................................................................................................8C-8

8C-i

### APPENDIX 8C. LIFETIME DISTRIUBTIONS

8C.1 INTRODUCTION

For each product class, DOE characterized the product lifetime using a Weibull

probability distribution that ranged from the minimum to maximum lifetime estimates described

in chapter 8, section 8.2.3. The Weibull distribution is recommended for applying lifetime data

because it can be shaped to match low, most likely or average, and high values, yet still allow

some probability of exceeding the high value. The Weibull distribution has a long tail, which the

analyst can use to specify a distribution with some chance of exceeding the high value. 1,2



### 8C.2 DERIVATION OF WEIBULL DISTRIBUTION PARAMETERS

DOE utilized an approach for calculating the Weibull distribution using product lifetime

data to determine low, average, high value, and percentile of high value. A similar approach is

described by Crystal Ball Technical Note, which uses most likely value instead of average. 3



### Available data is used to assign low, average, and high values to a random variable with

unknown distribution parameters.

The Weibull distribution can be defined as:

Eq. 1

where

L = location

Alpha = scale

Beta = shape

Therefore, the cumulative distribution is

Eq. 2

Based on available data, Weibull distribution parameters are specified in the following

manner:

1. The output deviates must be greater than the expert opinion for low value,

2. The average X

avg 

### , must be equal to the average value from the available data,

3. The high value, xb, must correspond to some particular percentile point (e. g., 95%, or 90%)

8C-1

### The values for the parameters in Eq.1 and Eq. 2 are determined using the approach

outlined in Crystal Ball’s Technote. 3



### This solution can be checked using Crystal Ball by specifying a Weibull distribution with

the calculated parameters (location, scale, and shape) in an assumption cell and generate a

forecast that equals the assumption. Forecast histogram and statistics verify that the Weibull

distribution matches the desired shape.

8C.3	 COOKING PRODUCTS – COOKTOP AND OVEN LIFETIME

DISTRIBUTIONS

Table 8C.3.1 shows the average, minimum, maximum lifetime, and maximum percentile

values used to determine the Weibull distribution parameters alpha and beta. DOE estimated

that the maximum lifetime percentile for both electric and gas cooking products was 99 percent.

Table 8C.3.1 Cooktops and Ovens

Product Class Expert Opinion Values Weibull

Parameters

Minimum

years Average

years Maximum

years Maximum

percentile

% Alpha

(scale) Beta

(shape)

Electric Cooktops and Ovens 10 19 28 99 10.126 2.655

Gas Cooktops and Ovens 12 19 26 99 7.876 2.655



### Figures 8C.3.1 to 8C.3.2 show the Weibull distribution as well as the cumulative Weibull

distribution for electric cooktops and ovens. Figures 8C.3.3 to 8C.3.4 show the Weibull

distribution as well as the cumulative Weibull distribution for gas cooktops and ovens.

8C-20% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11%

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

Age of Equipment (Years)Probability of Failure (%)



### Figure 8C.3.1 Fraction of the Electric Cooktops and Ovens Failing

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

Age of Equipment (Years)Probability (%)



### Figure 8C.3.2 Cumulative Lifetime Length of Electric Cooktops and

Ovens

8C-30% 5% 10% 15%

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Age of Equipment (Years)Probability of Failure (%)



### Figure 8C.3.3 Fraction of the Gas Cooktops and Ovens Failing

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Age of Equipment (Years)Probability (%)



### Figure 8C.3.4 Cumulative Lifetime Length of Gas Cooktops and

Ovens

8C-4

### 8C.4 MICROWAVE OVEN LIFETIME DISTRIBUTIONS

Table 8C.4.1 shows the average, minimum, maximum lifetime, and maximum percentile

values used to determine the Weibull distribution parameters alpha and beta. DOE estimated

that the maximum lifetime percentile for microwave ovens was 99 percent.

Table 8C.4.1 Microwave Ovens

Product Class Expert Opinion Values Weibull

Parameters

Minimum

years Average

years Maximum

years Maximum

percentile

% Alpha

(scale) Beta

(shape)

Gas-Fired Pool Heaters 7 9 10 99 2.183 4.806



### Figures 8C.4.1 to 8C.4.2 show the Weibull distribution as well as the cumulative Weibull

distribution for microwave ovens.

0% 10% 20% 30% 40% 50% 60% 70% 80% 90%

0 1 2 3 4 5 6 7 8 9 10 11

Age of Equipment (Years)Probability of Failure (%)



### Figure 8C.4.1 Fraction of the Microwave Ovens Failing

8C-50% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

0 1 2 3 4 5 6 7 8 9 10 11

Age of Equipment (Years)Probability (%)



### Figure 8C.4.2 Cumulative Lifetime Length of Microwave Ovens

8C.5 COMMERCIAL CLOTHES WASHER LIFETIME DISTRIBUTIONS

Table 8C.5.1 shows the average, minimum, maximum lifetime, and maximum percentile

values used to determine the Weibull distribution parameters alpha and beta. For commercial

clothes washers, DOE developed two lifetime estimates based on product application—one for

multi-family applications and another for Laundromats. DOE estimated that product lifetimes

did not vary based on whether the washer was top-loading or front-loading. DOE estimated that

the maximum lifetime percentile for both product applications was 99 percent.

Table 8C.5.1 Commercial Clothes Washers

Product Application Expert Opinion Values Weibull

Parameters

Minimum

years Average

years Maximum

years Maximum

percentile

% Alpha

(scale) Beta

(shape)

Multi-Family 7 11.3 15.5 99 4.782 2.655

Laundromat 5 7.1 9.3 99 2.391 2.655



### Figures 8C.5.1 to 8C.5.4 show the Weibull distribution as well as the cumulative Weibull

distribution for each commercial clothes washer product application.

8C-60% 5% 10% 15% 20% 25%

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Age of Equipment (Years)Probability of Failure(%)



### Figure 8C.5.1 Fraction of the Commercial Clothes Washers Failing,

Multi-Family

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Age of Equipment (Years)Probability (%)



### Figure 8C.5.2 Cumulative Lifetime Length of Commercial Clothes

Washers, Multi-Family

8C-70% 5% 10% 15% 20% 25% 30% 35% 40% 45%

0 1 2 3 4 5 6 7 8 9 10 11

Age of Equipment (Years)Probability of Failure (%)



### Figure 8C.5.3 Fraction of the Commercial Clothes Washers Failing,

Laundromat

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

0 1 2 3 4 5 6 7 8 9 10 11

Age of Equipment (Years)Probability (%)



### Figure 8C.5.4 Cumulative Lifetime Length of Commercial Clothes

Washers, Laundromat

8C-8

### REFERENCES

1 

### Barnes, P.R., J. W. Van Dyke, B. W. McConnell, S. M. Cohn, and S. L. Purucker, The

Feasibility of Replacing or Upgrading Utility Distribution Transformers During Routine

Maintenance . 1995, Oak Ridge National Laboratory: Oak Ridge, TN.

2 

### Karr, T., Making the Most of Life Test Data , in Appliance Magazine . 2003.

3 

### Crystal Ball. Technote: Derivation of Weibull Distribution Parameters When the Minimum,

Most Likely Value, and a Percentile are Known . 2007 [cited; Available from:

http://www.crystalball.com/support/simulation/cbl_gen_021A.html

.

8C-9