APPENDIX 8-C. LIFETIME DISTRIBUTIONS
 

TABLE OF CONTENTS
 

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

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

8-C.3 WATER HEATER LIFETIME DISTRIBUTIONS .............................................. 8-C-2 

8-C.4 GAS-FIRED POOL HEATERS LIFETIME DISTRIBUTIONS.......................... 8-C-7 

8-C.5 DIRECT HEATING EQUIPMENT LIFETIME DISTRIBUTIONS.................... 8-C-8 

LIST OF TABLES 

Table 8-C.3.1 Water Heaters............................................................................................. 8-C-2
 

Table 8-C.4.1 Gas-Fired Pool Heaters.............................................................................. 8-C-7
 

Table 8-C.5.1 Direct Heating Equipment ......................................................................... 8-C-8
 

LIST OF FIGURES 

Figure 8-C.3.1 Fraction of the Gas-Fired Storage Water Heaters Failing ......................... 8-C-3
 

Figure 8-C.3.2 Cumulative Lifetime Length of Gas-Fired Storage Water Heaters ........... 8-C-3
 

Figure 8-C.3.3 Fraction of the Electric Storage Water Heaters Failing............................. 8-C-4
 

Figure 8-C.3.4 Cumulative Lifetime Length of Electric Storage Water Heaters............... 8-C-4
 

Figure 8-C.3.5 Fraction of the Oil-Fired Storage Water Heaters Failing........................... 8-C-5
 

Figure 8-C.3.6 Cumulative Lifetime Length of Oil-Fired Storage Water Heaters ............ 8-C-5
 

Figure 8-C.3.7 Fraction of the Gas-Fired Instantaneous Water Heaters Failing................ 8-C-6
 

Figure 8-C.3.8 Cumulative Lifetime Length of Gas-Fired Instantaneous Water
 

Heaters ....................................................................................................... 8-C-6
 

Figure 8-C.4.1 Fraction of the Gas-Fired Pool Heater Failing........................................... 8-C-7
 

Figure 8-C.4.2 Cumulative Lifetime Length of Gas-Fired Pool Heaters........................... 8-C-8
 

Figure 8-C.5.1 Fraction of the Direct Heating Equipment Failing .................................... 8-C-9
 

Figure 8-C.5.2 Cumulative Lifetime Length of Direct Heating Equipment ...................... 8-C-9
 

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APPENDIX 8-C. LIFETIME DISTRIBUTIONS 

8-C.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, Life Cycle Cost and Payback Period Analyses. The Weibull distribution is 

recommended for application to lifetime data because it can be shaped to match low, average, 

and high values while still allowing some probability of exceeding the high value.1, 2 

8-C.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 in the Crystal Ball Technical Note, which uses the most likely value instead of an 

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: 

β ⎛

⎜

⎝

x L ⎞

⎟

⎠

β −1 −

⎛

⎜

⎝

⎞

⎟

⎠

β − x − L
 α f (x) =
 exp
 

α
 α
 

Where: 

L = location 

α = scale 

β = shape 

Therefore, the cumulative distribution is: 

β ⎛

⎜

⎝

⎞

⎟

⎠

x−L − α F(x) = 1− exp
 

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 Xavg, 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 percent, 

or 90 percent) 

The values for the parameters in the above equations are determined using the approach 

outlined in the Crystal Ball Technical Note.3 

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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 generating a 

forecast equal to the assumption. The forecast histogram and statistics verify that the Weibull 

distribution matches the desired shape. 

8-C.3 WATER HEATER LIFETIME DISTRIBUTIONS 

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

percentile values used to determine the Weibull distribution parameters alpha and beta for water 

heaters. The analysis utilized a maximum percentile of 95 to 99 percent. 

Table 8-C.3.1 Water Heaters 

Product Class 

Expert Opinion Values 

Weibull 

Parameters 

Minimum 

(years) 

Average 

(years) 

Maximum 

(years) 

Maximum 

percentile 

(%) 

Alpha 

(scale) 

Beta 

(shape) 

Gas-Fired/LPG Storage 6 13 20 95 7.8903 1.9134 

Electric Storage 6 13 20 95 7.8903 1.9134 

Oil-Fired Storage 6 13 20 95 7.8903 1.9134 

Gas-Fired Instantaneous 8 20 30 99 13.4203 3.0897 

Figures 8-C.3.1 to 8-C.3.8 show the Weibull distributions for each water heater product 

class. 

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0%

2%

4%

6%

8%

10%

12%

14%

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

Age of Equipment (Years) 

Probability of Failure (%) 

Figure 8-C.3.1 Fraction of the Gas-Fired Storage Water Heaters 

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

Age of Equipment (Years) 

Probability (%) 

Figure 8-C.3.2 Cumulative Lifetime Length of Gas-Fired Storage 

Water Heaters 

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0%

2%

4%

6%

8%

10%

12%

14%

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

Age of Equipment (Years) 

Probability of Failure (%) 

Figure 8-C.3.3 Fraction of the Electric Storage Water Heaters 

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

Age of Equipment (Years) 

Probability (%) 

Figure 8-C.3.4 Cumulative Lifetime Length of Electric Storage 

Water Heaters 

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0%

5%

10%

15%

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

Age of Equipment (Years) 

Probability of Failure (%) 

Figure 8-C.3.5 Fraction of the Oil-Fired Storage Water Heaters 

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

Age of Equipment (Years) 

Probability (%) 

Figure 8-C.3.6 Cumulative Lifetime Length of Oil-Fired Storage 

Water Heaters 

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0%

2%

4%

6%

8%

10%

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 8-C.3.7 Fraction of the Gas-Fired Instantaneous Water 

Heaters 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 8-C.3.8 Cumulative Lifetime Length of Gas-Fired 

Instantaneous Water Heaters 

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8-C.4 GAS-FIRED POOL HEATERS LIFETIME DISTRIBUTIONS 

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

percentile values used to determine the Weibull distribution parameters alpha and beta for gasfired pool heaters. The analysis utilized a maximum percentile of 99 percent. 

Table 8-C.4.1 Gas-Fired Pool Heaters 

Product Class 

Expert Opinion Values 

Weibull 

Parameters 

Minimum 

(years) 

Average 

(years) 

Maximum 

(years) 

Maximum 

percentile 

(%) 

Alpha 

(scale) 

Beta 

(shape) 

Gas-Fired Pool Heaters 3 8 20 99 5.4466 1.3417 

Figure 8-C.4.1 and Figure 8-C.4.2 show the Weibull distributions for pool heaters.
 

0%

5%

10%

15%

20%

25%

0 2 4 6 8 10 12 14 16 18 20

Age of Equipment (Years) 

Probability of Failure (%) 

Figure 8-C.4.1 Fraction of the Gas-Fired Pool Heater Failing
 

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0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 2 4 6 8 10 12 14 16 18 20

Age of Equipment (Years) 

Probability (%) 

Figure 8-C.4.2	 Cumulative Lifetime Length of Gas-Fired Pool 

Heaters 

8-C.5 DIRECT HEATING EQUIPMENT LIFETIME DISTRIBUTIONS 

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

values used to determine the Weibull distribution parameters alpha and beta. The analysis 

utilized a maximum percentile of 99 percent. 

Table 8-C.5.1	 Direct Heating Equipment 

Product Class 

Expert Opinion Values 

Weibull 

Parameters 

Minimum 

(years) 

Average 

(years) 

Maximum 

(years) 

Maximum 

percentile 

(%) 

Alpha 

(scale) 

Beta 

(shape) 

All DHE Equipment 10 15 20 99 5.6257 2.6548 

Figure 8-C.5.1 and Figure 8-C.5.2 show the Weibull distributions for direct heating 

equipment. 

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0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

20%

0 2 4 6 8 10 12 14 16 18 20 22

Age of Equipment (Years) 

Probability of Failure (%) 

Figure 8-C.5.1 Fraction of the Direct Heating Equipment 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

Age of Equipment (Years) 

Probability (%) 

Figure 8-C.5.2 Cumulative Lifetime Length of Direct Heating 

Equipment 

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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. Report No. Report 

No. ORNL-6804/R1. <http://www.ornl.gov/~webworks/cpr/v823/rpt/78562.pdf> 

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

<http://www.appliancemagazine.com/print.php?article=197&zone=1&first=1> 

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

Minimum, Most Likely Value, and a Percentile are Known, 2007. 

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

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