Using Excel to Implement Software Reliability Models
• Norman F. Schneidewind
• Naval Postgraduate School
• 2822 Racoon Trail,
• Pebble Beach, California, 93953, USA
• Voice: (831) 656-2719
• Fax: (831) 372-0445
• nschneid@nps.navy.mil
1
Outline
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Introduction
Characteristics of Excel Implementation
Combined Software Reliability Tools – Excel Approach
Structure of Combined Approach
Notation for Prediction Worksheet
Equations for Prediction and Comparison Worksheets
Example Prediction Worksheet
Analysis of Prediction Worksheet
Notation for Actual – Prediction Comparisons Worksheet
Example Actual – Prediction Comparisons Worksheet
Analysis of Comparison Worksheet
Cumulative Failure Prediction Plots
Validation of Failure Count Predictions
Time to Failure Plot
Validation of Time to Failure Predictions
Conclusions
Excel Demo
2
Introduction
• CASRE and SMERFS, hereafter referred to as SRT
(software reliability tools), were developed prior to the
availability of mature spreadsheet programs.
– Programs like Excel were not an option, but things have
changed.
• In Excel, the user can create equations, do data and
statistical analysis, make plots, an do programming, using
Visual Basic.
• In SRT, the programming of the models has been done for
the user, but the functionality is fixed until the next
revision.
3
Characteristics of Excel Implementation #1
• Advantages:
– Almost all practitioners have Excel. A minority of
practitioners have SRT.
– Easier for practitioners to use than SRT.
– Typically, failure data is provided by practitioners in
Excel.
– Improve technology transfer:
• Predictions can be made by the researcher in the
spreadsheet and returned to the practitioner in the
same spreadsheet.
– Formatted Excel data can be imported into Word and
PowerPoint for creating reports and presentations.
4
Characteristics of Excel Implementation # 2
• Advantages:
– User has more control over formatting of data,
prediction results, and plots.
– A large set of built-in mathematical and statistical
functions are available for reliability analysis.
• SRT limited to functions like Chi-square.
– User can construct his own reliability equations.
• SRT equations are fixed, based on the models
implemented.
– More flexibility in changing term in equations.
• Change cell values; copy and paste equations.
5
Characteristics of Excel Implementation # 3
• Disadvantages:
– Column and cell orientation of spreadsheets is
cumbersome.
• It is not a natural mathematical format.
• Need to repeat parameter entries for
iterations of equations.
• Variable names are not case sensitive.
• Variable names cannot be the same as
column or cell names.
– Thus, some variables must renamed to avoid
naming conflicts.
6
Characteristics of Excel Implementation # 4
• Disadvantages:
– Mathematical library is not as extensive as
Fortran and C++ libraries used in SRT.
– Does not have sophisticated model evaluation
criteria of SRT.
• However, error analysis between actuals and
predictions (i.e., validation) can be done in
Excel.
7
Combined Software Reliability Tools – Excel Approach
• Best approach may be to combine SRT with Excel.
• SRT provides model parameter estimation.
– Beyond the capabilities of Excel unless programmed in
Visual Basic.
– Copy and paste parameters from SRT into spreadsheet.
• Excel extends capabilities of SRT by allowing user
provided equations, statistical analysis, and plots.
8
Structure of Combined Approach
• Worksheets:
– Definitions:
• Notation
• Equations
– Predictions
• Analysis
– Actual – Prediction Comparisons
• Analysis
• Plots
• Validation
• Examples of this approach follow.
9
Notation for Prediction Worksheet
s
starting interval for using observed failure data in parameter
estimation

alpha: failure rate at the beginning of interval s

beta: negative of derivative of failure rate divided by failure rate
t
interval when time to next failure prediction made
Xs-1
observed failure count in the range [1,s-1].
Xs,t
observed failure count in the range [s,t]
Xt
observed failure count in the range [1,t]
Ft
given number of failures to occur after interval t
TF(t)
time to next failure(s) predicted at time t
r(t)
remaining failures predicted at time t
T
test or operational time
D(T)
cumulative number of failures detected at time T
D(TL)
cumulative number of failures detected over life of software TL
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Equations
Time
to
for
Prediction
Next
and
Failure(s)
Comparison
Predicted
Worksheets
at
Time
t
TF(t) = [(log[ /(  (Xs,t + Ft ))]) / ]  (t  s+1)
Remaining Failures Predicted at Time t:
r(t) = (/) – Xs,t
Cumulative Number of Failures Detected at Time T:
D(T) = (α/β)[1 – exp (-β ((T –s + 1)))] + Xs-1
Cumulative Number of Failures Detected Over Life of Software TL:
D(TL) = / + Xs-1
References: [1, 2, 3].
11
Example Prediction Worksheet
Project
Satellites
Project 1
Project 2
s
1
12
12
12
12
12
12
12
12
12
12
12
12
12


20.950000 0.15854
3.447300
3.447300
3.447300
3.447300
3.447300
3.447300
3.447300
3.447300
3.447300
3.447300
3.447300
3.447300
3.447300
0.044796
0.044796
0.044796
0.044796
0.044796
0.044796
0.044796
0.044796
0.044796
0.044796
0.044796
0.044796
0.044796
t
Xs-1
Xst
Xt
Ft
TF(t)
r(t)
T
D(T)
26
0
130
130
1
3.96
2.14
27 130.32 132.14
23
23
23
23
23
23
23
23
23
23
23
23
23
39
39
39
39
39
39
39
39
39
39
39
39
39
32
32
32
32
32
32
32
32
32
32
32
32
32
71
71
71
71
71
71
71
71
71
71
71
71
71
5
7
9
13
14
15
17
20
2.63
3.78
4.99
7.62
8.33
9.06
10.60
13.14
44.96
23
24
25
26
27
28
29
30
31
32
33
34
35
71.00
72.97
74.85
76.65
78.37
80.02
81.60
83.10
84.54
85.92
87.23
88.49
89.69
12
D (TL)
115.96
Analysis of Prediction Worksheet # 1
• s, , and  obtained from SMERFS.
• One interval = one week of calendar time.
• Project 1:
– Optimal s = 1 for both failure count and time to failure
predictions.
– t=26: interval when time to next failure prediction made
This is also the last interval of observed failure data.
– X26 = 130: observed failure count in the range [1,26].
– F1 = 1: given number of failures to occur after interval
26.
– TF(26) = 3.96 intervals: time to next failure predicted at
time 26 intervals.
13
Analysis of Prediction Worksheet #2
• Project 1:
– r(26) = 2.14: remaining failures predicted at time 26
intervals.
– T = 27 intervals: test time.
– D(27) = 130.32: cumulative number of failures detected
at time 27 intervals.
– D() = 132.14: cumulative number of failures detected
over life of software (conservatively, infinity).
• r(26) = D() - X26 = 132.14 – 130 = 2.14 remaining
failures, as in the above.
14
Analysis of Prediction Worksheet #3
• Project 2:
– Total range of 35 weeks divided into Parameter
Estimation Range = 1, 23 weeks and Prediction Range
= 24, 35 weeks for the purpose of model validation.
• Model fit using historical data does not demonstrate
validity!
– Estimate model parameters in range 1, 23 weeks.
• Accuracy of future predictions demonstrates
validity.
– Predict in range 24, 35 weeks and compare with
actuals.
– Optimal s = 12 for both failure count and time to failure
predictions.
15
Analysis of Prediction Worksheet #4
• Project 2:
– t=23: interval when time to next failure prediction made
– X11 = 39: observed failure count in the range [1,11].
– X12,23 = 32: observed failure count in the range [12,23].
– X23 = 71: observed failure count in the range [1,23].
– F1 = 5, …, 20: given number of failures to occur after
interval 23.
– TF(23) = 2.63, …, 13.14 intervals: time to next failures
predicted at time 23 intervals.
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Analysis of Prediction Worksheet #5
• Project 2:
– r(23) = 44.96: remaining failures predicted at time 23
intervals.
– T = 23, …, 35 intervals: test time.
– D(23, …, 35) = 71.00, …, 89.69 cumulative number of
failures detected at time 23, …, 35 intervals.
– D() = 115.96: cumulative number of failures detected
over life of software (conservatively, infinity).
• r(23) = D() - X23 = 115.96 –71 = 44.96 remaining
failures, as in the above.
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Notation for Actual – Prediction Comparisons Worksheet
• Parameter Estimation Range = 1, 23 weeks; Prediction Range = 24, 35
weeks; s = 12 weeks.
• D(T) Actual = Actual Cumulative Count, from Interval 1, in Prediction
Range
• D(T) Pred = Predicted Cumulative Count, from Interval1, in Prediction
Range
• Interval Actual = Difference in D(T) Actual
• Interval Pred = Difference in D(T) Pred
• Int Act Cum = Interval Actual Cumulative Count, from Interval 24, in
Prediction Range
• Int Pred Cum = Interval Predicted Cumulative Count, from Interval 24,
in Prediction Range
• TF(t) Actual = Actual Time to Next Given Number of Failures in the
Int Act Cum column
• TF(t) Pred = Predicted Time to Next Given Number of Failures in the
Int Act Cum column
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Example Actual – Prediction Comparisons Worksheet
Interval Failure Count D(T) Actual D(T) Pred Interval Actual Interval Pred Int Act Cum Int Pred CumTF(t) Actual TF(t) Pred
23
0
71
71.00
24
0
71
72.97
0
1.97
0
1.97
25
5
76
74.85
5
1.88
5
3.85
2
2.63
26
0
76
76.65
0
1.80
5
5.65
2
2.63
27
2
78
78.37
2
1.72
7
7.37
4
3.78
28
2
80
80.02
2
1.65
9
9.02
5
4.99
29
4
84
81.60
4
1.57
13
10.60
6
7.62
30
0
84
83.10
0
1.51
13
12.10
6
7.62
31
1
85
84.54
1
1.44
14
13.54
8
8.33
32
1
86
85.92
1
1.38
15
14.92
9
9.06
33
2
88
87.23
2
1.32
17
16.23
10
10.60
34
3
91
88.49
3
1.26
20
17.49
11
13.14
35
0
91
89.69
0
1.20
20
18.69
11
13.14
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Analysis of Comparison Worksheet # 1
• Project 2
– D(T) Actual is compared with D(T) Prediction.
• Failure counts are accumulated from Interval1in the
parameter estimation range, but are compared in the
prediction range.
– Interval Actual is compared with Interval Prediction.
• Interval failure counts are compared in the
prediction range.
– Int Act Cum is compared with Int Pred Cum.
• Interval failure counts are accumulated from Interval
24 in the prediction range and compared in the
prediction range.
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Analysis of Comparison Worksheet # 2
• Project 2
• Make plots in prediction range:
– Actual and Predicted Cumulative Failures in Range 1,
35 Weeks.
– Actual and Predicted Cumulative Failures in Range
24,35 Weeks.
– Validation of Failure Count Predictions.
• Residuals: (Predicted – Actual) versus week.
– Residuals do not show bias (i.e., trend in either
positive or negative direction).
– Average Residual = -0.55 failures indicates
optimistic prediction on average.
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Cumulative Failures in Range 1, 35 Weeks:
Parameter Estimation Range plus Prediction Range
Cumulative Failures in Range 1, 35 Weeks (Project 2)
95
90
85
Actual
Predicted
80
75
70
22
24
26
28
Week
30
32
34
36
22
Cumulative Failures in Range 24,35 Weeks:
Prediction Range
Cumulative Failures in Range 24, 35 Weeeks (Project 2)
25
20
15
10
5
0
Actual
Predicted
22
24
26
28
30
32
34
36
Week
23
Validation of Failure Count Predictions
Cumulative Failure Residuals: Predicted-Actual (Project 2)
3.00
Failures
2.00
1.00
0.00
-1.00 22
24
26
28
30
32
34
36
-2.00
-3.00
Week
Average Residual = -0.55 failures
24
Analysis of Comparison Worksheet # 3
• Project 2
• Make plot in prediction range:
– Actual and Predicted Time to Next Failures versus
given number of failures.
– Validation of Time to Failure Predictions.
• Residuals: (Predicted – Actual) versus given number
of failures.
– Residuals show bias starting at 15 failures (week
32) as it becomes difficult to predict further out
into the future.
– Average Residual = 0.87 weeks indicates
optimistic prediction on average.
25
Time to Given Number of Failures
Time to Failure(s) (Project 2)
Weeks
15
Actual
10
Predicted
5
0
0
5
10
Failures
15
20
25
26
Validation of Time to Failure Predictions
Time to Failures Residuals: Predicted-Actual (Project 2)
3.00
Weeks
2.00
1.00
0.00
-1.00
0
5
10
15
20
25
Given Number of Failures
Average Residual = 0.87 weeks
27
Conclusions
• Spreadsheet technology can effectively support software
reliability modeling and prediction.
• Advantages relative to SRT are:
– Easier transfer of technology to practitioners.
– More user control of program’s operation.
– Many built-in mathematical and statistical functions.
• Disadvantages relative to SRT are:
– Cell format is not conducive to mathematical modeling.
– No built-in model evaluation criteria.
• SRT and Excel can be combined to advantage:
– SRT for reliability model parameter estimation.
– Excel for reliability prediction.
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References
• [1] Norman F. Schneidewind, "Reliability Modeling for
Safety Critical Software", IEEE Transactions
on
Reliability, Vol. 46, No.1, March 1997, pp.88-98.
• [2] Norman F. Schneidewind, "Software Reliability Model
with Optimal Selection of Failure Data", IEEE
Transactions on Software Engineering, Vol. 19, No. 11,
November 1993, pp. 1095-1104.
• [3] Norman F. Schneidewind and T. W. Keller,
"Application of Reliability Models to the Space Shuttle",
IEEE Software, Vol. 9, No. 4, July 1992 pp. 28-33.
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