pareto software quality assessmentshodhganga.inflibnet.ac.in/bitstream/10603/71425/11/11_chapter...

52

CHAPTER 4

MONITORING PARETO TYPE IV SOFTWARE QUALITY USING SPC

4.1 Introduction

Statistical Process Control (SPC) is a scientific, data-driven methodology for

quality analysis and improvement. It is known to be a powerful tool to improve

process, to enhance quality and productivity [Florac, 1999]. It is a group of tools

such as check sheet, Run chart, Histogram, Pareto chart, Scatter diagram/chart and

control chart used for quality improvement, stability and predictability of a process.

During the manufacturing process SPC acts as an industry-standard methodology for

measuring and controlling quality. Quality data in the form of Product or Process

measurements are obtained in real-time during manufacturing. This data is then

plotted on a graph with pre-determined control limits.

Control limits are determined by the capability of the process, whereas

specification limits are determined by the client's needs. Data that falls within the

control limits indicates that everything is operating as expected. Any variation within

the control limits is likely due to a common cause—the natural variation that is

expected as part of the process. If data falls outside of the control limits, this

indicates that an assignable cause is likely the source of the product variation, and

something within the process should be changed to fix the issue before defects occur.

To identify and eliminate errors in software development process and also to

improve software reliability, the Statistical Process Control concepts and methods

are the best choice. Our effort is to apply SPC techniques in the software

development process so as to improve software reliability and quality.

S - Statistical, because we use some statistical concepts to help us understand

processes

P - Process, because we deliver our work through processes. i.e., how we do things

C - Control, by this we mean predictable.

The objective of SPC is to establish and maintain statistical control over a

random process. It is a powerful tool to optimize the amount of information needed

53

for use in making management decisions. Statistical techniques provide an

understanding of the business baselines, insights for process improvements,

communication of value and results of processes, active and visible involvement.

SPC provides real time analysis to establish controllable process baselines; learn, set,

and dynamically improve process capabilities; and focus business areas needing

improvement. The early detection of software failures will improve the software

reliability. The selection of proper SPC charts is essential to effective statistical

process control implementation and use.

Control Chart:

Control charts are used to monitor a process to determine whether or not the

process is in statistical control, to evaluate a process and determine normal statistical

control parameters and to identify area of improvement in process. Control charts

attempt to differentiate between the types of process variation:

Common cause variation: It is intrinsic to the process and will always be presents. It

is also known as chance cause variation.

Special cause variation: Special cause variation stems from external source and

shows that the process is out of statistical control. It is also known as assignable

cause variation or Out of Statistical Control.

In the present chapter we proposed a control mechanism based on the

cumulative quantity observations of interval domain data using mean value function

of Pareto Type IV distribution, which is based on Non-Homogenous Poisson Process

(NHPP). The MLE method is used to derive the point estimators of the unknown

parameters of the proposed model under consideration. The content of this chapter is

published in the following journal.

Dr R.Satya Prasad, G.Sridevi . “MONITORING PARETO TYPE IV SRGM

USING SPC”, International Journal of Computers & Technology (IJCT). Pg:2161-

2168, Volume 11-No. 1, September 2013, Published by Council for Innovative

Research. ISSN: 2277-3061.

54

4.2 Pareto Type IV Model Development

Software reliability models can be classified according to probabilistic

assumptions. When a Markov process represents the failure process; the resultant

model is called Markovian Model. Second one is fault counting model which

describes the failure phenomenon by stochastic process like Homogeneous Poisson

Process (HPP), Non Homogeneous Poisson Process (NHPP) and Compound Poisson

Process etc. A majority of failure count models are based upon NHPP described in

the following lines.

A software system is subjected to failures at random times caused by errors present

in the system. Let��, � > 0� be a counting process representing the cumulative

number of failures by time ‘t’. Since there are no failures at t=0 we have

��0 = 0

It is to assume that the number of software failures during non-overlapping time

intervals do not affect each other. In other words, for any finite collection of times �� < �� < ⋯ �7. The ‘n’ random variables { } { }1 2 1 1( ), ( ) (t ) ,... ( ) ( )n nt N t N N t N t −− −

are independent. This implies that the counting process ��, � > 0� has

independent increments.

Let �� represent the expected number of software failures by time ‘�’. The

mean value function �� is finite valued, non-decreasing, non-negative and

bounded with the boundary conditions.

0, 0

( ),

tm t

a t

= =

= → ∞

Where ‘$’ is the expected number of software errors to be eventually detected.

55

Suppose �� is known to have a Poisson probability mass function with parameters

�� i.e.,

��U = UA� = [��]7. ��A! , A = 0,1,2 … ∞

then �� is called an NHPP. Thus the stochastic behavior of software failure

phenomena can be described through the �� process. Various time domain models

have appeared in the literature (Kantam and Subbarao, 2009) which describe the

stochastic failure process by an NHPP which differ in the mean value function ��.

We consider �� as given by

( ) 1 1 , 0

b

tm t a t

c

− = − + ≥

Where [��/$] is the cumulative distribution function of Pareto type IV

distribution (Johnson et al., 1994) for the present choice.

��U = UA� = [��]7. ��A!

Lim7→_ �� = A� = ��c . $7A!

This is also a Poisson model with mean ‘a’.

Let �� be the number of errors remaining in the system at time ‘t’.

�� = ��∞ − �� e[��] = e[��∞] − e[��] = $ − ��

1 1

b

ta a

c

− = − − +

1

b

ta

c

− = +

56

4.3 Maximum Likelihood Estimation

Based on the inter failure data given in this Chapter, we demonstrate the

software failure process through Mean Value Control chart. We used cumulative

failures data for software reliability monitoring. The use of cumulative quality is a

different and new approach, which is of particular advantage in reliability. ’a’,‘b’

and ‘c’ are Maximum Likelihood Estimates (MLEs) of parameters and the values

can be computed using iterative method for the given cumulative time between

failures data. Using the estimators of ‘a’,‘b’ and ‘c’ we can compute m�� .

Mathematical derivation for parameter estimation

Parameter estimation is important in software reliability prediction. In this

chapter we discuss the MLE technique to estimate the unknown parameters for the

software reliability model. Assuming that the data are given for the cumulative

number of detected errors ni in a given time-interval (0, ti) where ) = 1,2, … , @

and0 < �� < �� < ⋯ . < �, then the log likelihood function (LLF) takes the

following form:

1 = 3 2$&��4��J�oR/45�

Take the natural logarithm on both sides, The Log Likelihood function is given as:

( ) ( ) ( ) ( )1 1

1

logk

i i i i k

i

Log L n n m t m t m t− −=

= − − − ∑ (4.3.1)

Take the mean value function of Pareto Type IV is of the form

( ) 1 1

b

tm t a

c

− = − + (4.3.2)

By substituting Equation (4.3.2) in Equation (4.3.1), we get

57

( ) 11

1

b b bki i k

i i

i

c t c t c tLog L n n Log a Log a a

c c c

− − −

−−

=

+ + + = − + − − +

∑ (4.3.3)

( )1

1

10 1

bkk

i i

i

c tLog Ln n

a a c

−

−=

+∂ = − + − + ∂ ∑

0Log L

a

∂=

∂

( )1

1

1 1 0

bkk

i i

i

c tn n

a c

−

−=

+ ⇒ − − + = ∑

( ) ( )( )1

1

bk

k

i i b bi k

c ta n n

c t c−

=

+∴ = −

+ −∑ (4.3.4)

( ) ( ) ( ){ } ( ) ( )1 1 1

1

log log logk

b b

i i i i i i

i

b

k

Log L n n Log a b Log c c t c t b c t b c t

c ta a

c

− − −=

−

= − + + + − + − + − +

+ − +

∑

( ) { }1

1

1 1 1

1

0 log log( ) log( )

01( ) log( ) ( ) log( )

( ) ( )

log

i ik

b bi i

i i i i ib b

i i

b

k k

c c t t cLogL

n nc t c t c t c tb

c t c t

c ca

c t c t

−

−= − −

−

+ − + − + ∂ = − − + + + − + +∂

+ − +

+ + +

∑

( )

( )

1 11

1 1

1

1

1

( 1) log( 1) ( 1) log( 1)log( 1) log( 1)

( ) ( 1) ( 1)

1 1log

( 1) 1 1

b b

i i i iki i b b

i i i i

i

k

i i bi k k

t t t tt tLogL

g b n n t tb

n nt t

− −−

− −=

−=

+ + − + +− + − + +∂

= = − + − + ∂

+ − + − +

∑

∑ (4.3.5)

58

( )( ) ( )

( ) ( )

( ) ( ) ( ) ( )

( )

112

'

1 221

1

1 21

12 1 1 ( 1)

1( )

1 1

1 11

1 1

b b ii i ik

i

i ib b

ii i

bk

k k

i i kb

ik

tt t Log t Log

tLog Lg b n n

b t t

t Log tn n Log t

t

−−

−=

−

−=

++ + + +∂ = = − +

∂ + − +

+ + − + + −

∑

∑

(4.3.6)

( ) 0Log L

g cc

∂= =

∂

( )( ) ( )

( ) ( )( ) ( )

( )( )

11 1

1

1 21 1 1

(1)bb b

ki i k

i i b bi i i ki i k

b t c b t c t c cLog L b b b cn n ab

c c t c t c t ct c t c t c

−− −

−−

= − −

+ − + + − ∂ = − − − + + ∂ + + ++ − + +

∑

( )( ) ( )

( )( )1 1

1 11

1 1 1 1( )

k k

i i i i

i ii i k

Log Lg c n n n n

c c t c t c t c− −

= =−

∂ = = − − − + −

∂ + + + ∑ ∑ (4.3.7)

2'

2( ) 0

Log Lg c

c

∂= =

∂

( )( ) ( )

( )( )

'

1 12 2 221 11

1 1 1 1( )

k k

i i i i

i ii i k

g c n n n nc t c t c t c

− −= =−

∴ = − − + + − −

+ + + ∑ ∑ (4.3.8)

4.4 Interval domain Failure Data Sets

Dataset #1a: On-line Data Entry Software Package Test Data

The small on-line data entry software package test data are available in Japan since

1980 (Ohba, 1984a). The size of the software is approximately 40,000 LOC. The

59

testing time is measured on the basis of the number of shifts spent running test cases

and analyzing the results. The pairs of the observation time and the cumulative

number of faults detected are presented in table 4.4.1

Dataset #2a and #3a: Telecommunication System Data

The data set was reported by Zhang et al. (2002) based on system test data for a

telecommunication system. System test data consisting of two releases (Phases 1and

2) are shown in Tables 4.4.2 and 4.4.3. In both tests, automated test and human

involved tests are executed on multiple test beds.

Dataset #4a and #5a: Failure Data from Misra (1983)

A set of failure data taken from Misra (1983), given in Tables 4.4.4 and 4.4.5

consists of the observation time (week) and the number of failures detected per week

are errors: major and minor.

Datasets Release #1, #2 and #3 from Alan Wood Tandem Computers (1996)

A set of failure data taken from Wood (1996) given in Tables 4.4.11, 4.4.12 and

4.4.13 consists of the observation time(week), CPU Hours and the number of

failures detected per week :defects found .

Table 4.4.1: Dataset #1a (Ohba, 1984a)

Testing

Time

(day)

Failures Testing Time

(day) Failures

Testing

Time (day) Failures

1 2 8 1 15 1

2 1 9 7 16 6

3 1 10 3 17 1

4 1 11 1 18 3

5 2 12 2 19 1

6 2 13 2 20 3

7 2 14 4 21 1

60

Table 4.4.2: Dataset #2a Zhang et al. (2002)

Week

Index Fault Week Index Fault Week Index Fault

1 1 8 3 15 3

2 0 9 1 16 0

3 1 10 2 17 1

4 1 11 2 18 1

5 2 12 2 19 0

6 0 13 4 20 0

7 0 14 0 21 2

Table 4.4.3: Dataset #3a Zhang et al. (2002)

Testing

time

(day)

Failures Testing time

( day) Failures

Testing

time(day) Failures

1 3 8 3 15 4

2 1 9 4 16 1

3 0 10 2 17 2

4 3 11 4 18 0

5 2 12 2 19 0

6 0 13 5 20 3

7 1 14 2 21 1

61

Table 4.4.4: Dataset #4a Misra (1983)

Week Major

Errors Week

Major

Errors Week

Major

Errors

1 6 14 0 27 2

2 2 15 2 28 2

3 1 16 5 29 1

4 1 17 5 30 2

5 3 18 2 31 2

6 1 19 2 32 0

7 2 20 2 33 2

8 3 21 1 34 2

9 2 22 3 35 3

10 0 23 2 36 1

11 3 24 4 37 2

12 1 25 1 38 1

13 3 26 2

Table 4.4.5: Dataset #5a Misra (1983)

Week Minor

Errors Week

Minor

Errors Week

Minor

Errors

1 9 14 5 27 0

2 4 15 3 28 2

3 7 16 3 29 3

4 6 17 3 30 6

5 5 18 4 31 3

6 3 19 10 32 1

7 2 20 3 33 1

8 5 21 1 34 4

9 4 22 2 35 3

10 2 23 4 36 2

11 4 24 5 37 11

12 7 25 0 38 9

13 0 26 2

62

In the process of developing the points on the Mean Value Control chart Successive

Differences of Mean Value Function are made use of. There is a possibility of some

differences becoming zero for equal values of Time between Failures. All such

sample points are deleted and only reduced sample is considered for the sake of

effective non zero points on the control chart.

Table 4.4.6: Dataset #1b

Testing

time

(day)


( day) Failures

Testing

time(day) Failures

1 2 8 1 15 1

2 1 9 7 16 6

3 1 10 3 17 1

4 1 11 1 18 3

5 2 12 2 19 1

6 2 13 2 20 3

7 2 14 4 21 1


Week

Index Fault Week Index Fault Week Index Fault

1 1 6 1 11 3

2 1 7 2 12 1

3 1 8 2 13 1

4 2 9 2 14 2

5 3 10 4

63


Testing

time

(day)


( day) Failures

Testing

time(day) Failures

1 3 7 4 13 4

2 1 8 2 14 1

3 3 9 4 15 2

4 2 10 2 16 3

5 1 11 5 17 1

6 3 12 2


Week Major

Errors Week

Major

Errors Week

Major

Errors

1 6 13 2 25 2

2 2 14 5 26 2

3 1 15 5 27 1

4 1 16 2 28 2

5 3 17 2 29 2

6 1 18 2 30 2

7 2 19 1 31 2

8 3 20 3 32 3

9 2 21 2 33 1

10 3 22 4 34 2

11 1 23 1 35 1

12 3 24 2

64


Week Minor

Errors Week

Minor

Errors Week

Minor

Errors

1 9 13 5 25 2

2 4 14 3 26 3

3 7 15 3 27 6

4 6 16 3 28 3

5 5 17 4 29 1

6 3 18 10 30 1

7 2 19 3 31 4

8 5 20 1 32 3

9 4 21 2 33 2

10 2 22 4 34 11

11 4 23 5 35 9

12 7 24 2

65

Table 4.4.11: Dataset Release #1 (Alan Wood Tandem Computers -1996)

Test

Week CPU Hours

Percent

CPU Hours Defects Found

Predicted

Total Defects

1 519 - 16 -

2 968 - 24 -

3 1,430 - 27 -

4 1,893 - 33 -

5 2,490 - 41 -

6 3,058 - 49 -

7 3,625 - 54 -

8 4,422 - 58 -

9 5,218 - 69 -

10 5,823 58 75 98

11 6,539 65 81 107

12 7,083 71 86 116

13 7,487 75 90 123

14 7,846 78 93 129

15 8,205 82 96 129

16 8,564 86 98 134

17 8,923 89 99 139

18 9,282 93 100 138

19 9,641 96 100 135

20 10,000 100 100 133

66


Test

Week CPU Hours

Percent CPU

Hours Defects Found

Predicted

Total Defects

1 384 - 13 -

2 1,186 - 18 -

3 1,471 - 26 -

4 2,236 - 34 -

5 2,772 - 40 -

6 2,967 - 48 -

7 3,812 - 61 -

8 4,880 - 75 -

9 6,104 - 84 -

10 6,634 65 89 203

11 7,229 70 95 192

12 8,072 79 100 179

13 8,484 83 104 178

14 8,847 86 110 184

15 9,253 90 112 184

16 9,712 95 114 183

17 10,083 98 117 182

18 10,174 99 118 183

19 10,272 100 120 184

- - - - -

67


Test

Week CPU Hours

Percent CPU

Hours Defects Found

Predicted

Total Defects

1 162 - 6 -

2 499 - 9 -

3 715 - 13 -

4 1,137 - 20 -

5 1,799 - 28 -

6 2,438 - 40 -

7 2,818 - 48 -

8 3,574 71 54 163

9 4,234 84 57 107

10 4,680 93 59 93

11 4,955 98 60 87

12 5,053 100 61 84

4.5 Estimated parameters and the control limits

The estimated parameters and the calculated control limits of the Mean Value Chart

for Dataset #1b to Dataset #5b are given in Table 4.5.2. Using the estimated

parameters and the estimated limits, the control limits UCL =m�� , CL = ��

and LCL = ��P are calculated. They are used to find whether the software process

is in control or not. The values of control limits are tabulated as follows.

68

Table 4.5.1: Estimation of Parameter values of Interval domain data

Dataset Number of Samples Estimated Parameters

a b c

#1b 21

66.190046 0.978993 9.541455

#2b 14

37.628137 0.973637 6.766191

#3b 17

59.481269 0.976195 7.953206

#4b 35

105.329506 0.985185 15.109772

#5b 35 200.558922 0.985185 15.109772

Release

#1 20 123.844535 0.978352 9.144224

Release

#2 19 158.153536 0.977674 8.74581

Release

#3 12 83.720313 0.971698 5.978218

Table 4.5.2: Parameter estimates and Control limits of Interval domain data.

Data

Set

No.

of

Samp

les

Estimated Parameters Control Limits

a b c UCL CL LCL

#1b 21 66.190046 0.978993 9.541455 66.100786 33.095023 0.089356

#2b 14 37.628137 0.973637 6.766191 37.57734 18.81407 0.050800

#3b 17 59.481269 0.976195 7.953206 59.40097 29.74063 0.080300

#4b 35 105.32950 0.985185 15.10977 105.187311 52.664753 0.142194

#5b 35 200.55892 0.985185 15.10977 200.288167 100.27946 0.270754

Release

#1 20 123.84453 0.978352 9.144224 123.677344 61.922267 0.167190

Release

#2 19 158.15353 0.977674 8.74581 157.940028 79.076768 0.213507

Release

#3 12 83.720313 0.971698 5.978218 83.607290 41.860156 0.113022

69

4.6 Distribution of failure counts

Based on the failure count data given in Dataset #1b to Dataset #5b, the

software failures process is demonstrated through Mean Value Control chart.

Cumulative failure count data is used for software reliability monitoring. The use of

cumulative quantity is a different and new approach, which is of particular

advantage in reliability. ‘ a’ , ‘ b’ and ‘c’ are Maximum Likelihood Estimates (MLEs)

of parameters and the values can be computed using iterative method for the given

cumulative failure count data. Using ‘a’, ‘b’ and ‘c’ values m(t) can be computed.

These limits are converted to ��, �� and ��Pform. They are used to find

whether the software process is in control or not by placing the points in Mean value

charts.

The control limits are calculated as follows.

1 − C1 + D1 + F�GHI�JK = 0.99865

D1 + F�GHI�J = 0.00135

1 + F�GH = �0.00135��/J

�G = �0.00135�l� − 1

� = G[�0.00135��/J − 1] = (� � = G[�0.99865��/J − 1] = (P � = G[�0.5��/J − 1] = (�

The mean value successive differences of failure count cumulative data of

the considered data sets are tabulated in Table 4.6.1 to 4.6.8. Considering the mean

value successive differences on y axis and failure numbers on x axis and the control

limits on Mean Value chart, Figure 4.6.1 to Figure 4.6.8 are obtained. A point below

the control limit ��P indicates an alarming signal. A point above the control limit �� indicates better quality. If the points are falling within the control limits it

indicates the software process is stable.

70

Table 4.6.1: Successive differences of mean values of Dataset #1b

TT

(day) CF m(t)

Successive

Differences

TT

(day) CF m(t)

Successive

Differences

1 2 11.250777 4.292283 12 25 47.405365 1.007121

2 3 15.543061 3.664488 13 27 48.412486 1.719008

3 4 19.207549 3.165402 14 31 50.131495 0.378542

4 5 22.372952 5.193436 15 32 50.510038 1.940020

5 7 27.566388 4.083476 16 38 52.450059 0.277170

6 9 31.649864 3.295811 17 39 52.727230 0.767628

7 11 34.945676 1.420664 18 42 53.494858 0.236594

8 12 36.366340 7.181037 19 43 53.731452 0.659179

9 19 43.547377 2.110547 20 46 54.390631 0.204340

10 22 45.657925 0.617899 21 47 54.594972

11 23 46.275825 1.129540

Figure 4.6.1 : Mean Value Chart for Dataset #1b

UCL 66.1008CL 33.0950

LCL 0.0894

0.000010

0.000100

0.001000

0.010000

0.100000

1.000000

10.000000

100.000000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Su

cce

ssiv

e D

iffe

ren

ces

of

Me

an

Va

lue

s

Failure Number

Mean Value Chart

71


TT

(day) CF m(t)

Successive

Differences

TT

(day) CF m(t)

Successive

Differences

1 1 4.725773 3.660098 8 13 24.378376 1.186849

2 2 8.385872 2.919377 9 15 25.565225 1.827249

3 3 11.305250 4.366752 10 19 27.392475 1.040808

4 5 15.672002 4.355693 11 22 28.433284 0.300893

5 8 20.027696 1.087839 12 23 28.734178 0.281583

6 9 21.115535 1.812668 13 24 29.015761 0.512248

7 11 22.928204 1.450171 14 26 29.528010


UCL 37.5773CL 18.8141

LCL 0.0508

0.000010

0.000100

0.001000

0.010000

0.100000

1.000000

10.000000

100.000000

1 2 3 4 5 6 7 8 9 10 11 12 13

Su

cce

ssiv

e D

iffe

ren

ces

of

Me

an

Va

lue

s

Failure Number

Mean Value Chart

72


TT

(day) CF m(t)

Successive

Differences

TT

(day) CF m(t)

Successive

Differences

1 3 15.961150 3.557847 10 25 44.631457 1.912900

2 4 19.518997 7.846727 11 30 46.544357 0.632570

3 7 27.365724 3.703953 12 32 47.176928 1.094333

4 9 31.069678 1.545907 13 36 48.271262 0.243499

5 10 32.615586 3.761695 14 37 48.514761 0.456239

6 13 36.377281 3.622716 15 39 48.971001 0.616630

7 17 39.999997 1.412429 16 42 49.587631 0.189593

8 19 41.412427 2.283075 17 43 49.777225

9 23 43.695503 0.935954


UCL 59.4010CL 29.7406

LCL 0.0803

0.000010

0.000100

0.001000

0.010000

0.100000

1.000000

10.000000

100.000000

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

Su

cce

ssiv

e D

iffe

ren

ces

of

Me

an

Va

lue

s

Failure Number

Mean Value Chart

73


TT

(day) CF m(t)

Successive

Differences

TT

(day) CF m(t)

Successive

Differences

1 6 29.563231 6.464201 19 47 79.163165 1.188191

2 8 36.02743 2.832736 20 50 80.351357 0.733532

3 9 38.860170 2.608712 21 52 81.084889 1.344151

4 10 41.468882 6.720029 22 56 82.429041 0.312905

5 13 48.188912 1.934352 23 57 82.741946 0.600660

6 14 50.123264 3.498247 24 59 83.342607 0.569320

7 16 53.621511 4.483421 25 61 83.911927 0.540375

8 19 58.104933 2.577939 26 63 84.452303 0.260016

9 21 60.682872 3.375948 27 64 84.712319 0.500938

10 24 64.058821 1.013888 28 66 85.213257 0.477002

11 25 65.072709 2.761415 29 68 85.690260 0.454746

12 28 67.834125 1.638323 30 70 86.145006 0.434016

13 30 69.472449 3.527542 31 72 86.579022 0.615159

14 35 72.999991 2.891741 32 75 87.194182 0.196116

15 40 75.891733 1.015912 33 76 87.390298 0.379686

16 42 76.907646 0.947656 34 78 87.769985 0.183835

17 44 77.855303 0.886071 35 79 87.953821

18 46 78.741374 0.421791


UCL 105.1873CL 52.6648

LCL 0.1422

0.000010

0.000100

0.001000

0.010000

0.100000

1.000000

10.000000

100.000000

1000.000000

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33Su

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TT

(day) CF m(t)

Successive

Differences

TT

(day) CF m(t)

Successive

Differences

1 9 73.994022 17.762946 19 89 170.606852 0.280758

2 13 91.756969 21.404917 20 90 170.887610 0.545903

3 20 113.161887 12.580994 21 92 171.433514 1.033272

4 26 125.742881 7.999296 22 96 172.466786 1.192186

5 31 133.742177 4.023065 23 101 173.658973 0.448815

6 34 137.765243 2.421515 24 103 174.107789 0.433977

7 36 140.186759 5.303732 25 105 174.541767 0.624719

8 41 145.490492 3.612055 26 108 175.166487 1.162963

9 45 149.102547 1.632795 27 114 176.329450 0.542152

10 47 150.735343 2.971294 28 117 176.871603 0.175326

11 51 153.706637 4.422721 29 118 177.046930 0.172731

12 58 158.129359 2.677080 30 119 177.219661 0.666092

13 63 160.806439 1.448940 31 123 177.885753 0.474966

14 66 162.255380 1.346318 32 126 178.360720 0.305662

15 69 163.601698 1.254248 33 128 178.666382 1.540322

16 72 164.855947 1.544758 34 139 180.206705 1.106809

17 76 166.400705 3.330804 35 148 181.313514

18 86 169.731509 0.875342


UCL 200.2882CL 100.2795

LCL 0.2708

0.000010

0.000100

0.001000

0.010000

0.100000

1.000000

10.000000

100.000000

1000.000000

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33

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Table 4.6.6: Successive differences of mean values of Dataset Release #1

TT

(day) CF m(t)

Successive

Differences

TT

(day) CF m(t)

Successive

Differences

1 16 77.808782 10.902146 11 81 110.643761 0.679099

2 24 88.710928 2.855628 12 86 111.322861 0.494472

3 27 91.566557 4.503169 13 90 111.817333 0.345705

4 33 96.069727 4.343190 14 93 112.163039 0.326185

5 41 100.412917 3.159073 15 96 112.489225 0.207416

6 49 103.571990 1.571890 16 98 112.696641 0.100862

7 54 105.143881 1.090659 17 99 112.797504 0.099033

8 58 106.234540 2.429100 18 100 112.896537 0

9 69 108.663641 1.059895 19 100 112.896537 0

10 75 109.723536 0.920224 20 100 112.896537

Figure 4.6.6 : Mean Value Chart for Dataset Release #1

UCL 123.6773CL 61.9223

LCL 0.1672

0.000010

0.000100

0.001000

0.010000

0.100000

1.000000

10.000000

100.000000

1000.000000

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Failure Number

Mean Value Chart

76


TT

(day) CF m(t)

Successive

Differences

TT

(day) CF m(t)

Successive

Differences

1 13 93.240043 11.890836 11 95 144.064207 0.6336774

2 18 105.130880 11.968978 12 100 144.697884 0.466909

3 26 117.099858 7.528576 13 104 145.164793 0.642012

4 34 124.628435 4.040183 14 110 145.806806 0.199979

5 40 128.668618 4.070693 15 112 146.006785 0.193533

6 48 132.739312 4.641541 16 114 146.200319 0.278883

7 61 137.380853 3.401823 17 117 146.479202 0.090059

8 75 140.782676 1.649871 18 118 146.569262 0.175968

9 84 142.432548 0.786679 19 120 146.745231

10 89 143.219228 0.844978


UCL 157.9400CL 79.0768

LCL 0.2135

0.000010

0.000100

0.001000

0.010000

0.100000

1.000000

10.000000

100.000000

1000.000000

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

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Failure Number

Mean Value Chart

77


TT

(day) CF m(t)

Successive

Differences

TT

(day) CF m(t)

Successive

Differences

1 6 41.106293 8.318942 7 48 73.852276 0.960630

2 9 49.425235 7.046375 8 54 74.812906 0.412582

3 13 56.471611 7.164666 9 57 75.225489 0.254178

4 20 63.636277 4.611570 10 59 75.479667 0.121390

5 28 68.247848 3.939911 11 60 75.601058 0.117816

6 40 72.187759 1.664516 12 61 75.718875


UCL 83.6073CL 41.8602

LCL 0.1130

0.000010

0.000100

0.001000

0.010000

0.100000

1.000000

10.000000

100.000000

1 2 3 4 5 6 7 8 9 10 11

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Mean Value Chart

78

4.7 Conclusion

The parameter estimation is carried out by Newton Raphson Iterative

method. Dataset #5b, Release #1 and Release #2 have shown that, some of the mean

value successive differences have gone out of control limits i.e., below LCL at

different instant of time. Data Sets #1b, #2b,#3b, #4b and Release #3 has shown that

all the mean value successive differences are within the control limits i.e., in

between UCL and LCL, which indicates a stable process control. Hence it is

concluded that the proposed method of estimation and the control chart are giving a

positive recommendation for their use in finding out preferable control process or

desirable out of control signal. When the successive differences of failure counts are

less than LCL, it is likely that there are assignable causes leading to significant

process deterioration and it should be investigated. On the other hand, when the

successive differences of failure counts have exceeded the UCL, there are probably

reasons that have lead to significant improvement.

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