A testing-effort dependent software reliability model and its application

Microelectron. Reliab., Vol. 27, No. 3, pp. 507-522, 1987.

0026-2714/8753.00 + .00 © 1987 Pergamon Journals Ltd.

Printed in Great Britain.

A TESTING-EFFORT DEPENDENT SOFTWARE RELIABILITY MODEL AND ITS APPLICATION SHIGERU YAMADA, HIROSHI OHTERA and HIROYUKI NARIHISA Department of Electronic Engineering, Faculty of Engineering, Okayama University of Science, Okayama-shl 700, Japan (Received for publication 3 March 1987) Abstract-

We

discuss

a software

reliability

witfi t e s t l n g - e f f o r t

based on a nonhomogeneous

and its a p p l i c a t i o n

to a t e s t i n g - e f f o r t

tlme-dependent

behaviour

is i n c o r p o r a t e d by a W e i b u l l number sets

curve

due

of t e s t l n g - e f f o r t

of a c t u a l

examples

reliability

to the

error

of a t e s t l n g - e f f o r t

control

process

problem.

The which

growth is expressed in d e s c r i b i n g

patterns.

data,

model

expenditures

flexibility

expenditure

software

Polsson

control

of t e s t i n g - e f f o r t

into software

growth

the

Using

model

a

several

fitting

and

p r o b l e m are illustrated.

I. INTRODUCTION In the l a s t d e c a d e , have

been

studies

software. growth

One

model

phenomenon i[3],

by

of

those

which

in the

software [4],

between

software

models

describes

Musa

reliability

relationship

software

for q u a n t i t a t i v e

Littlewood

software

many

The model

measures

such

software

system

as

software

[5] and

Yamada

model

the c u m u l a t i v e and

(e.g.

the

expected

and

the

reliability

error Goel

detection

and

Okumoto

et al.

[8]).

The

concerned

with

the

number of errors detected

time

can estimate the

is

model

of a computer

a software

testing

growth

testing

testing.

evaluation

is a

reliability

span

several initial

time-interval

of

the

software error

software

reliability

content

between

of

a

software

failures. Generally, detect

and

a

correct

lot

latent

software

testing

software

reliability

with

testing

the

of

phase

testing errors

resources

in a software

in the s o f t w a r e for

resource

the

system

development.

The

spent

system

is c l o s e l y

expenditures. 507

are

to

during Then,

associated

behaviour

of

508

S. YAMADAet al.

testing resource observed

as

expenditures

a

consumption

testing-effort

is m e a s u r e d

man-power executed The

spent test

existing

during

curve

of

testing-effort.

by the f o l l o w i n g :

the

testing

phase,

the

number

software

reliability

growth

models

above

such

testing-effort we discuss

considers

the

a software

in s o f t w a r e

testing-effort

spent

Assuming

testing

the software

a nonhomogeneous

on

Poisson process

behaviour

that

dependent

a p p l y this m o d e l

to a t e s t i n g - e f f o r t

specified

number

of

to the

reliability

model.

control

errors

by

is called

Further,

we

problem which

of t e s t i n g - e f f o r t

software

on

testing

is modelled

The model

of

error

is dependent

phenomenon

(NHPP).

amount

the

at an arbitrary

a testing-effort

the r e q u i r e d

have

growth

is p r o p o r t i o n a l

testing

error detection

of

so on.

reliability

time-dependent

expenditures.

of

expenditures.

current error content and the proportionality

the

the a m o u n t

and

rate

determines

The

hours,

which

time,

can be

of CPU

testing-effort

the

period

the n u m b e r

In this paper,

detection

the testing

cases,

not considered

model

over

to d e t e c t

during

a

given

testing time interval. In

section

2,

we

discuss

a

Weibull

function describing the time-dependent effort

expenditures,

reliability

and

section

3.

A

it

The estimation

effort parameters and r e l i a b i l i t y in

behavlour

incorporate

growth model.

testing-effort of testing-

into

a

methods

software

of testing-

growth parameters are given

testing-effort

control

problem

as

an

application of the testing-effort dependent r e l i a b i l i t y model is d i s c u s s e d several

sets

illustrations effort control

in s e c t i o n 4. of

actual

2. TESTING-EFFORT During subject in

the

the

system.

of

and

in section

5.

a

caused

by

failure

is

testing-

MODEL

software the

system

errors

is

remaining

defined

operation

to

numerical

analyses

testing

program

the m o d e l

data,

DEPENDENT RELIABILITY

software

departure

error

are presented

failures A

applying

reliability

software

to software

unacceptable

software

of software problems

Finally,

caused

as by

an a

Software reliabilitymodel software

error

remaining

testing-effort incorporates

in

dependent

the

testing-effort

2.1 Testing-effort

we d i s c u s s

software

testing

a

which

on s o f t w a r e

testing

growth.

phase.

of d e v e l o p m e n t

( B a s i l i and Z e l k o w i t z

Commonly,

effort

behaviour

the

in the

during

time-dependent

software

development

by an exponential or aRayleigh

[I] and P u t n a m

[6]).

curve

Then, Y a m a d a et

[10] have proposed a testing-effort dependent r e l i a b i l i t y

model

on the a s s u m p t i o n

during

software

curve

or

testing

that the t e s t i n g - e f f o r t

the

can

Rayleigh

effort.

describe

testing-effort

the

and R a y l e i g h

be described

curve

development

However,

curves

the

a

number

the

as

behaviour exponential

the

Then,

software

difficult

by only

data

we offer a Weibull

which

testing-effort

has

to

exponential

testing-effort

function,

of

well

by

it is s o m e t i m e s

since a c t u a l

testing-effort

describing

as

expenditures

various expenditure patterns. as

spent

develop

model

the t e s t i n g - e f f o r t

process has been expressed

al.

We

function

First,

behaviour

system.

reliability

into the software r e l i a b i l i t y

the

the

509

show curve

flexibility

expenditure

in

patterns,

as w(t)

where

= ~.B.m.tm-l.exp[-Btm],

a , B ,

and

function form,

and

=

m

m

are constant

parameters

=

the scale parameter,

m

=

the

I and

shape m

= 2,

we

have

exponential

respectively.

parameters

(I) can be estimated

m ,B, and m in

of testing-effort W(t)

to specify the

r e q u i r e d by

parameter.

testlng-effort f u n c t i o n s ,

least-squares.

(I)

testing,

B

=

B > O, m > O,

the t o t a l a m o u n t of t e s t i n g - e f f o r t software

When

~ > O,

The

and

Rayleigh

testing-effort by a method of

F u r t h e r f r o m (I), the total a m o u n t f u n c t i o n W(t) spent in the time interval

~ Stw(x)dx 0

= ~(! - e x p [ - B t m ] ) .

(0, t] is (2)

510

S. YAMADAet al. 2.2 R e l i a b i l i t y

growth model

Let(N(t), the

t ~ O} denote a counting process r e p r e s e n t i n g

cumulative

point such

t.

number

Then,

an e r r o r

NHPP as

the

of errors software

detection

detected

reliability

phenomenon

(Goel and Okumoto

up

to

testing

growth

time

model

can be d e s c r i b e d

[3] and Yamada and Osaki

for

by an

[7])

{m(t)} n Pr {N(t)=n}

=

exp[-m(t)](n

•

= O, I, 2 . . . .

), (3)

n!

where

m(t)

is

a

mean

expected

cumulative

interval

(0,

errors

to

t].

the

proportional

value number

Now,

which

of e r r o r s

we assume

current

to the

function

detected

that

error

in the

the number

testing-effort

current

indicates

the time

of d e t e c t e d

expenditures

content,

is

i.e.

dm(t) w(t)

= r(a - m(t)),

a > O,

I > r > O,

(4)

dt

where

a

and

r

is the expected is

the

teatlng-effort following

error

initial

detection

at t e s t i n g

relationship

time

error content rate t).

per

in the system

error

Solving

(per

unit

(4) y i e l d s

the

b e t w e e n m(t) and w(t) ( Y a m a d a et al.

[10]):

m(t)

Substituting

m(t)

From

(6)

(5)

= a(1 - e x p [ - r W ( t ) ] ) .

(2) for W(t)

= a(1

the

in (5), we have

- exp[-r.a (I - exp[-stm])]).

error

detection

rate

per

(6)

remaining

error

at

t e s t i n g t i m e t is g i v e n by

dm(t) d(t) ~ dt

Equation

(7)

remaining

error

expenditures

/ /

(a - m(t))

indicates is

a

(Yamada

that

function [9]).

= r. w(t).

the of

error the

(7)

detection

current

rate

per

testing-effort

Software reliability model 3. E S T I M A T I O N 3.1

Testing-effort The

defined

investigated time

tk

(k

=

I, ,

the f o l l o w i n g

can

m

be

2,

in the

...

,

testing-effort

estimated

by

a

for

,

and

a , 8

testing-effort

~

~

and

estimators

current

^

B,

(I)

The for

estimators

a , by

least-squares.

OF P A R A M E T E R S

parameters

parameters

function

METHODS

511

n).

wk

Thus,

method m

of are

s p e n t at t e s t i n g

the

least-squares

" ,

and

m

can

be o b t a i n e d

by m i n i m i z i n g

equation: n

S(a,

8, m) =

[ { inw k k=l

- i n a - i n 8 - into - ( m - l ) i n t k + Bt~) 2. Thus,

the least-squares

estimators

~ ,

,

(8)

and m are

given

by s o l v i n g

a^ = e x p [

1 01

-

(~_1)C

2

+ in~) + ~GI(~)],

(in8

-

A

[n(m-1)F(m)

A

+

{C 1 -

n

(m-1)C2}C'l(m)

(9) ^

- n [ Clkgl(m, k=l

13=

k)]

(10) (nC12 - C1C 2) ^

-

-

(m-1)(n-1)C22

^

IBC2GI(m)

+ ;In

^

^

+ ;2F(m)GI(m)

+ n8

- n; 2 ~ C2kg2(m, k)

+ {C 1 -

(m-1)C2}]F(m)

]~

k=l

{(m-1)C2k

- Clk}f(.~,

= O,

(117

k=l where n f(m,

k) = (in t k ) . t ; ,

F(m)

:

(in t k ) . t ~, k=1 n

g1(m,

m k) = tk,

01(m ) =

[

m tk,

k=1 n

g2(m,

k)

= t k2m ,

Clk = in Wk,

G2(m ) =

C2k

n

C1 =

MR 27:3-H

[ k=1

[ k=1

t k2m ,

= in t k, n

inwk,

k)

C2 =

[ k=1

in t k,

S. YAMADA et al.

512

n

C12 =

n

[ k=1

(in Wk).(in

3.2 R e l i a b i l i t y Using Weibull

tk) ,

parameters

testing-effort A

and

[3] and Yamada

detected

function,

r

and Osaki that

errors

unknown

m(t),

, and

the

in

the

growth

function

joint

data

on

the

a

and

is given

L = Pr{N(t 1) = Yl,

(Goel and Okumoto

N(t2)

cumulative

time i n t e r v a l

number

in the

mass

of

(0, tk](k = I,

are o b s e r v e d . r

probability

l i k e l i h o o d function,

m(t) can be

[7]).

Yk, in a g i v e n

parameters

the

m

reliability

in the mean v a l u e

2, ... , n; 0 < t I < t 2 < ... < tn), the

a ,

by a method of m a x i m u m - l i k e l i h o o d

Suppose

(in tk)2

^

a

estimated

[ k=1

growth parameters

the e s t i m a t e d

parameters

C22 =

NHPP

Then, model

function,

for with

i.e.

the

by

= Y2 . . . .

, N(tn)

= Yn}

Yk-Yk-1 =

where

n H k=1

{ m(t k) - m(tk_1) ) . exp[-m(tn)] ,

t O £ 0 and YO

£ O. Thus,

s i n c e the m a x i m u m

likelihood

^

estimates alnL/aa

of r e l i a b i l i t y =

(12)

(Yk - Yk-1 )!

alnL/ ar

growth

parameters,

= O, we h a v e

the

^

a and

following

r, satisfy likelihood

equations:

Yn

'

=

(I

- exp[-rW(tn)]),

(13)

a

aW(tn).exp[-rW(tn) ]

(Yk

n

=

- Yk-1)'(W(tk) exp[-rW(tk)]

k-~1

exp[-rW(tk_1)]

- W(tk_ I) exp[-rW(tk_ 1)])

- exp[-rW(tk)]

(14.) which can be s o l v e d

numerically.

4. T E S T I N G - E F F O R T Using with

the t e s t i n g - e f f o r t

the W e i b u l l

CONTRO~PROBLEM dependent

testing-effort

reliability

function

defined

model by

(2) -

Software reliability model (5),

we

consider

the

following

513

testlng-effort

control

problem: I. Software

testing

2. The initial is

error

estimated

manager

number

the

testing

in f i g u r e

value

two

at time

to

increased

to be spent

).

expected

interval

(TI,

is assumes

the

error

the

goal

of A,

at

the

growth

for the estimated

curve

goal

curve

of

of a*

is

Then,

w(t) at time

the T 1 is

(TI, T 2] (figure

of detected

Weibull

at

on s o f t w a r e

T 2.

errors

T 2] can be shown as C in figures

that the estimated

content

for the case of B,

time

in the time i n t e r v a l

is G% of

growth

expenditures

of a*

to the

of A and B as shown

for an e x p e c t e d case

the

a.

initial

realizations

a,

on

errors

(0, T2] , w h i c h content

testing-effort

system,

model.

software

On the o t h e r hand,

testlng-effort

schemed The

of

In the

T 2.

satisfy

error

typical

errors.

to c o n t r o l

testing

initial

time T 2.

T I based

dependent

to detect

1-a is c o n s i d e r e d

satisfied

time

of a * in the i n t e r v a l

TI,

the d e t e c t e d

we h a v e

arbitrary

decides

estimated time

at specified

in the software

reliability

the estimated From

content

at

testing-effort 3. The

is terminated

in the

1-a.

1-b time

Then,

testing-effort

it

function

at time T I is

w1(t ) = ~181mtm-1.exp[-81tm],

a I > O, 81 > O,

(15)

m > O, 0 < t _< TI,

and

the expected

interval

Welbull

testing-effort

function

in the time

(TI, T 2] is

al 81 expr-'~iT I tD w2(t)

= u282mtm-l.exp[-B2tm],

a2 =

]

> 8 2 exp[-82T I]

0

82 > O, m > O, T I < t % T 2,

where

82

parameter

is B2

a

constant represents

parameter a

modified

to

be

estimated.

expenditure

rate

The of

514

S. YAMADAet al.

~

a~ t~ no

A I

B

LAJ LU nl: 0 (2)

0

T2 (a)

I

I

I I

I I

I I

I I

I

~

n~ o u_ u_ u~ i

I I

11

W2(~)

Wl(f') !

z uu

0

I

I

I

I

I

I

I

I)

T1

T2

TIME (b)

Fig.

I.

The

testing-effort

control.

SoRwarcrcliabilitymodcl testing-effort (16) we have

a

in the

time

the f o l l o w i n g

interval

T2 ~TI w2(x)dx

a2(exp[-~T where

m(T I)

detected

is

the

the f o l l o w i n g

T 1.

Then,

using

- e x p [ - r W 2 ( T 2 - TI)]),

(17)

=

(18)

1] - exp[-B2T2]) ,

expected

up to time

(TI, T2].

relationship:

= m(T I) + (a - m(T1)).(1

W2(T 2 - T I) =

515

cumulative

Thus, 82

number

of

can be obtained

errors

by s o l v i n g

equation n u m e r i c a l l y : I

a 2 ( e x p [ - B 2 T I] - exp[-B2T2])

a - a in

-

[

r

].

(19)

the

time

a - m(T I ) A

The

modified

testing-effort

function

w2(t)

in

^

interval (19).

(TI,

T 2]

Therefore,

time

interval

can by

(TI,

be

obtained

increasing T2]

by u s i n g

the

according

B 2 satisfying

testing-effort

to

the

Weibull

in

the

testing-

^

effort

function

can a c h i e v e errors

w2(t) , the m a n a g e r

the, goal

of a

to be detected

which

In this s e c t i o n , to

show

numerical

dependent

testing-effort error data

DSI :

problems.

sets with

(tk'

Wk'

(months) time. wall DS2:

(~k,

wall

is

software

for

the

and indicate Consider

(k

=

I,

measured

error data

testing-effort

the e x a m p l e s

the f o l l o w i n g

of the

software

data:

2, on

The t e s t i n g - e f f o r t

... the

, 35) basis

where of

tk

calendar

d a t a are the n u m b e r

of

clock hours. Wk,

(months) time.

actual

testing-effort

Yk)

of

EXAMPLES

illustrations model,

number

T 2.

we a n a l y z e

reliability

development

is the c u m u l a t i v e

up to time

5. N U M E R I C A L

of s o f t w a r e

Yk ) (k is

=

measured

I,

2, on

The t e s t i n g - e f f o r t clock hours.

... the

,

13)

basis

where of

tk

calendar

d a t a are the n u m b e r

of

516

S. YAMADAet al. DS3:

(tk,

Wk,

y k ) (k

(months) time. CPU

is

=

1,

measured

2, on

...

the

The t e s t i n g - e f f o r t

,

12)

basis

where of

tk

calendar

d a t a are the n u m b e r

of

hours.

[14].

w h i c h were cited by Brooks and M o t l e y

5.1 Data a n a l y s e s First,

we e s t i m a t e

testing-effort squares

the p a r a m e t e r s

function

estimators

of (I).

~,

,

and

a, 6,

and

m

in the

F r o m (9) - (11), the l e a s t m can be o b t a i n e d .

For the ^

data sets DSI

- DS3,

the l e a s t - s q u a r e s

estimators

^

o,s,

6,s,

^

and

m's are ^

^

DS I:

= 2253.2,

~

B = 4.5343x10

-4,

m = 2.2580,

(20)

^

DS 2 :

a

= 259.68,

= 2.5052x10

-2,

m = 1.8087,

(21)

= 3.7032xlO

-2,

m = 0.97559.

(22)

^

DS 3 :

~

= 30991,

B

^

The e s t i m a t e d

testing-effort

DS I are p l o t t e d effort

function

in f i g u r e 2 a l o n g

w(t) for the d a t a

w i t h the a c t u a l

set

testing-

data. ^

Using

the

estimated

testing-effort

parameters

^

a,s, B 's,

^

and

m's for

the d a t a

sets

DSI

- DS3,

the s i m u l t a n e o u s

non-

150 Actual F-r~

o

hi_ i, ILl

100

I

CD Z p-O,

50 F i tied

0

I0

20

30

40

TIME(MONTHS) Fig. 2.

The estimated

testing-effort

actual data set DS I.

^ f u n c t i o n w(t)

for the

Software reliability model linear the

equations

data

sets

(13) and

DSI

517

(14) can be s o l v e d

- DS3

to

obtain

the

numerically

for

maximum-likelihood

A

estimates r.

a

and

r

of r e l i a b i l i t y

The e s t i m a t e d p a r a m e t e r s A

for the data

r

A A

=

The e s t i m a t e d

-3 ,

(23)

r = 2 . 8 2 9 7 ~ 0 -3 ,

(24)

mean

in f i g u r e

r = 1 . 1 0 6 0 ~ 0 -4.

value

function

3 along

with

data. The K o l m o g o r o v - S m i r n o v

effort

1.5791x10

^

DS 3: a = 3850.1,

0saki

sets DSI - DS3 are

^

DS 2: a = 2511.8,

and

a and

A

DS I: a = 1 3 9 7 . 6 ,

plotted

growth p a r a m e t e r s

[7]

and

dependent

Yamada

(25)

f o r the d a t a

the

actual

software

goodness-of-fit [9])

reliability

s e t s DSI

test

can

show

that

models

with

the

is

error

(see Yamada

the

testing-

estimated

mean

A

v a l u e f u n c t i o n s m(t),s are w e l l - f l t t e d DS3.

Also,

figure

4 shows

to the data sets DSI -

the e s t i m a t e d

error

detection

rate

A

function

d(t) in (7) for the d a t a

set DS1.

1500

O0 C~ 0 OC Lul

1000

r'~"

Lu 0 09

u_ 0

500

ctual

n~ UJ ~3D D Z

I

I

I

I

10

20

3O

40

TIME(MONTHS) ^

Fig.

3.

The

estimated

actual

5.2 E x a m p l e s We software where

data

mean set

DS

of t e s t i n g - e f f o r t

apply error

the

sets that

DSI

function

m(t)

for

the

I.

control p r o b l e m

testing-effort

data

it is a s s u m e d

value

control

problems

- DS3 a n a l y z e d

in

to

the

section

5.1

518

S. YAMADAet al.

.18 U_l l---

zo

.10

(J u.J l-u.J

c~

o~ o c~ L~J

.05

0

I

I

l

I

10

20

30

40

TIME(MONTHS)

Fig. 4.

The

estimated

error

for

the a c t u a l

data

detection set DS

rate

function

d(t)

I.

DSI:

T I = 35,

T 2 = 50,

a

= 1360

(G = 9 7 . 3 % ) ,

(26)

DS2:

T I = 13,

T 2 = 20,

a

= 1400

(G = 5 5 . 7 % ) ,

(27)

DS3:

T I = 12,

T 2 = 20,

a

= 3200

(G = 8 3 . 1 % ) .

(28)

for

sets

Then,

the

DSI

DS3

are

DSI:

B2

= 2"77356xi0-4'

(29)

DS2:

B2

= 3"71777xi0-3'

(30)

DS3:

B2

= I"07166xi0-3"

(31)

The -

-

modified

modified

DS3

are

expenditure

estimated

~2,s

the

data

as

testing-effort

plotted

rates

in f i g u r e s

functions 5 - 7.

for

the

data

sets

DSI

Software reliability model

519

1500 tn r~

o

1000

,,¢:

500

~-

~rj

0

10

20

30 T 1 40

50

100

I

I--. 0:: I.II.I.. LI.I I Z

(t) 50

I-U') IJJ I.--

I

0

Fig.

5.

I

10

The m o d i f i e d

I

I

20 30 T 1 40 TIMEIMONTHS)

testing-effort

data set DS I.

I

50

f u n c t i o n for the actual

520

S. YAMADAet al.

1500 rj 3

o

,=.,

looo

,'-;-, 50o

y, / i '

J 0

,

,

5

10

I

T.

,

15

20

30

I-0 I.ILI. LLI I (...q Z I--LIJ

20

w2(t )

10

0

Fig. 6.

The modified

5

10 T 1 15 T [ ME(MONTHS)

testing-effort

data set DS 2.

function

20

for the actual

Software reliability model

521

4000 "

o ~ n," 0 n," I.U

3000

I~

2000

I-LI_

o

U')

1000 I

5

0

I

10 T

I

I

15

20 I I I I I I I I I I I I I I

1500

I---

o

i, I.L

/.,w2(t)

1000

I Z I---

w1(t)

500

l.U

I

I I I I I I

15

20

I--

I

0

Fig.

7.

The data

5

mbdifled set

I

10 T I TIME(MONTHS)

testing-effort

DS 3.

function

for

the

actual

522

S. YAMADAet al. REFERENCES [1]

V.R. Basili and M.V. Zelkowitz, "Analyzing Medium-Scale Software Development," Engineering,

[2]

Proc. 3rd Int. Conf.

pp. 116-123

(1978).

W.D. Brooks and R.W. Motley, Software

Reliability

Software

"Analysis

of

Discrete

Models," Technical Report RADC-

TR-80-84, Rome Air Development Center, New York (1980). [3]

A . L . Goel

and

K.

Detection

Rate

Okumoto,

Model

for

"A Time

Dependent

a Large

Scale

Error

Software

System," Proc. 3rd USA-Japan Computer Conf., pp. 35-40

(1978). [4]

B. Littlewood, "Theories o f Software Reliability: How Good

Are

Trans. [5]

They

and

How

Can

They

Software Eng., Vol. SE-6,

J.D. Musa,

Be

Improved?

pp. 489-500

"IEEE (1980).

"The Measurement and Management of Software

Reliability,"

Proc.

IEEE,

Vol.

68,

pp.

1!31-1143

(1980). [6]

L. Putnam, "A General Empirical S o l u t i o n to the Macro Software Sizing and Estimating Problem," IEEE Software Eng., Vol. SE-4,

[7]

pp. 345-361 (1978).

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"Software

Modeling:

Applications,"

Models

and

Trans.

Reliability IEEE

Gowth Trans.

Software Eng., Vol. SE-11, pp. 1431-1437 (1985). [8]

[9]

S. Yamada, M. Ohba and S. Osaki, "S-Shaped R e l i a b i l i t y Growth M o d e l i n g for Software

Error Detection,"

Trans.

pp. 475-478 (1983).

Reliability,

S. Yamada, Japanese),

[10]

S.

Yamada,

Vol. R-32,

"Software

Reliability

Soft Research Center, H.

Ohtera

and

R.

Evaluation,"

IEEE

(in

Tokyo (1985). Narihisa,

"Software

R e l i a b i l i t y Growth M o d e l s with Testing-Effort," IEEE Trans. Reliability, Vol. R-35, pp. 19-23 (1986).