Software Reliability Model Deterministic Models Probabilistic Models Halstead’s software metric...

Software Reliability Model

•Deterministic Models

•Probabilistic Models

Halstead’s software metric

McCabe’s cyclomatic complexity metrix

Error seeding

Failure rate

Curve fitting

Reliability growth

Program structure

Input domain

NHPP

Markov


Halstead’s Software Metric (1977)

nsinstructio machine ofnumber

program in the errors ofnumber

program theofvolumn

program theoflength

soccurrence operand ofnumber totalthe

soccurrenceoperator ofnumber totalthe

program ain appear that operandsdistinct of numbe the

program ain appear that operatorsdistinct of numbe the

2

1

2

1

I

B

V

N

N

N

n

n

3000

)(log)( 21221

21

VB

nnNNV

NNN


McCabe’s Cyclomatic Complexity Metrix

One should start with 1 for the straight path through the module or subroutine . Then you should add 1 when each time you see one ofthe following keywords: IF, REPEAT, WHILE, FOR, OR, AND.Also you should add 1 for each case in a case statement. In addition,you also should add 1 more if the case statement lacks a default case.If the total score is less than 10, then the code is considered to be of high quality. (McConnel, 1993)

a

b c

ed



The cyclomatic number V(G) = e - n + 2pwhere e is the number of edged in the control graph in which an edge is equivalent to a branching point in the program (IF, WHILE, REPEAT, and CASE) n is the number of vertices in the control graph and a vertex is equivalent to a sequential block of code in the program, and p is the number of connected elements usually 1

6-5+2*1=3



McCabe (1976) notes that when used in the context of the basispath testing method V(G) provide an upper bound for the number of independent paths in the basis set of a program and an upper bound on the number of tests that must be conducted to ensurethat all program statements have been executed at least once.

Cyclomatic Number Number of Errors

Predict


Error Seeding Models

Operation Profile

Indigenous Error

Pseudo(seeded) Error

Introducing pseudo errors into the program



r

nN

kr

N

k

n

rnnNkp1

1

11 ),,;(

Where N = number of indigenous errors n1= number of induced errors r = number of errors removed during debugging k = number of induced errors in r removed errors r-k = number of indigenous errors in r removed errors

The probability of k induced errors in r removed errors followsa hypergeometric distribution.



Mills, 1970

k

krnN

)(ˆ 1

Maximum likelihood estimate

Type, location, difficulty level

Indigenous errors

Induced errorsSame?



Basin (1980) – two step procedure

One programmer detects n1 errors and a second programmerindependently detects r errors from the same program. Let kbe the common errors found by two programmers.

r

N

kr

nN

k

n

rnNNkp

11

1 ),,;(k

rnN 1ˆ


Failure Rate Models

Assumptions

•The program contains N initial faults•Each error in the program is independent and equally likely to cause a failure during test •Time intervals between occurrences of failure are independent of each other•Whenever a failure occurs, a corresponding fault is removed with certainty•The fault that causes a failure is assumed to be instantaneously removed and no new errors are introduced during the error removal process•The software failure rate during a failure interval is constant and is proportional to the number of faults remaining in the program


Failure Rate Models

Jelinski and Moranda Model (1972)

The program failure rate at the ith failure interval is

)]1([)( iNti

ψ= a proportional constantN = the number of initial errors in the programti = the time between the (i-1)th and ith failure


Failure Rate Models


The pdf and cdf of ti are

)))1((())1(()( itiNi eiNtf

itiNi etF ))1((1)(


Failure Rate Models


itiNi etR )1()(

Note:

dtttR

tRtdt

tdRtRttf

tfdt

tdR

tR

tft

)()(ln

)()()(

)()()(

)()(

;)(

)()(


Failure Rate Models


Suppose that the failure data set {t1,t2,…,tn} is given and ψis known

n

ii

n

i

tiN 11 1

1 N can be solved by

If the parameter ψis also not known

0))1(()(ln

0)1(

1)(ln

1

1 1n

ii

n

i

n

ii

tiNnNL

tiNN

NL


Failure Rate Models

Schick and Wolverton Model (1978)

Assume the failure rate at the ith time interval increases withtime since the last debugging. The program failure rate functionbetween the (I-1)th and the ith failure can be expressed as

ii tiNt )]1([)(

Note: ti is the test time since the (i-1)st failure


Failure Rate Models


2

)1( 2

)1()(itiN

ii etiNtf

2

)1()(

2

0)(iit

iitiN

dtt

i eetR


Failure Rate Models


Maximum likelihood estimate

n

ii

n

ii

n

i

tT

T

ti

T

nN

TiN

1

2

1

2

1

where

)1(2

)1(

12


Failure Rate Models

Sukert Modified Schick and Wolverton Model (1977)

Allow more than one failure at each time interval.

iii tnNt ][)( 1

2][

2

1

)(i

it

nN

i etR


Failure Rate Models

Jelinski-Moranda Geometric Model (1979)

Assume the program failure rate function is initially a constantD and decreases geometrically at failure time. The program failurerate and reliability function of time between failures at the ithfailure interval are

)(

)(1

1

ii tDK

i

ii

etR

Dkt

1)(0function geometric ofparameter

where

kk


Failure Rate Models

Lipow Modified Version of the J-M Geometric Model

Allow multiple errors removal in a time interval. The programfailure rate becomes

1)( ini Dkt

where ni-1 is the cumulative number of errors found up to the(i-1)st time interval.


Failure Rate Models

Goel and Okumoto Imperfect Debugging Model

Assume a fault is removed with probability p whenever a failureoccurs.

)]1([)( ipNti

itipNi etR )]1([)(

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Software Reliability Model Deterministic Models Probabilistic Models Halstead’s software metric...

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