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 Total Downloads : 631
 Authors : Rahul Chaudhary , Darothi Sarkar
 Paper ID : IJERTV1IS10446
 Volume & Issue : Volume 01, Issue 10 (December 2012)
 Published (First Online): 28122012
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
A TestingEffort on Software Reliability Estimation Model
Rahul Chaudhary , Darothi Sarkar
Department of Computer Science, Amity University, Noida, Uttar Pradesh, India
Abstract The software reliability growth models have become prominent in use. Although we know that the use of such model is delaying the SDLC (software development life cycle) i.e. during testing because these models requires failure time data in order to determine the models parameters. At a time we have more than one model such as Basic Execution Time model which can apply to SDLC in beginning, as its parameters can be directly depends on software characteristics although it only implies uniform execution of program instruction i.e. program without loops and branches. We have another model Extended Execution Time model that consider this non uniformity of program, which is related to measurable characteristics of software product, and this help to identify
which model is best to apply for Reliability estimation according to degree of uniformity i.e. Uniform execution
reliability. These various models which are used are analytical in nature; generally have one or more variable whose values are unknown. Hence for the application of this model, one must first estimate values of unknown parameters. The only software reliability model that has usefully in predicting reliability prior the availability of all data (that are required to estimate reliability) from the test the Musas Basic Execution Time model [MIO]. Parameters of this model can be similar to such measurable software product and process characteristics as faults density, number of newly developed source instructions, the number of machine instruction and speed of the machine on which the product runs. Same as BET (Basic Execution Time) model, EET model accept two parameter as in BET model and the third parameter is a ,
a
Basic Execution Time Model, Nonuniform execution Musa okomoto logarithmic model, Our work is to perform EET
Table 1
Types of execution
Model (Extended Execution Time) for software reliability estimation with testing side by side i.e. implementation of Functional Testing (part of Black Box Testing (Structural testing)) with Representative Testing in parallel over the result of EET model.
This helps us as follows:
Applying functional testing together with Representative testing after or parallel to EET model help us to detect and remove the concrete faults and develop concretely reliable software Structural Testing Methods have been underestimated due to less deterministic results [1] although it supports automated constraint solving capacity and on another hand Representative Testing [3] that allow reliability estimation modeling, to give the desired quantification or determination. If we use both testing in parallel or conjunction then results from Structural Testing can help us to update the reliability estimates from EET model that are conventionally connected with Representative testing. Here we use order statistics to merge the observed failure rates of faults, nothing to attention that how those faults were identified [3].
Index Terms Basic Execution Time model (BET model), Extended Execution Time Model (EET Model), fault detection, faults spraying, Functional Testing (Structural testing)), imperfect ordering, nonuniform execution, perfect ordering, Representative testing, Sorted activity profile, test case, uniform execution.
I. INTRODUCTION
P
ositively towards the development of reliable software we use various reliability growth models these various models estimate the appearance of failure occurrence during testing, and removal of these faults helps in improvement of
>4 very non uniform, characterized by many infrequently
extended path associated with branches and loops with many iterations.
.2, 4 moderately uniform.
< .2 nearly uniform, few infrequently executed path associated with branches, few loops with few iterations associated with each loop.
evolved related to a measure of uniformity of instruction execution. Where a is defined as using Table1 [2]: The change in the table 1 is in upper limit which is reduced to 4, because each program complexity increases rapidly with little
change. I.e. When a =0 all instructions are executed
uniformly, at that time EET model identical to BET model and for large value of a parameter show very non uniformity
in execution of instruction then, [MIO] the logarithmic model is best applied i.e. for non uniform execution in case of moderately uniform condition, we refer to apply logarithmic model as we are certain on that the, there is obviously degree of non uniformity.
The focus of EET model is its capability to estimate the parameter before the collection of failure data. Two parameter can be predicted in precisely the same manner as for BET model as parameter related to program execution hence the
a parameter can be derived from instruction execution
profile data.
EET model over comes one problem faced by users in deciding whether to use the BET model or the logarithmic model. The a parameter is decided for program using
UNIX profile facility.
NHPP model as it incorporates both model depending on value of the parameter a . Measure of uniformity done by
finding linear execution frequency [1 section2] and measure
of non uniformity by activity profile and sorted activity profile [1section 5]. Hence various models simulates failures occurrences during testing and whose removal result growth in reliability is already defined, now applying functional testing together with Representative testing help us to detect and remove the concrete faults and develop concretely reliable software. In functional/ structural testing test cases are created using functional specification [6] or program requirement. Example when I option used in CMD line of UNIX sorting program it causes sort to ignore characteristics outside ASCII Range. Hence we to point that different individual are able to construct different test sets for testing a single program in hand. Thus, it is also likely that the designed test sets will cause to detect different tests sequences which are further used in designing to variations of functional testing used to elaborate more to reliability test. All this testing done with representative testing which gives the required quantification and when it is merged with structural testing, this combination is shifting the observed random variable from inter failure time to a detailed analysis of debugged faults. Representative testing helps to find most commonly occurring faults, first hence time to detect new faults that are generated due to new innovations increases over time. Hence merge of these two testing methods i.e. structural and representative testing may do well because
Structural testing considered as defensive approach against potential uncertainly [3 section 1], in the operational profile and catalyses the faults detection states in detection of testing process.
Representative testing can give quantified measures of progress being made using order statistics to combine the observed failure of faults. Section 3 explains the summary of EET model, section 4 explains the parallel implementation of
component to construct an efficient reliability estimation model? And (2) how do we deal with the uncertainties that stay in this model due to the absence of operational profile?
B. Estimating and analyzing software reliability: during teting is a branch with over 30 years of past experience. Many reliability models have been given: Software Reliability Growth Models i.e. SRGMs which are used to estimate software reliability using statistical approaches [6, 8, 9, and 10]. The common theme across all of these approaches, however, is their possibilities to implementationlevel artifacts, and reliability estimation during testing. At architectural level, existing reliability estimation approaches consider only the structure of the system in hand. The exceptions are [7, 11, 12, and 13]. However, none of them approaches consider the effect of a components internal behavior on its reliability.

SUMMARY OF EET MODEL
EET modal summarized as follow first divide the system into parts (cells) of equal size, although it should be focused that there is uniform execution of the instruction in each cell. The failure caused by fault occurrence can be modeled with the help of BET model[1 section 2] , considering a unique functional form to show the amount of time spent in each cell that allow the bet models for all the cell to be merged to give a compound model for the software in whole. Consider our
division consists of m cells. Let e i represent the execution time utilized for instruction in a cell i and Fi (ei ) are the total failure that occurs in execution of instruction in cells i .Now Fi (ei ) is random variable whose distribution [1 section 2]
testing paradigms, and section 5 is conclusion.
II. RELIABILITY AND TESTING
Prob{ F(e i
n (e )
) = n} = i i e
n!
i (ei )
2.1
Reliability is a famous concept that has been celebrated for
(e ) – means value of F (e )
i
years as a necessary attribute of a product or a person. Its i i i i
beginning was in 1816, far before than any one guess. The word reliability was first given by Samuel Taylor Coleridge
e – Execution time for machine instruction in cell i .
who was a poet. In statistics, reliability is the consistency of a set of parameters, which are used to satisfy a test. Reliability is
Since execution of instruction is uniform
In each cell, then according to BET model
i ei
given as
inversely related to unknown error. In Psychology, reliability
(e ) =
w (1 eiei )
2.2
refers to the consistency of a symptom. A test is considered reliable if we get the same result in every pass of test. For example, if a test is created to measure introversion, then the time the test is heading to a subject, the results should be
i i oi
o
w – Faults in cell i at time ti 0
i
i
– The rate of failure for each fault in i th cell.
approximately the same every time.
Testing is a necessary and critical part of the software
Recursively both parameter
w and
o
i
i can further be
development process, on which the reliability and quality of the product completely depend. Testing is not bounded to the finding of bugs in the software, but also boost up confidence
allocated in terms of software characteristics of cell i [1 section 2]
w = DS 2.3
in its proper working and with the selection of functional and nonfunctional properties. Testing related activities measure the complete progress of process and may consume an enough effort required for producing software.

Challenges in reliability estimation: (1) how can we effectively use the architectural specifications of a
oi Ii
R
i = K
I
M
i
Where
I
S – Source instruction cell i
i
2.4
I
M – Machine instruction in cell i
i
D – Fault density
analog of (a + 1).1 ma , and similarly dx continuous analogue of 1 .
i
Let e = m e
i1
(i.e. Total execution time of instructions in
m
Consider the density of fault D is equal is equal for each cell
cells of software components) and express
e as e . The
the S I
i.e. source instructions and the
M I i.e. machine
i i i i
i
factor is proportion of time spent in i th cell. Since the cell
instructions are uniformly distributed over the cells, the fault
m
i give a partition of the software component [1section 2].
i =1
number w
o
i
S
I
in cell i can be given by
w
m
i 1
Let F(e) =
F (e ) using the super position property of
wo = D o
i
m m
2.11
i i
i 1
Where
wo = DSI
, total software component fault. The fault
Poisson probability distributions, the distribution of F(e) is also derived same i.e. as Poisson random variable with mean
exposure ratios i are same to each cell i , the per fault
intensity [1] [sec4] can be given as
m m i
(e) i (ei ) i ( i ei )
2.5
2.12
i1
i1
R
m
M
Now equation 2.5 is for any of the software partition and for any execution time distribution in partition of each cell. Consider a single distribution of time utilized in each cell for
the division. Division of software component in to m equal
cells, in such a way that there each cell contains equal number of machine instruction. Hence
i
I
m
Where R
M I
1 e
i
i
2.13
Indicates per fault failure intensity of total software component, in case where execution is uniform. Thus
Ii
m
M = M I 2.6
Without generality loss, sorting the partition according to time
w
m
(e) o
m i1
e
spent in each cell. It is a Sorted Activity Profile [2 section 2]. Let time proportion spent in cell i be
1 i a
(e) wo
m 1 eiie 1 i1 m
i
2.7
m i a
c m
(e) w
1 1 ei (a1) m e 1
2.14
Since
m
=1, then the constant(c) can be evaluated as:
i1 m
o
i
i 1
Hence equation 2.14 summation is just integral approximation
i a
c m
1
(e) e
i (a1) xae
dx
m i a 1
0
m
m
c m
i1
2.8
(e) , can be expressed as:
Equation 2.8 is identified as the numeric approximation of the
integral xa dx , where 0 x 1is partitioned into m cells
1
a
o
1 (e) w
0
1 e
0
i (a1) x e dx
2.15
of size 1 and the area under curve x a is probabilistic by the
m
Equation 2.15 can further compressed as
sum of individual cell area. Hence equation 2.8 written as
(e) wo 1 H (a, l)
2.16
1
c m x a dx
0
m a 1
2.9
1
H (a,l) elxa dx,l (a 1)..e
0
2.17
Thus, equation 2.7 written as
i
1
a
i (a 1) m m
2.10
Now equation 2.17 is for sorted activity profile (SAP) that can be draw with function x a . Execution time e 0 , correspond to l 0 which on placing in equation 2.17 give H (a,0) 1,
i , play vital role in formation of Sorted Activity Profile of the software component. Note, (a + 1).x a is continuous
hence equation 2.16 after substitution of above value becomes
(0) 0 .
Similarly, when e reaches , then equation 2.17 gives
In this section, we can reach very near to find the reliable
H (a, ) 0 , so then equation 2.16 becomes
and the failure intensity
() wo
software as by findingequations that can be implemented to draw sorted activity profile and cumulative failure versus time
o .H.(
(e) w . a a 1
,l)
2.18
graphs.


PARALLEL IMPLEMENTATION OF TESTING
Hence, the above result not depend on a i.e. factor of uniformity, hence model reduced to BET model, properties of
H (a, l) are summarized in appendix A.
PARADIGAM
Here comes the most crucial step for your research publication. Ensure the drafted journal is critically reviewed
Now, suppose a in equation 2.17 and 2.16 then by
execution of instruction is very non uniform. If l depend on
a and reaches , however let a with (a 1)
remaining constant. This show would have to approach 0, due to which wo approach 0. Unless wo approaches
This section is described in two parts 3.1) Step to implement functional testing 3.2) Representative testing

Step to implement functional testing:
All part is assumed to be conducted over the UNIX sorting
.Take
a limit such a way
(a 1) = 2
and
program [2] or variations obtained by adding faults. We have
wo 1
remain constant. From equation 2.18 for failure
Notations
e, e'
[Execution, exposure] timeintensity becomes
(e) 1 H (1,
2 e)
2.19
Perfault hazard rate
1 e2e
ei Time of failure i
(e) 1
2 e
EFi
Reliability estimates for functional testing (
Integrating last two equations, we get
i 1,2 )
(e) 1
2e 1 e x
dx
2.20
FR Fault reduction factor
2 0 x
The two functional testing usage causes explained in
(e) 1 o(
2
2 e)
introduction.
Hence we use two different specification sequences and also
Properties of o(x) function are summarized in Appendix A. Now equation 2.19 & 2.20 shares many same properties of Logarithmic NHPP model described in reference [MIO].This
lead to EET model to Logarithmic NHPP model.
The cumulative failure number and failure intensity may be expressed as
there combinations for functional testing. Testing that we use is referred as functional_test_1 & functional_test_2 below given discussion as follow: these two testing performed on data collected [summarized in appendixB], failure data gathered with the help of two different sets of functional test data from appendixB.
(e) 1
2
log( 2 e 1)
2.21
These failure data were used with EET model to compute
reliability estimates. Reliability, estimates that were obtained After K failures.
(e)
1
2 t 1
2.22
The execution time (eunits), and by comparing performance of different method of testing.
Due to asymptotic behavior of both models, cumulative number of failure and failure intensity as e
Select K=10 and e=12.3, as they were the smallest value at which FR . > 0. Exposure time > total execution time
(e) 1 log(
2
(e) 1
2e
2 e)
expanded after the software brought to its operational environment.
The above explanation is further used to draw graph between reliability and exposure time for the function_test_1, function_test_2, also for the block testing and random testing.
Similarly, for small e , the EET model cumulative number o
failure can be
l 2 l 3

Representative Testing (as explained in introduction) 3.2.1) Perfect Ordering
(e) 1 l
2
2.2!
3.3!
… ,
Let start with the consideration that is common to most of reliability models, that are used during the testing, faults
This is much same as Logarithmic NHPP expansion for small
e for the cumulative number of failure
detected in nonincreasing order by operational failure rate.
I.e. our debugging and testing process remove the most
1
l 2 l 3
common faults first. The relation between total set of faults
(e)
l 2
3 …
and observed faults, which is then given by:
2
i
k i1.k
m1i.m
,1 i k
3.2.1
i =operational failure rate.
And then after K faults the program failure rate is
mk
this paper.
.
k i m
i1
APPENDIX A
Properties of H(a, l)
We have to follow little hood model [9] and we use to take the effect of fault ordering as more visibly by explicit use of order statics.
This appendix drives and summarized properties of function
H (a, l) . The definition of H (a, l) is
Although we dont know value of for faults that have not
yet been detected, but we try to compute their expected estimate.
1
H (a,l) e
0
lxa dx
A.1
Note, that H (a,0) 1
and
H (a, ) 0 with a little
E(r.m ) 0 . f r.m ()d
change of variable (exchange lxa with x ) during integral in
Using density function [4]
mk
E(k ) . f j.m ()d
3.2.2
(A, 1), the following alternate expression for obtained.
H (a, l) can be
j 1 0
1 1 1 1
A.2
This is resulting estimation for k .
3.2.2) imperfect ordering
H (a, l)
1
al a
1
e x x a dx
0
1
Here is assumption that decreasing order of failure rate is only an approximation detected by representative testing. For any
H (a, l)
(
1
a
al a
, l)
two faults, there must be a certain probability that less common fault happened to found firstly. A perusal of data that
Where
( , l)
1
a
is the incomplete gamma function [AB
publish sets such as [5] reveals although failure rate range may indeed several order span of magnitude, hence the expected decrease in failure rate from one to the next fault is quiet small when compare to overall program failure rate, i.e. some fault which are out of order are found in any particular test history.
page number 260].
If the exponent function in (eqnA.1) is integrated step by step and expanded in its power series, the following power series expansion for H (a, l) results.
We would think that order of this sort odd deviation or fault
H (a, l)
(l)m
A.3
detection will be comparatively small. I.e. a fault is not to be found more than a few positions out of expected order. Thus if, we were to sort fault into non increasing order by failure rate, the reconstruction of resulting sequence is almost happen expected order of fault detection. We modeled by assuming that
m0 (ma 1)m!
l ; a 0
For specific value of a, functions:
H (a, l) reduces to simple
i k i1.k
m1i.m ,1 i k , is a much weak
1
l l
A.4
condition that the equation 3.2.1 although equation still hold
as an estimate of s .
H (a, l) e
0
1
dx e
1 el


CONCLUSION
We have given reliability growth model which allow quantification of the effect of applying directed tested method
H (a, l) el dx
0 l
For integer values of 1 (e.g.
a
A.5
1 k ), equation A.2 can be
a
over the EET model. By shifting directly observed inter failure
repeatedly integrated by parts to obtain
time quantity to the individual faults failure rate. This model allows all test plans (that are the combination of representative
H( , l)
1
k
k!el (el E (l))
k
lk
A.6
and directed (functional) testing methods and contribute) to
Where E k l) are partial sums of the power series for the
attain a common goal of reliability, although doing the EET model with parallel implementation of representative testing
exponential function i.e.
k 1 (l)m
A.7
and directed (functional) testing is great idea. This is also a way to explore more in reliability estimation. Implementation
Ek (l)
m0 m!
, E0 (l) 0
of this is done future trends since it combines the advantages of both EET model and testing techniques, thereby providing optimal solution of reliability estimation. According to Recent researches, by the end of 2013 there will be many new methods are evolved in reliability estimation field by using
Differentiating the power series expansion (A.3) for
H (a, l) step by step, the result obtained is:
dH (a, l) (l)m1
dl m0 (ma 1)(m 1)!
Using the relation
(ma 1) (a
1)(m
1)
a 1
a 1
A.9
Obtain the sequence s1 , s2 ,…, s29 then,
^
An asymptotic expansion for
H(a, l)
valid for large values
s ^ ^ ^
= t1 , t2 ,…, t ~ ,
of positive l can be obtained by first rewriting (A.2) as
ki
1 1
1 1
29
H (a, l)
1
e x x a
dx
A.10
k gilb ki
29
al a
a l
i1
Integrating by parts in the integral in (A.10) results in
^ l
t j tij
l, j 1,2,…,k
H (a, l)
1 1 1
i1
a
1
l a
A.11
l max( m  km j);
e l 1
1 a (1 a)(1 2a)
…
All failure data obtained while testing are used to calculate reliability estimates from the EET model and described in
al
( al) 2
(al)3
Section 2. Parameter in EET model were estimated with the
We define the function o(l) as
l l 1 e x
o(l) H (1, x)dx x dx
A.12
help of failure data, if debugging is imperfect, implies that even in case on which program failed, testing is continued without repairing the fault.9 faults were sprayed in to the Unix
sorting program ,the fault type are a sample from commonly
0 0
This function appears in the limiting version of the EET model where a > . Integrating the power series expansion for the exponential function results in
x2 x3
occurred faults that are entitled by other researchers. The sorting program itself is about 1000 lines of executable c language code. The failure data was generated is by executing the below given sequences of steps.
1: Ten faulty versions were conducted on sorting as follow
o(l) x
2.2!
3.3!
…
A.13
Sort0: sorting having all 9 faults
Sort1: sorting contains faults f2,,f9
Following a similar approach as was done in deriving A.11, the following asymptotic expression for
o(l) log(l) 0.5172 ez 0! 1! 2! …
SortI: (2<=i<=8): sorting having faults f1 , , fi1, fi+1,
,f9
Sort9: sorting having f1,,f8 faults
l
APPENDIX B
l 2 l 3
2: Repeat this step for each of 3 testing methods until every fault is not found.
Failure data collection
Experiment conducted to obtain failure data using 3 testing methods i.e. random testing, block testing and functional testing, all parts of this experiment have been implemented using Unix sorting program or variants of this program obtained by spraying faults.
For random testing, we come with different seeds to get 29
[Initialize] faultsdetected to 0 [testcase construction] construct a test case C, this is manual work for functional & block testing, and by program in case of random testing. In any case, output correctness is checked automatically by an oracle. [Program execution] determine if test case C causes at least one fault to remove. Execute sorting and sort 0 against C. if~ ~ result of sort 0 is correct then C is success with respect to
different sequences of failure data si , i 1,2,3,…, 29. si is obtained by executing sorting on a total of Ti test cases. Testing stop after all 9 faults in sorting were removed.
Notations:
~ ~ ~ ~
UNIX sorting program. Repeat step 2b. If the output of sort 0 is not correct then a failure has occurred. Note the total execution time b/w previous & this failure.
[Fault removal] imperfect debugging procedure is used. Execute sort 1 to sort 9 on C constructed in step 2b as theoutput of all sort i is reviewed sequentially, starting with
si ti , ti.2 ,…, t ~ ,1 i 29
i, i [1,9]
i ki
i=1. The first correct sort
implies that
f i is the
~
ki Total failure required before all the 9 faults revealed from sort
^
s Sequence of average failure interval, which help to
obtain estimation of reliability
f i Fault i,i [1,9]
Rearrange the sequence such that k1 k2 … k29 to
fault which is responsible for the failure. In this case f i is the fault which is responsible for the failure. In this case f i is considered removed. Replaced sort 0 by sorts i, increment
faultsfounded by 1 and go to step 2e. If none result from all
9 sort is correct then assume that the fault is not removed and repeat from step 2b.
[Check for termination of the experiment] if faults founded< 9, then repeat from step 2b; otherwise this procedure
terminates for one testing method
Step 2d implements prefect debugging by analyzing the outputs of all variants of the sort program since sort 0 having all 9 faults correct behavior of this program says that test case C cannot remove any of these faults. However, if sort 0 not work or fails then we need to select which fault, or a combination results in failure.
To do so, we execute the other remaining 9 versions of sort. If exactly 1 sort out of 9 sorts generates correct output on Test case C, we say that \
The fault that caused sort to failure is f i . The fault selected & corrected.
If none of all version of sorts i, generate correct output. Then
it is obvious that failure is caused due to combination of all 9 faults and not by only one fault. In this case no fault is corrected.
All experiment were conducted on a sunsparcmachine ([2] section 4) and method used is one of perfect debugging assumption. But it is not found yet that how imperfect debugging actually work in real environment ([2] section 3).
ACKNOWLEDGMENT
The authors are very thankful to their respected Ms. Darothi Sarkar, faculty, computer science department, Amity University, Noida, Uttar Pradesh, Authors also pay their regards to Dr. Abhay Bansal, professor & Head of ASET, Amity University,Noida, Uttar Pradesh for giving their moral support and help to carry out this research work.
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AUTHORS
First Author Rahul Chaudhary, pursuing master of technology (1st year), from computer science department, Amity University, Noida, Uttar Pradesh, and done there Bachelor of technology degree from IAMR college of engineering, Meerut, Uttar Pradesh, His area of interest includes Software engineering and Reliability, Email id: rahulcp69@yahoo.in
Second Author Darothi Sarkar, M.Tech, faculty, computer science department, Amity University, Noida, Uttar Pradesh, Email id: dsarkar@amity.edu.
Correspondence Author Rahul Chaudhary, pursuing master of technology (1st year), from computer science department, Amity University, Noida, Uttar Pradesh, and done there Bachelor of Technology degree from IAMR college of Engineering, Meerut, Uttar Pradesh, His area of interest includes Software engineering and Reliability, Email id: rahulcp69@yahoo.in.