IJCET Volume 4 Issue 5 HLSRGM Software Reliability

International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
INTERNATIONAL JOURNAL OF COMPUTER ENGINEERING &
ISSN 0976 - 6375(Online), Volume 4, Issue 5, September - October (2013), © IAEME

TECHNOLOGY (IJCET)

ISSN 0976 – 6367(Print)
ISSN 0976 – 6375(Online)
Volume 4, Issue 5, September – October (2013), pp. 277-284
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IJCET
©IAEME

SPRT -TWO STEP APPROACH: HLSRGM
Dr. R. Satya Prasad1, Shaheen2, G. Krishna Mohan3
1

(Associate Professor, Dept. of Computer Science & Engg., Acharya Nagrjuna University
Nagarjuna Nagar, Guntur, Andhrapradesh, India)
2
(Asst. Professor, IPE, Osmaina university campus, Hyderabad)
3
(Reader, Dept. of Computer Science, P.B.Siddhartha College, Vijayawada, Andhrapradesh, India)

ABSTRACT
In Classical Hypothesis testing volumes of data is to be collected and then the conclusions are
drawn, which may need more time. But, Sequential Analysis of Statistical science could be adopted
in order to decide upon the reliability / unreliability of the developed software very quickly. The
procedure adopted for this is, Sequential Probability Ratio Test (SPRT). It is designed for continuous
monitoring. The likelihood based SPRT proposed by Wald is very general and it can be used for
many different probability distributions. The parameters are estimated using two step approach. In
the present paper, the HLSRGM (Half Logistic Software Reliability Growth model) is used on five
sets of existing software reliability data and analyzed the results.
Keywords: HLSRGM, Two step approach, Decision lines, Software testing, Software failure data.
I.

INTRODUCTION

Wald's procedure is particularly relevant if the data is collected sequentially. Sequential
Analysis is different from Classical Hypothesis Testing were the number of cases tested or collected
is fixed at the beginning of the experiment. In Classical Hypothesis Testing the data collection is
executed without analysis and consideration of the data. After all data is collected the analysis is
done and conclusions are drawn. However, in Sequential Analysis every case is analyzed directly
after being collected, the data collected upto that moment is then compared with certain threshold
values, incorporating the new information obtained from the freshly collected case. This approach
allows one to draw conclusions during the data collection, and a final conclusion can possibly be
reached at a much earlier stage as is the case in Classical Hypothesis Testing. The advantages of
Sequential Analysis are easy to see. As data collection can be terminated after fewer cases and
277


decisions taken earlier, the savings in terms of human life and misery, and financial savings, might
be considerable.
In the analysis of software failure data we often deal with either Time Between Failures or
failure count in a given time interval. If it is further assumed that the average number of recorded
failures in a given time interval is directly proportional to the length of the interval and the random
number of failure occurrences in the interval is explained by a Poisson process then we know that the
probability equation of the stochastic process representing the failure occurrences is given by a
Homogeneous Poisson Process with the expression
e − λt ( λ t )
P  N (t ) = n =


n!

n

(1.1)

Stieber (1997) observes that if classical testing strategies are used, the application of software
reliability growth models may be difficult and reliability predictions can be misleading. However, he
observes that statistical methods can be successfully applied to the failure data. He demonstrated his
observation by applying the well-known sequential probability ratio test (SPRT) of Wald (1947) for
a software failure data to detect unreliable software components and compare the reliability of
different software versions. In this paper we consider popular SRGM HLSRGM and adopt the
principle of Stieber (1997) in detecting unreliable software components in order to accept or reject
the developed software. The theory proposed by Stieber (1997) is presented in Section 2 for a ready
reference. Extension of this theory to the SRGM – HLSRGM is presented in Section 3. Application
of the decision rule to detect unreliable software with respect to the proposed SRGM is given in
Section 4. Analysis of the application of the SPRT on five data sets and conclusions drawn are given
in Section 5 and 6 respectively.
II.

WALD'S SEQUENTIAL TEST FOR A POISSON PROCESS

The sequential probability ratio test (SPRT) was developed by A.Wald at Columbia
University in 1943. Due to its usefulness in development work on military and naval equipment it
was classified as ‘Restricted’ by the Espionage Act (Wald, 1947). A big advantage of sequential tests
is that they require fewer observations (time) on the average than fixed sample size tests. SPRTs are
widely used for statistical quality control in manufacturing processes. An SPRT for homogeneous
Poisson processes is described below.
Let {N(t),t ≥0} be a homogeneous Poisson process with rate ‘λ’. In our case, N(t)= number
of failures up to time ‘ t’ and ‘ λ’ is the failure rate (failures per unit time ). Suppose that we put a
system on test (for example a software system, where testing is done according to a usage profile and
no faults are corrected) and that we want to estimate its failure rate ‘ λ’. We can not expect to
estimate ‘ λ’ precisely. But we want to reject the system with a high probability if our data suggest
that the failure rate is larger than λ1 and accept it with a high probability, if it’s smaller than λ0. As
always with statistical tests, there is some risk to get the wrong answers. So we have to specify two
(small) numbers ‘α’ and ‘β’, where ‘α’ is the probability of falsely rejecting the system. That is
rejecting the system even if λ ≤ λ0. This is the "producer’s" risk. β is the probability of falsely
accepting the system .That is accepting the system even if λ ≥ λ1. This is the “consumer’s” risk.
With specified choices of λ0 and λ1 such that 0 < λ0 < λ1, the probability of finding N(t) failures in
the time span (0,t ) with λ1,λ0 as the failure rates are respectively given by

278


e− λ1t [ λ1t ]

Q1 =

N (t )

(2.1)

N (t )!

Q0 =

e− λ0t [ λ0t ]

N (t )

(2.2)

N (t )!
The ratio

Q1
at any time ’t’ is considered as a measure of deciding the truth towards λ0 or λ1
Q0

, given a sequence of time instants say t1 < t2 < t3 < ........ < tK and the corresponding realizations
N (t )

λ 
Q
Q
N (t1 ), N (t2 ),........N (tK ) of N(t). Simplification of 1 gives 1 = exp(λ0 − λ1 )t +  1 
Q0
Q0
 λ0 
The decision rule of SPRT is to decide in favor of λ1 , in favor of λ0 or to continue by
Q
observing the number of failures at a later time than 't' according as 1 is greater than or equal to a
Q0
constant say A, less than or equal to a constant say B or in between the constants A and B. That is,
we decide the given software product as unreliable, reliable or continue the test process with one
more observation in failure data, according as
Q1
≥A
Q0

(2.3)

Q1
≤B
Q0

(2.4)

B<

Q1
<A
Q0

(2.5)

The approximate values of the constants A and B are taken as A ≅

1− β

α

, B≅

β
1−α

Where ‘ α ’ and ‘ β ’ are the risk probabilities as defined earlier. A simplified version of the above
decision processes is to reject the system as unreliable if N(t) falls for the first time above the line

NU ( t ) = a.t + b2

(2.6)

To accept the system to be reliable if N(t) falls for the first time below the line

N L ( t ) = a.t − b1

(2.7)

To continue the test with one more observation on (t, N(t)) as the random graph of [t, N(t)] is
between the two linear boundaries given by equations (2.6) and (2.7) where

279


a=

λ1 − λ0
λ 
log  1 
 λ0 

(2.8)

1 − α 
log 

 β 
b1 =
λ 
log  1 
 λ0 

(2.9)

1 − β 
log 
 α 

b2 =
 λ1 
log  
 λ0 

(2.10)

The parameters α , β , λ0 and λ1 can be chosen in several ways. One way suggested by Stieber

λ.log ( q )

λ.log ( q )

λ1
q −1
q −1
λ0
If λ0 and λ1 are chosen in this way, the slope of NU (t) and NL (t) equals λ. The other two ways
of choosing λ0 and λ1 are from past projects and from part of the data to compare the reliability of
different functional areas.
(1997) is λ0 =

III.

, λ1 = q

where q =

HLSRGM

One simple class of finite failure NHPP model is the HLSRGM, assuming that the failure
intensity is proportional to the number of faults remaining in the software describing an exponential
failure curve. It has two parameters. Where, ‘a’ is the expected total number of faults in the code and
‘b’ is the shape factor defined as, the rate at which the failure rate decreases. The cumulative
1 − e−bt
distribution function of the model is: F ( t ) =
.The expected number of faults at time‘t’ is
1 + e−bt

(
(

denoted by m (t ) =

IV.

a (1 − e − bt )

(1 + e )
− bt

)
)

, a > 0, b > 0, t ≥ 0 .

SEQUENTIAL TEST FOR SRGMS

In Section II, for the Poisson process we know that the expected value of N(t) = λt called
the average number of failures experienced in time 't' .This is also called the mean value function of
the Poisson process. On the other hand if we consider a Poisson process with a general function m(t)
as its mean value function the probability equation of a such a process is

P[ N (t) = Y ]

[ m(t)]
=

y

.e−m(t ) , y = 0,1,2, −− − −

y!
Depending on the forms of m(t) we get various Poisson processes called NHPP.
280


We may write

Q1 =

Q0 =

e − m1 (t ) .[ m1 (t )]

N (t )

N (t )!
e − m0 (t ) .[ m0 (t )]

N (t )

N (t )!
Where, m1 (t ) , m0 (t ) are values of the mean value function at specified sets of its parameters

indicating reliable software and unreliable software respectively. Let P , P be values of the NHPP at
0
1
two specifications of b say b0 , b1 where ( b0 < b1 ) respectively. It can be shown that for our models
m ( t ) at b1 is greater than that at b0 . Symbolically m0 ( t ) < m1 ( t ) . Then the SPRT procedure is as follows:

Accept the system to be reliable if
i.e.,

e − m1 (t ) .[ m1 (t )]

Q1
≤B
Q0

N (t )

e − m0 (t ) .[ m0 (t )]

N (t )

≤B

 β 
log 
 + m1 (t ) − m0 (t )
 1−α 
i.e., N (t ) ≤
log m1 (t ) − log m0 (t )
Decide the system to be unreliable and reject if

(4.1)
Q1
≥A
Q0

1− β 
log 
 + m1 (t ) − m0 (t )
 α 
i.e., N (t ) ≥

(4.2)

Continue the test procedure as long as

 β 
 1− β 
log 
log 
 + m1 (t ) − m0 (t )
 + m1 (t ) − m0 (t )
 1−α 
 α 
< N (t ) <

(4.3)

Substituting the appropriate expressions of the respective mean value function – m(t) of HLSRGM,
we get the respective decision rules and are given in followings lines
Acceptance region:

2 a ( e − b0t − e − b1t )
 β  

log 
+

− t b −b
 1 − α   1 + e − b0t + e − b1t + e ( 0 1 ) 


(4.4)
N (t ) ≤
 1 − e − b1t   1 + e − b0t  
log 
− b1t  
− b0 t  
 1 + e   1 − e  
281


Rejection region:
 1− β
log 
 α
N (t ) ≥


2a e − b0t − e − b1t
 

+

− t b −b
 1 + e − b0t + e − b1t + e ( 0 1 ) 



(

)

(4.5)

 1 − e − b1t   1 + e − b0t  
log 
− b1t  
− b0 t  
 1 + e   1 − e  

Continuation region:
−b0t
−b1t

 β   2a e − e
log 
+  −b t −bt −t(b −b ) 

 1−α  1+ e 0 + e 1 + e 0 1 



(

1
 1− e−bt   1+ e−b0t 
log 
−b1t  
−b0t 
 1+ e   1− e 

)

−b0t
−bt
1

 1− β   2a e − e
log 
+  −b t −b t −t(b −b ) 

 α  1+ e 0 + e 1 + e 0 1 



(

< N(t) <

1
 1− e−bt   1+ e−b0t 
log 
−b1t  
−b0t 
 1+ e   1− e 

)

(4.6)

It may be noted that in the above model the decision rules are exclusively based on the
strength of the sequential procedure (α,β ) and the values of the respective mean value functions
namely, m0 (t ) , m1 (t ) . If the mean value function is linear in ‘t’ passing through origin, that is, m(t) =
λt the decision rules become decision lines as described by Stieber (1997). In that sense equations
(4.1), (4.2), (4.3) can be regarded as generalizations to the decision procedure of Stieber (1997). The
applications of these results for live software failure data are presented with analysis in Section 5.

V.

SPRT ANALYSIS OF LIVE DATA SETS

The developed SPRT methodology is for a software failure data which is of the form [t, N(t)].
Where, N(t) is the failure number of software system or its sub system in ‘t’ units of time. In this
section we evaluate the decision rules based on the considered mean value function for Five different
data sets of the above form, borrowed from (Xie, 2002), (Pham, 2006) and (LYU,1996). The
procedure adopted in estimating the parameters is a two step approach (Satyaprasad et al.,2013).
Based on the estimates of the parameter ‘b’ in each mean value function, we have chosen the
specifications of b0 = b −δ , b1 = b +δ equidistant on either side of estimate of b obtained through a
Data Set to apply SPRT such that b0 < b < b1. Assuming the value of δ = 0.025 , the choices are given
in the following table.

Table 5.1: Estimates of a, b & Specifications of b0, b1 for Time domain
Data Set
Estimate of ‘a’ Estimate of ‘b’
b0
b1
31.524466
0.029012
0.004012
0.054012
XIE
28.767254
0.075071
0.050071
0.100071
NTDS
18.610725
0.105451
0.080451
0.130451
IBM
23.655141
0.026521
0.001521
0.051521
AT&T
27.139750
0.470899
0.445899
0.495899
LYU
Using the selected b0 , b1 and subsequently the m0 (t ), m1 (t ) for the model, we calculated the
decision rules given by Equations 4.4 and 4.5, sequentially at each ‘t’ of the data sets taking the
strength ( α, β ) as (0.05, 0.2). These are presented for the model in Table 5.2. The following
consolidated table reveals the iterations required to come to a decision about the software of each
Data Set.
282


Table 5.2: SPRT analysis for 5 data sets of Time domain data
Data Set
T
N(t) Acceptance region (≤) Rejection Region (≥) Decision
2.096161
3.892803 Accept
Xie
30.02
1
3.289091
10.007634 Accept
NTDS
9
1
9.537308E-01
1.161761E+01
10
1
-1.043887E+00
1.484484E+01
19
2
-9.708590E+00
2.581256E+01
32
3
-2.609443E+01
5.180791E+01
43
4
-8.665526E+01
1.568921E+02
58
5
-2.236631E+02
3.995060E+02
70
6
-9.364992E+02
1.667050E+03
88
7
-3.110639E+03
5.535421E+03 Continue
IBM
103
8
-1.819048E+04
3.236855E+04
125
9
-1.357475E+05
2.415514E+05
150
10
-6.257971E+05
1.113554E+06
169
11
-6.991078E+06
1.244004E+07
199
12
-9.174644E+07
1.632552E+08
231
13
-6.856016E+08
1.219971E+09
256
14
-1.712571E+10
3.047378E+10
296
15
0.346755
1.578489
5.5
1
AT&T
Reject
0.549381
1.782911
7.33
2
-1.197242E+01
2.915316E+01
0.5
1
-9.029384E+00
3.620727E+01
1.7
2
-1.761479E+01
6.114396E+01
4.5
3
-5.297372E+01
1.243779E+02
7.2
4
-1.608839E+02
3.145575E+02
10
5
-5.273155E+02
9.641456E+02
13
6
-1.085730E+03
1.956316E+03
14.8
7
-1.563321E+03
2.805418E+03
15.7
8
-2.768828E+03
4.949402E+03
17.1
9
-1.181216E+04
2.103861E+04
20.6 10
-4.952239E+04
8.813843E+04
24 11
-8.249745E+04
1.468140E+05
25.2 12
LYU
Continue
-1.211256E+05
2.155489E+05
26.1 13
-2.508898E+05
4.464526E+05
27.8 14
-4.581086E+05
8.151803E+05
29.2 15
-1.470789E+06
2.617159E+06
31.9 16
-5.904594E+06
1.050674E+07
35.1 17
-1.757032E+07
3.126493E+07
37.6 18
-4.214090E+07
7.498623E+07
39.6 19
-3.036268E+08
5.402786E+08
44.1 20
-1.417638E+09
2.522570E+09
47.6 21
-1.407760E+10
2.504991E+10
52.8 22
-3.410741E+11
6.069131E+11
60 23
-3.942281E+13
7.014960E+13
70.7 24

From the Table 5.2, a decision of either to accept, reject the system or continue is reached
much in advance of the last time instant of the data.
283

VI.

CONCLUSION

The above consolidated table of HLSRGM as exemplified for five Data Sets indicates that the
model is performing well in arriving at a decision. The model has given a decision of acceptance for 2
Data Sets i.e Xie and NTDS at 1st and 1st instances respectively, a decision of rejection for 1 Data Sets i.e
AT&T at 2nd instance and a decision of continue for 2 Data Sets i.e IBM and LYU. Therefore, we may
conclude that, applying SPRT on data sets we can come to an early conclusion of reliability / unreliability
of software.
VII.

REFERENCES

1. Lyu, M.R. (1996). “The hand book of software reliability engineering”, McGrawHill & IEEE Computer
Society press.
2. Pham. H., (2006). “System software reliability”, Springer.
3. Satya Prasad (2007).”Half logistic Software reliability growth model “Ph.D Thesis of ANU, India.
4. Satyaprasad. R, Shaheen and Krishna Mohan. G., (July-August, 2013). “Two Step approach for Software
Process control:HLSRGM”, International Journal of Emerging Trends & Technology in Computer
Science(IJETTCS), Volume 2, Issue 4.
5. Stieber, H.A. (1997). “Statistical Quality Control: How To Detect Unreliable Software Components”,
Proceedings the 8th International Symposium on Software Reliability Engineering, 8-12.
6. Wald. A., 1947. “Sequential Analysis”, John Wiley and Son, Inc, New York.
7. Xie, M., Goh. T.N., Ranjan.P. (2002). “Some effective control chart procedures for reliability monitoring”
-Reliability engineering and System Safety 77 143 -150.
8. Sandeep P. Chavan, Dr. S. H. Patil and Amol K. Kadam, “Developing Software Analyzers Tool using
Software Reliability Growth Model for Improving Software Quality”, International Journal of Computer
Engineering & Technology (IJCET), Volume 4, Issue 2, 2013, pp. 448 - 453, ISSN Print: 0976 – 6367,
ISSN Online: 0976 – 6375.

AUTHOR’S PROFILE

Dr. R. Satya Prasad Received Ph.D. degree in Computer Science in the faculty of
Engineering in 2007 from Acharya Nagarjuna University, Andhra Pradesh. He received
gold medal from Acharya Nagarjuna University for his outstanding performance in a
first rank in Masters Degree. He is currently working as Associative Professor in the
Department of Computer Science & Engineering, Acharya Nagarjuna University. His
current research is focused on Software Engineering. He published 70 research papers in
National & International Journals.

Miss. Shaheen working as Assistant Professor, Institute of Public Enterprise (IPE),
Osmania Univeristy Campus. She did her Master of Computer Science. She was
associated with INFOSYS, Gacchibowli, Hyderabad and Genpact, Uppal as an
Academic Consultant training fresh recruits on technical subjects. She is currently
pursuing Ph. D. in Computer Science from Acharya Nagarjuna University.
Mr. G. Krishna Mohan, working as a Reader in the Department of Computer
Science, P.B.Siddhartha College, Vijayawada. He obtained his M.C.A degree from
Acharya Nagarjuna University, M.Tech from JNTU, Kakinada, M.Phil from Madurai
Kamaraj University and pursuing Ph.D from Acharya Nagarjuna University. He
qualified, AP State Level Eligibility Test. His research interests lies in Data Mining and
Software Engineering. He published 19 research papers in various National and
International journals.
284

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