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Detection of Reliability using Sprt: S-Shaped Models
S.Chitti Babulu
1
, Dr. R. Satya Prasad
2
, Dr. K.Raja Sekhara Rao
3
1
Research Scholar, Department of CSE, JNTUK, Kakinada, India.
2
Professor, Dept. CSE, ANU, Guntur, India.
3
Professor, Dept. CSE, KLEF, Guntur, India.
DOI:
https://doi.org/10.51583/IJLTEMAS.2026.150100092
Received: 28 January 2026; Accepted: 02 February 2026; Published: 16 February 2026
ABSTRACT
Traditional hypothesis testing may cause delay to take important decisions, as it depends on collecting a lot of
evidence before conclusions are drawn. An alternative method for assessing software reliability is the sequential
analysis, the Sequential Probability Ratio Test (SPRT). SPRT provides the mechanism of continuous monitoring
which made it possible to attribute reliable or unreliable software rapidly. A framework of SPRT is proposed by
Wald for different probability distributions. This paper proposed to use SPRT on ungrouped software failure
data of six datasets collected from literature categorized as ungrouped with two popular S-shaped models. Real
Valued Genetic Algorithm is proposed for parameter estimation to assess the performance evaluation.
Keywords: SRGMs, S-shaped models, SPRT, RVGA.
INTRODUCTION
An alternative to classical hypothesis testing is the SPRT proposed by Wald (1947) that evaluates data
sequentially. comparing accumulated evidence with predefined thresholds allows continuous assessment and
early termination of testing. This is very effective for decisions which involves binary classification such as
reliability or unreliability. Time Between Failures (TBF) and Failure counts (FC) are the failure patterns
commonly found in software reliability analysis. A process is modelled as a Homogeneous Poisson Process
when failures occur with a constant rate and follows Poisson distribution which describes the occurrence of
failures. Assuming a constant failure rate λ following a Poisson distribution the stochastic behaviour of software
is given as follows which enables to gain insight.
(1.1)
The limitations of traditional testing methods were highlighted by Stieber (1997) for Software Reliability Growth
Models (SRGMs) for reliability predictions which leads to misleading decisions. He proposed the use of SPRT
for failure data analysis. The present work applies two popular S-shaped models to assess reliability of software
following the principle proposed by Stieber. Application of this principle for software acceptance or rejection is
explained in section 2. It is extended to the considered NHPP-based SRGMs in section 3. Section 4 presents the
decision rule being applied for identifying unreliable software. Section 5 describes the proposed Real-Valued
Genetic Algorithm for parameter estimation, with estimated model parameters given in Section 6. Section 7
evaluates the models on the datasets and discusses their effectiveness in detecting unreliable software.
Wald's Sequential Test
The SPRT, developed by A. Wald, originated from wartime research and was initially classified due to its
military applications. Its key advantage over fixed-sample tests is the reduced average number of observations
required, leading to efficient decision-making. Consequently, SPRT is widely used in statistical quality control
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and manufacturing as shown in homogeneous Poisson processes, it offers an effective framework for sequential
hypothesis testing with optimized resource use.
Let {O(x), x 0} be a HPP with rate ‘’. In our case, O(x) = count of failures up to ‘x’ time and ‘’ is the rate
of failure. Suppose a system is placed on test and aimed to estimate its ‘’. it cannot expect to exactly estimate.
The system is rejected with a likelihood high if data under consideration recommend that the rate of failure is
greater than
1
. if it is smaller than
0
accept it with probability high. As statistical tests, have some risk getting
the answers misleading. Two small numbers ‘u’ and ‘v’ are specified representing the probability of falsely
refusing, if λ ≤
0
, considered as "producer’s" risk. ‘v’ is the probability of falsely accepting, if λ ≥
1,
considered
as the “consumer’s” risk.
The relative risk choice essential in the description of the alternative hypothesis will influence highly the Wald’s
SPRT. Continuously at each point of time x > 0 classical SPRT tests are performed, as data is collected
additionally. With choices of
0
and
1
such that 0 <
0
<
1
, the probability of finding O(x) failures in the time
span (0, x ) with
1
,
0
as failure rates is given by (Murali mohan et al., 2013)
(2.1)
(2.2)
At any time ’x’,
decides the truth towards
or
, given a sequence of time, say
<…<
and the corresponding realizations
of
.
simplification gives
The rule of decision is to decide in favor of
, in favor of
, or to continue by observing the number of
failures later than 'x'. As
is greater than or equal to a constant say R, less than or equal to a constant say S or
in between the constants R and S. The product under consideration is decided as continue, reliable, unreliable of
the test process with another observation in data.
The values of R and S are approximately taken as
,
Where ‘u’ and ‘v’ are the probabilities of risk. A test that minimizes both the (u) and (v) errors as much as
possible is considered as good. The procedure in general is to fix the v error and then choose a critical region to
maximize or minimize the error. A easier version of the above decision is to reject, as unreliable if above the
line,
falls for the first.
(2.6)
if
falls for the first below the line, accept as reliable
(2.7)
To proceed the test with another on
as the random graph of
is between the two linear
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boundaries given by equations (2.6) and (2.7) (Satya prasad et al., 2013).
The parameters chosen in multiple ways i.e u, v,
0
and
1
. Stieber suggested as,
where
the slope of
and
= , If
and
are above.
NHPP based SRGMs
SRGMs grounded in Non-Homogenous Poisson Processes (NHPP) have demonstrated considerable efficacy in
real-world software reliability engineering, as evidenced by Haidry et al. (2023). The main problem with NHPP
models being used in reliability lies in a suitable Mean Value Function (MVF). As all the models are based on
assumptions, various NHPP SRG models are constructed, and their parameter estimation is a huge task. Various
algorithms were proposed in the literature to facilitate model parameters estimation. Huang et al. (2022)
proposed debugging models with imperfection acknowledging the possibility that fault removal may not be
immediate and may introduce new faults.
In SRGMs the cumulative count of software failures by time ‘x’ is given by
. The MVF
representing the cumulative count of expected software failures detected up to a given time is represented by
. Predicting and assessing software reliability during testing phases is a critical concept in SRGMs. The
failure intensity
is proportional to residual fault content. These include models like:
Delayed S-shaped
The Delayed model, based on NHPP theory, describes software error detection as an S-shaped growth curve,
with slow initial detection, rapid discovery in the middle phase, and a gradual slowdown as remaining errors
become harder to find. The MVF is given as
Were, . ‘z’ is the error
detection rate per error in state of steady. ‘y’ is the count of initial faults (Pełka, 2012).
Inflection S-shaped
Ohba’s inflection model (1984) characterizes an accelerating fault detection rate during testing, controlled by an
inflection parameter that reflects the proportion of detectable faults while limiting unrealistic exponential growth.
Its mean value function is
,where ‘x’ is time,
is the expected cumulative failures by
time ‘x’, ‘y’ is the total expected failures, ‘z’ is the detection rate of failure, and ‘c’ is the factor of inflection
indicating the transition from rapid to slower fault detection.
Sequential Test for Software Reliability Growth Models
In Section 2, the expected value of , which represents the average count of failures in time , is
referred to as the MVF of the Poisson process. On the other hand, if we consider a Poisson process with a
funnction as its MVF, the probability equation of such a process is given by:
various Non-homogeneous Poisson processes (NHPP) are obtained based on the form of . For a delayed
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S-shaped model it is given as (Yamada et al., 1984).
Where,
,
are values of the MVF at specified sets of its parameters representing reliable and
unreliable software respectively. Let
,
be values of the NHPP at two specifications of ‘z’ say
, where
. It can be shown that for the model m
at
is greater than that at
. Symbolically
. Then the SPRT procedure is as follows:
if
, the system is considered as reliable and is accepted.
i.e.,
(4.1)
if
, the system is considered as unreliable and is rejected.
i.e.,
(4.2)
Continue the test as long as
(4.3)
By substituting the respective MVF for the delayed S-shaped model, the corresponding conclusion rules
are obtained, as presented in the following lines.
Acceptance region:
(4.4)
Rejection region:
(4.5)
Continuation region:
(4.6)
Similar rules are derived for Inflection S-shaped model. The judgement rules are determined solely by the
strength of the sequential procedure
and the corresponding MVFs
and
. Stiber described that
the judgement rules become conclusion lines when MVF is linear in ‘x’ passing through origin, that is,
. In that sense equations (4.1), (4.2), (4.3) can be regarded as generalizations to the Stieber decision procedure.
The applications of these for software failure data are obtained with analysis. Let
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H₀: The S-shaped model adequately fits the observed failure data
H₁: The S-shaped model fails to adequately fit the observed failure data.
For each dataset, the cumulative inter-failure time is plotted against:
Acceptance boundary
Rejection boundary
Decision Rules:If the cumulative curve crosses the accept boundary, H₀ is accepted. If it crosses the reject
boundary, H₀ is rejected. If it remains between both boundaries, the test is inconclusive and sampling continues.
METHODOLOGY
Data Collection
Software failure data analysis is a critical process in understanding and improving software reliability. This
analysis is essential for understanding the behaviour of software systems, enabling targeted fault detection and
correction strategies. For the improvement of quality, the analysis of reliability growth and robust data sets
provide valuable insights for effective use of strategies.
Parameter Estimation:
SRGMs rely on techniques of parameter estimation like Maximum Likelihood Estimation, Least Square
Estimation to predict software reliability. The methods proposed by Koren and krishna (2021) and Latha
Shanmugam et al. (2013) necessitate the collection of failure data to estimate the unknown parameters. The
potential constraints related to unimodality, continuity and the existence of derivatives within complex likelihood
functions is highlighted by Costa et al. (2010). LSE proves more suitable for small to medium sample sizes as
suggested by Mahmood et al. (2022). Genetic Algorithm (GA) based parameter estimation for Hyper-Geometric
Distribution is proposed by Minohara and Tohma (1995). Zhen's (2020) application of Particle Swarm
Optimization (PSO) for parameter estimation have demonstrated limitations.
Genetic Algorithm (GA):
Jiang (2006) and McMinn (2011) proposed parameter estimation can be effectively conceptualized as a search
within the solution space. The incorporation of GA into software reliability modelling is a potentially
advantageous approach. Lopes et al. (2024) demonstrated the application of a refined RVGA coupled with
empirical data through the S-shaped models elucidates both the parameter estimation process and the resultant
assessments. This integrated methodology offers a robust framework for improving software reliability
predictions.
Real Valued Genetic Algorithm
For a large set of diverse problems, the GAs flexibility offers a versatile approach to optimization. Their strength
towards quality solutions is ease of implementation, effectiveness and rapid convergence in large search spaces.
A significant adaptation of the traditional GA for continuous variables is RVGA tailored for optimization
problems. RVGAs uses real numbers to avoid the encoding and decoding in handling continuous values with
binary strings.
Real Valued Genetic Algorithm Parameter Estimation
A RVGA is used to estimate the SRG models parameters. The efficacy of this approach is demonstrated through
Table 6.1 presenting the parameters estimated for six data sets available from the literature. This research
contributes software reliability prediction through the application of RVGA and the comparative analysis of the
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delayed and inflection models under consideration.
Table 6.1: Estimated Parameters
Data Set
Delayed
Inflection
Y
Z
y
Z
c
DS1 (LYU)
69.994993
0.110755
70.000000
0.180372
15.962549
DS2 (IBM)
295.999928
0.187389
296.000000
0.357895
9.544106
DS3 (AT&T)
680.00000
0.109179
679.999127
0.276183
15.826329
DS4 (Xie)
737.891028
0.063255
729.866440
0.134056
25.262465
DS5 (NTDS)
249.999901
0.097633
250.000000
0.159340
18.053581
DS6 (SONATA)
1831.982958
0.112970
1831.928582
0.176066
16.183552
SPRT-Based Analysis
In this section, we evaluate the conclusion rules using the selected MVF across six different data sets obtained
from Pham (2006), Xie et al. (2002), and SONATA (Ashoka, 2010). Based on the estimated value of the
parameter z from each MVF, the specifications
and
, were chosen and positioned
equidistantly on either side of the estimated z obtained from the dataset. These choices ensure that
and enable the application of the SPRT. The decision rules are computed using Equations 4.4 and 4.5 assuming
a value of for each model using the corresponding mean value functions
and
. These rules are
evaluated sequentially at each in the datasets, with the strength parameters
set to
. The
resulting decision rules for the models are presented graphically in Table 7.1.
Table 7.1: SPRT analysis for 6 sets
Data Set
Delayed S-shaped
Inflection S-shaped
LYU
IBM
AT&T
Xie
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NTDS
SONATA
Fig: Graphical representation of SPRT.
The results presented in the above table clearly indicate that the decision regarding system acceptance or
rejection is made significantly earlier than the final observation time. To statistically validate the applicability
of this model, the SPRT is employed. This section presents a comprehensive SPRT-based evaluation of the
Delayed and Inflection S-shaped SRGM using six real-world software failure datasets: LYU, IBM, ATT, NTDS,
SONATA, and Xie.
7.2 Comparative Summary of SPRT Results
Dataset
Delayed
Inflection
SPRT Decision
Suitability
SPRT Decision
Suitability
LYU
Inconclusive / Weak
Poor
Inconclusive
Weak
IBM
Accept H₀
Good
Accept H₀
Good
ATT
Accept H₀
Excellent
Accept H₀
Excellent
NRDS
Accept H₀ (Borderline)
Reasonable
Reject H₀
Not Suitable
SONATA
Reject H₀
Not Suitable
Reject H₀
Not Suitable
Xie
Reject H₀
Not Suitable
Reject H₀
Not Suitable
CONCLUSION
SPRT-based results show that the Delayed S-shaped SRGM performs well for IBM and ATT datasets, which
exhibit delayed learning with mild acceleration, but fails to capture late-stage failure surges in SONATA and
Xie, while only partially fitting the inconsistent LYU data. The Inflection S-shaped model also fits IBM and
ATT well, showing significant inflection behavior and early SPRT acceptance. However, it is rejected for Xie,
SONATA, and NTDS due to excessive acceleration or erratic failures, and yields inconclusive results for LYU
owing to the absence of a clear inflection point.
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