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CHARACTERIZATION OF POISSON TYPE LENGTH
BIASED EXPONENTIAL CLASS SOFTWARE RELIABILITY GROWTH MODEL AND PARAMETER
ESTIMATION
Rajesh Singh1, Kailash R. Kale2 , Pritee Singh3
1. R. T. M. Nagpur University, Nagpur-440033,
rsinghamt@hotmail. com
2. G. N. A. ACS College, Barshitakli, Dist-Akola.
kailashkale10@gmail.com 3. Institute of Science, Nagpur; priteesingh25@gmail.com
Abstract
The authors of this study set out to build a software reliability growth model (SRGM). Software reliability is a crucial attribute that has to be quantified and evaluated. In most cases, software errors happen at unpredictable times. In this article, the failure intensity of the single parameter length-biased exponential class SRGM has been characterized taking into account the Poisson process of the incidence of software faults. The parameters of the proposed SRGM under investigation are the scale parameter (AjJ and the total number of failures (60). It is considered that the experimenter may have previous knowledge of the parameters from past or earlier experiences in the form of gamma priors. The posterior probability may be obtained by combining the prior probability with the likelihood of the data, and Bayes estimators can then be suggested.
Keywords: Binomial process, gamma prior, maximum likelihood estimator (MLE), Rayleigh class, software reliability growth model (SRGM), incomplete gamma function, confluent hyper-geometric function.
I. Introduction
Beginning in the early 1970s of the previous century, research on software reliability has advanced until the present. Various kinds of software have dominated many fields of the humanities, sciences, and technology as well as daily life for all people. The film industry, education sector, Ecommerce sector, medical and healthcare sectors, space agencies, banking sector and various government agencies, all employ different types of software for the convenience and development the fields.
The software is the end product of several intricate code sequences developed by the humans according to the needs of above sectors within stipulated time period. Due to huge magnitude of complicated code sequences, there is a greater likelihood of failures or ineffective performance. These software failures may result from a variety of issues, including memory faults, language-specific issues, calling third-party libraries, standard library issues, etc. Such flaws may have operational repercussions that cause system failure and unanticipated dangerous outcomes.
As a result, the aforementioned areas require software that operates reliably. Hence, the evaluation and quantification of the software's performance are therefore crucial. The reliability of the software is one of the performance indicators. To put it another way, it becomes crucial to create reliable software that serves the needs of users or systems.
Software reliability is now thought to be a crucial factor in determining customer satisfaction, along with software functionality and performance. Software reliability growth models (SRGM) outline the broad link between software failure occurrences and the key process influences (such as fault introduction, fault removal, operational profile, etc.). The statistical relation between data on defects and the known characteristics of probabilistic behavior is known as the SRGM. The basic goal of software reliability modeling is to represent a relationship in which, when defects are found and removed, there is a reduction in the number of failures per time interval or an increase in the time interval between failures. The SRGM is often characterized by the mean failure function or failure intensity function. The pattern of occurrence of software failure is its type, and the mathematical functional form of failure intensity is its class. The software reliability growth models are categorized according to the system described by [8].
The length-biased distributions have been presented by [2] and formalized by [9]. These distributions are sometimes referred to as size-biased probability distributions. Reliability theory may also use these distributions (see [4], [5], and [6]). Modeling software reliability may be done using length-biased distributions. In this study, using the Poisson pattern of occurrence of software failure and the length-biased exponential form of failure intensity, the Poisson type length-biased exponential class model is introduced as per the classification system provided by [8]. As this SRGM is being described, it is assumed that the failure occurring at time t has a Poisson occurrence (i.e., Type) and that the mean failure function's functional form is characterized by a length-biased exponential distribution (i.e., Class). The software failures in this model are presumed to be independent of one another and dependent on the duration of the time interval that comprises the same software failure. For the estimation part of parameters the gamma priors taking into account. The Bayes estimators of the parameters are obtained in this study by the methods of [7], [12], [10], and [11], and they are compared with MLEs in subsequent parts.
II. Model Section
Suppose time to failure follows length biased exponential distribution denoted by f(t) with scale parameter 91 and software failures occur in Poisson pattern then
_ (tdie-eit; t>0,61> 0,E[t] ± 0 ^
(.0 ; otherwise
Also let the total number of faults remaining in the software at time t = 0 is a Poisson random variate with mean 0O then the failure intensity A(t) = 90f(t) (cf. [7]) can be obtained as
A(t) = 80te^e-eit; t>0, e1>0, e0>0 and E[t] ± 0 (2)
The mean failures function at time t comes out to be
Kt) = 9o [1-(1 + 9it)e-9lt] (3)
The details about number of failures experienced by time t, performance of failure intensity A(t) and n(t) have been discussed in [13].
III. Maximum Likelihood Estimation The most important and extensively used technique of point estimation is maximum likelihood estimation when underlying distribution of data is known. The maximum likelihood estimation is considered for failure times. The base of maximum likelihood estimation is likelihood function
which can be obtained by assuming that me failures are experienced at times i = 1,2, ...,me up to execution time is te (> tm). Also using the failure intensity at each tt, i = 1,2, ...,me obtained in (2) and mean failure function at time te obtained by replacing t = te, the likelihood function of 90 and 91 can be obtained as L(90, 91) = [nT=eiKk)}exp (-l*(t e)) (cf. Musa et al. (1987)). The L(80,8-t) can take following form for this model
L(90,91) = e™eelme [nTA ti] e-T01e-e° M1*^)^1^ (4)
where Z=\ti = T
The Maximum Likelihood Estimators for the parameters 9 0 and d1 are
8m0 = me(l -(1 + emite)e-e^lt^) (5)
and
dmi = [e-0t-2(2me -T6mi)ed^lte]1/2 (6)
respectively. The values of 9m0 and 9m1 can be obtained after simultaneous solution of equations (5) and (6).
IV. Bayesian parameter estimation
The bayesian technique is used to put the subjective and objective data sources together into the analysis. In this technique the parameters are considered as a random variables having known probability pattern. This known probability pattern is termed as prior in Bayesian technique. Whole the analysis is based on this prior and using Bayes theorem combines this prior and likelihood of data. In present case, it is considered that the experimenter have prior information about both the parameters 0O and 91 in the form of gamma probability function. Then the following prior distributions g(90) and g(91) can be considered for parameters 90 and 9, respectively.
g(do) - {Q0 "
and
^ « (7)
- tiT'-* (8)
Now, Consider the total execution time is te and during this time me failures are experienced at times ti, i = 1,2, ...,me then, the joint posterior of 90 and 0,given t(= t^, i = 1,2,...,me) is
n(90,9,1$ - e^e+v-19lme+a-1e-(T+P)0ie-(^+1)9o e^1*6^-"1''] 90 > me,91 > 0 (9)
In this section, the point estimates (posterior mean) of both the parameters 90 and 91 under study are obtained by Bayesian technique considering the squared error loss as
8B0= D-1^^^^ tF(2me+a,2me + a+j + 1,rt-1) (10)
and
9Bi = t^IT-o^^1™ n2me +a + l,2me + a+j + 2,T*t-1) (11)
where *F(a, P\ x) is Confluent Hypergeometric Function (cf. [1] and [3]), normalizing constant is
D = J^o^+S;1™ H2me + a,2me+a+j + 1,T*t-1) and
T* = T + p + jte.
V. Discussion
I. Subsection One
The proposed Bayes estimators i.e. 9B0 and 9B1 are compared with corresponding maximum likelihood estimators i.e. 9m0 and 8m1 respectively on the basis of risk efficiencies REj = R'jRj1 where Rj = E[dBj - dj] and R' = E[9mj - 9j]2; j = 0,1. Here, the performance of proposed Bayes estimators 9B0 and 9B1 over MLEs 9m0 and 9m1 have been compared on the basis of risks efficiencies using Monte Carlo simulation technique. The risks efficiencies are obtained by generating sample of size, say me failures upto total execution time te and it was repeated 103 times from the length biased exponential distribution. Then, using Monte Carlo simulation technique risks efficiencies has been evaluated and is presented in the graphs Figure 1 to 9.
REn
RE,
40 20
2 1.5
3
40
45
50 2
80
2.5
2 30
35
40
50
80
3
0
1
Figure 1: Risk Efficiencies 6B0 and 6B1 for te = 100; d = 1, n = 1; a = 1, ft = 1,
Figure 2: Risk Efficiencies 9B0 and 9B1 te = 125; d = 1, n = 1; a = 1, ft = 1
Figure 3: Risk Efficiencies 9B0 and 9B1 te = 150; d = 1, n = 1; a = 1, ft = 1,
Figure 4: Risk Efficiencies 9B0 and 9B1 te = 200; d = 1, n = 1; a = 1, ft = 1
80
Figure 5: Risk Efficiencies 9B0 and 9B1 te = 100; d = 10, n = 1; a = 10, ft = 1,
10
RE0 5
300
45 2.2
2.4
2 ■
RE1 1.5
2 30
35 40
00
45
50
3
e
e
1
0
e
1
Figure 6: Risk Efficiencies 6B0 and 6B1 te = 100; d = 10, n = 5; a = 10, ft = 5,
REn
40
45
50 2
RE1 1.5
01
2 30
35 40
00
45
50
3
9
1
6
0
Figure 7: Risk Efficiencies 6B0 and 6B1 te = 100; d = 10, n = 10; a = 10, ft = 10,
Figure 8: Risk Efficiencies 9B0 and 9B1 te = 100; d = 1, n = 10; a = 1, ft = 10,
Figure 9: Risk Efficiencies 9B0 and 9B te = 100; d = 3, n = 10; a = 3, ft = 10,
II. Subsection Two
Based on the above graphical representation from Figure 1 to Figure 9 of performance of proposed Bayes estimators against their corresponding MLE, it can be seen that the risk efficiencies RE0 of 9B0 decrease as 90 and 91 increase. It can also be seen that for large values of 91 and 90 the proposed Bayes estimator of 90 is not better than MLE otherwise proposed Bayes estimator 9B0 is better than MLE. Moreover, when the value of te is small, the values of RE0 first increase, attain a maxima and then decrease as the value of te increase. Similarly, it can be observed that the risk efficiencies of 9B1 i.e. RE1 decrease as the value of 91 and 60increase but the values of risk efficiencies RE1 are almost constant for the increase in values of 90. Further, The values of RE1 are uniform over the variation in value of te. It is important to note that the proposed Bayes estimator 9B1 is always better than MLE. Due to increase in values of shape and scale parameter of both the priors the values of RE0 decrease for constant values of scale parameters.
On the basis of better performance of risk efficiencies of 9B0 and 9B1 over 9m0 and 9m0 following conclusions are drawn.
VI. Conclusions
After having experience or prior knowledge about the software failure process to researchers. These proposed Bayes estimators can perform better than their corresponding MLEs for the proper choices of prior parameters. The proposed Bayes estimator of 00 can be preferred over MLE if it is felt that total number of failures may not be very large and failure rate may be small. The proposed Bayes estimator of 91 can be preferred over MLE. Under this prior belief these estimators can be preferred for calendar time modeling.
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