BAYES ESTIMATOR OF PARAMETERS OF BINOMIAL TYPE EXPONENTIAL CLASS SRGM USING GAMMA

PRIORS

Rajesh Singh, 2Kailash R. Kale and 3Pritee Singh

!R. T. M. Nagpur University, Nagpur-440033. 2G. N. A. ACS College, Barshitakli, Dist-Akola. 3Institute of Science, Nagpur. rsinghamt@hotmail.com kailashkale10@gmail.com priteesingh25@gmail.com

Abstract

The Reliability is one of the key characteristics of software that operates flawlessly and in accordance with needs of users. The assessment of Reliability is very important but it is complicated. The one-parameter exponential class failure intensity function is used in this article to quantify the model and assess the software Reliability. The scale parameter and the number of existing total failures are the model's parameters. Using the Bayesian approach, the estimators of parameters are obtained under the assumption that gamma priors are suitable to provide prior information of the parameters. Using risk efficiencies computed under squared error loss, the performance of proposed estimators is studied with their corresponding maximum likelihood estimators. The suggested Bayes estimators are found to outperform over the equivalent maximum likelihood estimators.

Keywords: Binomial process, gamma prior, maximum likelihood estimator (MLE), Exponential class, software reliability growth model (SRGM), confluent hyper-geometric function.

I. Introduction

The modern computers are widely and extensively used worldwide for solving the majority of complex problems pertaining to a variety of fields due to their ability to perform intricate and time-consuming tasks quickly, accurately, and with effective global communication. When completing all of such tasks, very sophisticated computers are used, and they are guided by a series of input instructions, known as a program or Software. Since, Software are necessary components of every computer system, its performance is crucial and has significant role.

Software are developed by human and due to its complexity and size, faults are more likely to occur. As a result, it gets the user's acceptance or rejection. For acceptance of any Software, its reliability is probably most important feature. Reliable software possesses high-quality and meets the needs of users or industries or government organizations. Software must be of an acceptable quality, which is closely connected to reliability qualities, to please the consumers. Since the 1950s, the area of software reliability is being studied by researchers, and several significant results have

Rajesh Singh, Kailash R. Kale, Pritee Singh татя тчт пял ESTIMATION OF PARAMETERS OF BINOMIAL TYPE Kl&A, No 2 (/8) EXPONENTIAL CLASS SRGM USING BAYESIAN TECHNIQUE_Volume 19, June, 2024

been produced. A variety of modern Statistical methods may be applied to measure the Software

reliability. One of the strategy uses a zero-one method, where a flawed Software has a reliability of

zero and a faultless Software has a reliability of one. Another strategy focuses on testing of Software,

where software reliability is defined as the proportion of times a software executes an intended

function as predicted. Thus, the measurement of Software reliability may be done using this method.

The usage of Software Reliability Growth Models (SRGMs) in the evaluation of software reliability is highly beneficial. The operational profile, accessibility of limited failures and irrational assumptions provide significant difficulties for SRGMs in practice. When estimating the parameters of software reliability model, various fundamental techniques are used which are maximum likelihood method, least squares estimation, Bayesian estimation, the EM method, etc., [5], [6], [7] and [12] have presented the calculation of factors, such as failure rate and total number of failures incorporated in SRGMs.

Software Engineering is the process of developing a software that can balance the dependability, delivery time, and price of the developed software. Software reliability modelling, as described by [8] is another way to represent software reliability using a mathematical function of involve factors, such as fault introduction, fault removal, the operating environment, etc. Software reliability is evaluated mathematically and objectively using the body of Statistics and Probability theory.

The Binomial type models, as categorized by [10] and [11] have been taken into consideration for study in this paper. Here, an attempt is made to derive the Bayes estimators for the parameters of the Binomial type exponential class. The prior distribution for the total number of initial software failures and scale parameters have been taken as the gamma prior distribution assuming that the experimenter is having prior information about both the parameters c.f. [14]- [20].

II. Model Characterization

Based on the following assumptions [12] have modelled the process of software failure using Binomial type models.

• The defect that resulted in a software failure will always be immediately fixed.

• There are в0 intrinsic faults in the programme.

• The hazard rates Z(t) for all faults are same.

According to [12, 19], if f(t) is the class of the SRGM and A(t) = 60f(t) is failure intensity. Considering binomial process and solving the differential equations using boundary conditions P0(0) = 1 and Pn(0) =0 V n = 0,1,2,... 60, the following result is obtained.

P[M(t) =n] = (en0){1 - exp[-e1t]}n{exp[-e1t]}e°~n rn = 0,1,2, ...,в0.

The P[M(t) = n] gives the probability that M(t) = n number of failures encountered at time t has a binomial distribution i.e. Binomial type of the software reliability model [12]. The Binomial Type Exponential Class model has failure intensity

A(t) = doele~0lt ,t>0, в0>0 and вг>0 (1)

where в0, failure rate (в-J are the parameters of the model and t be the execution time. This model exhibits failure intensity similar to [4] and [13]. The function of the mean failures is given by

^(t) =в0[1- e~0lt] ,t>0, в0>0 and вг>0 (2)

The expected number of failures at time t is represented by the Binomial distribution with the mean failure function.

^(t) = в0{1 - exp[-e1te]} and variance of M(t) is

var[M(t)] = в0{1 - exp[-e1t]}{exp[-e1t]}. Let me be the number of failures that occurred upto execution time te , then the likelihood function of Binomial type exponential class model is

ФoAll) = е"е1£е(е°"те)вГее-е1гб0— ,te>0,e1>0, 60>0 and t0 = 0 (3)

where

T = Y™\ti

and

в0— are falling factorials (cf. [1], [2] and [3]). The MLEs of в0 and в1 are obtained by applying standard method of obtaining MLE from equation (3) which comes out to be

Z^rnO - i + I)"' = Smite (4)

and

Smi=me[te{Sm0-rne) + T]~1 (5)

The solution for 0mO and Sml can be obtained by solving (4) and (5) using any standard iterative method.

III. Priors for Model Parameters

If the software professional could somehow predict or guess the information about the total number of failures present in the software and the value of scale parameter Let's thus assume that gamma priors are seen to be appropriate for both в0 and в1 then it would be appropriate to use an informative prior for в0 and 01. The time-to-first failure distribution for a system with standby exponentially dispersed backups may naturally exhibit the gamma likelihood. Additionally, in practice, the gamma distribution may appear whenever items were tested, and whenever a part of an item or an entire item fails is replaced by an identical one having an exponential failure time distribution with parameter, the total amount of time on test could subsequently follow a gamma probability function. The gamma distribution is based on the fact that the total of i.i.d. random failure times after exponentials with parameters is distributed as gamma [9] and [24]. When an item may fail partially, or when a certain number of partial failures occur before an item fails (such as with redundant systems), the gamma distribution can be employed. Modelling the time to failure or failure rates for products with infant mortality may be done using the continuous Gamma probability.

Many studies have shown that the gamma distribution is feasible for failure rate and sufficiently adaptable for real-world hardware reliability applications in life testing. [7] created a Bayesian SRGM under the gamma prior assumption for the parameter of exponentially distributed periods between model failures. The Bayesian software reliability growth models established by [5] and [6] take into account the gamma prior distribution.

Thus, the gamma prior distributions may be used as informative priors in the current investigation for the parameters 0O, and в1. Therefore, в0а~1е~Рв° ,0<в0<™ 0 , Otherwise

g(90)a

and

g(61)a

(.0 , Otherwise

where a, r], and v are hyper parameters of considered priors for 0O and 01 respectively. The hyper parameters ^ and a are shape parameters and v and ¡3 are scale parameters of the prior distributions. The flexibility in choices of a, y, and v allows the researcher to select the prior model for parameters that best expresses the current state of knowledge about the number of failures and failure rate.

Hence, the joint prior distribution of both parameters 0O and 01 is given as

a_1 1 e e"vei , 0 < (6) (.0 , Otherwise

Rajesh Singh, Kailash R. Kale, Pritee Singh

ESTIMATION OF PARAMETERS OF BINOMIAL TYPE Kl&A, № 2 (/8) EXPONENTIAL CLASS SRGM USING BAYESIAN TECHNIQUE_Volume 19, June, 2024

IV. Joint Posterior and Marginal Posterior Distributions

Assuming the total execution time is te, during this time me failures are experienced at times tu i = 1,2, ...,me, 0O be the number of failures present in the software and в1 be the failure rate then, combining likelihood function (3) with joint prior given by (6), the joint posterior of в0 and в1 given t0i = 1,2,... ,me(= t) is

n(e0,ei\t) К e-ei[£e(eo-me)+(r+v)]6me+^ "l^a-lg-^m, ^ <во<т,0<в1<ю (7)

The constant of proportionality (normalizing constant) of above equation is

D = Qi™e-e i[te(Oo-me)HT+v)]eine+v -1в0а~1е-Рвов(^йв0йв1 ,me<e0<^,0<e1<^ (8) The above expression of D can be solved using the results given in [1], [2], and [3] as

n = rVme m^m+a-lfm+a-l\fmet£\~r .

u = LLm=0^meme Lr=0 К r J\(T+V)J 'l

where

_ = r(me+V)mea~1 ePmeTi(T+v)me+ V

I1 = Г(г+ 1)W (r + 1,r - me +

\ te j

Г(. ) is standard Gamma function and W(.,.,.) is confluent hyper-geometric function defined in [1], [2], and [3].

The marginal posterior of 60, say n(6011) can be obtained after integrating п(в0,6i |t) over the whole range of в1 and it is

n(60\t) к Г(те + г])в0а-1е-Рв-в0^^[1е(в0 - тв) + (Т + v)]-^^ (9)

where

в0 > me, (te(60 - me) + (T + v))>0 The marginal posterior of say n(d1lt) is the solution of п{в1\t) = п{в0, dd0 i.e.

(10)

where

¡2 = Crf e-^^e ^-i + e^r^-V

V. Bayes Estimates for Model Parameters

The Bayes estimators of в0 and в1 are posterior mean under the squared error loss function. The Bayes estimator for в0 i.e. the posterior mean can be obtained from (9) and is

q к Г(те+т1)теа „me r(m) , m Vm+a(m+a\ [те'е!"Г ,

°B0 К epmeTi(T+v)me+1Lm=0^me me Lr=0\ r J[(T+v)\ 'з

where

I3 = r(r + 1)W (r + 1,r-me+v + The Bayes estimator for is the posterior mean of its marginal posterior distribution (10) is

S К r{me+^+l)mea~1 yme c(m) „ mym+a-lfm+a-^\\mete'\~r .

epmeTi(T+v)me+V+^m=^m^ne L,r=0 У r )[(г+у)\ ^

Where

ц = Г(г + 1)v(r +1,r-me + v +

VI. Discussion

The proposed Bayes estimators i.e. SB0 and SB1 of total number of failures (в0) and failure rate (в-J for the parameters of Binomial type exponential class SRGM are obtained by considering gamma priors and are compared with corresponding maximum likelihood estimators 9m0 and Sml

respectively. The comparative performance of proposed Bayes estimators against corresponding

maximum likelihood estimators has been studied based on the risk efficiencies i.e. REn =

E(eB0-e0)

E(6mo~6o)

E(dn -9 )2 ~ ~

and RE1 = V 1—^2. The estimators §B0 and SB1 are based on the prior parameters a, p, tj, v and

s(Sml-Sl)

execution time te. The risk efficiencies are calculated by considering different values of these constants and arbitrary values of parameters 80 and 6i using the Monte Carlo Simulation technique by generating 103 samples. An execution time te is prefixed and up to this time, sample failures are generated and the risk efficiencies are presented in the following Figure 1 to Figure 14.

Consider the effect of variation in the value of te on the risk efficiencies of the proposed Bayes estimator 8B0, given in Figure 1 to Figure 4.

01 TirT^^ 23 24

28 29

25 26 "

Figure 1: Risk Efficiencies of 0BO and 0Bi, for 01(= 0.12(0.01)0.21), 0O(= 20(1)29), a = 30, 0 = 10, ц = 10, v ■

10 and t„ = 3.0

160 150 u0^7T20~2r~22 23

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e1 e0

24 25 26

27 28 29

13^225212223

Figure 2: Risk Efficiencies of0BO and 0B1,for 0^= 0.12(0.01)0.21), 0„(= 20(1)29), a = 30, /? = 10,r] = 10, v :

10 and t„ = 3.5

0 12 20

d,

Figure 3: Risk Efficiencies of 9B0 and 9B1, for 91(= 0.12(0.01)0.21), 90(= 20(1)29), a = 30, ¡3 = 10, r] = 10, v =

10 and = 4.0

Figure 4: Risk Efficiencies of 9B0 and 9Bi, for 9i(= 0.12(0.01)0.21), 90(= 20(1)29), a = 30, P = 10, T] = 10, v =

10 and t„ = 4.5

From these figures, it is seen that for the increase in value of te, the risk efficiencies RE0 of Bayes estimator 9B0 decrease as 90 and 91 increase. The point of maxima varies as the value of te changes. Particularly, the risk efficiency RE0 attains maxima at smaller values of &0 and for increasing values of te.

The variation of shape constant a(= 30(5)40) of proposed prior for the total number of failures, the risk efficiencies of Bayes estimators &B0 and 9B1 are presented in Figure 5 and Figure 6.

<?i 6

Figure 5: Risk Efficiencies of 0SO and §B1, for 0^= 0.12(0.01)0.21), 0O(= 20(1)29), te = 4.0, p = 10, ^ = 10, v =

10 and a = 35

REn

RE,

28 29

0.14

0.12 20 21

22 23

01

0,

Figure 6: Risk Efficiencies of 0SO and 0B1, for 0^= 0.12(0.01)0.21), 0O(= 20(1)29), te = 4.0 , 0 = 10, ^ = 10, v =

10 and a = 40

It is observed that the values of risk efficiencies of both the proposed estimators are increased for increasing the value of a. Here, in these figures the risk efficiencies of both the estimators are increasing for increasing values of a.

The risk efficiencies of Bayes estimators &B0 and 9B1 i.e. RE0 and RE1 calculated for different values of scale constant ^(= 1,10(5)20) of prior proposed for &0 are summarized in Figure 7 to Figure 8.

RE

0.12 20

200-i—

Figure 7: Risk Efficiencies of 0BO and 0B1, for 01(= 0.12(0.01)0.21), 0O(= 20(1)29), te = 4.0, a = 30,, r/ = 10, v -

10 and p = 1

Or

Figure 8: Risk Efficiencies of 0BO and 0Bi, for 0i(= 0.12(0.01)0.21), 80(= 20(1)29), te = 4.0, a = 30, y = 10, v =

10 and p = 15

Here, the risk efficiencies of both estimators are decreasing for increasing values of p. It is also seen that, both the proposed Bayes estimators 9B0 and 9Bi are becoming more inefficient than corresponding maximum likelihood estimators as p increases.

The risk efficiencies of Bayes estimators 8B0 and 8B1 i.e. RE0 and RE1 are also evaluated using various values of shape constant 1,10(5)20) of prior for 81 and are summarized in Figure 9 to Figure 11.

Figure 9: Risk Efficiencies of 9B0 and 0B1, for 91(= 0.12(0.01)0.21),0O(= 20(1)29),te = 4.0, a = 30, ¡3 = 10, v =

10 and r] = 1

Figure 10: Risk Efficiencies of 9B0 and 9B1, for 9X(= 0.12(0.01)0.21), 90(= 20(1)29)te = 4.0, a = 30, £ = 10, v =

10 and ^ = 15

Figure 11: Risk Efficiencies of 9B0 and 9B1, for 91(= 0.12(0.01)0.21), 90(= 20(1)29), te = 4.0, a = 30, ¡3 =

10, v = 10 and 1} = 20

Here, it is observed that the risk efficiencies of both estimators decrease for the increase in the values of It is also seen that, both the proposed Bayes estimators &B0 and 9B1 are becoming more inefficient than corresponding maximum likelihood estimators as ^ increasing.

The risk efficiencies of Bayes estimators &B0 and 9B1 i.e. RE0 and RE1 evaluated using various values of scale constant v(= 1,10(5)20) of prior 91 and are summarized from Figure 12 to Figure 14.

Figure 12: Risk Efficiencies of 9B0 and 9B1, for 91(= 0.12(0.01)0.21), 90(= 20(1)29), a = 30, ¡3 = 10, T) = 10, te =

4.0 and v = 1

Figure 13: Risk Efficiencies of 9B0 and SB1, for 91(= 0.12(0.01)0.21),0O(= 20(1)29), te = 4.0, a = 30, ¡3 = 10, tj

10 and v = 15

Figure 14: Risk Efficiencies of9B0 and 9B1,for 9X(= 0.12(0.01)0.21), 0O(= 20(1)29), te = 4.0, a = 30, /? = 10,

^ = 10, and v = 20

Here, it is seen that the risk efficiencies of 8B0 are increasing whereas the risk efficiencies of 8Bi are decreasing for increasing the values of v. It is also seen that the proposed Bayes estimator dB0 is becoming efficient as ^ increases whereas 8Bi becoming more inefficient than the corresponding maximum likelihood estimator.

VII. Conclusions

Both the proposed Bayes estimator of 80 and 8i i.e. 8B0 and 8Bi can be preferred over corresponding MLEs if the parameters of gamma priors for model parameters are properly chosen. The value of te should be small for moderate values of true parameters and prior constants. The values of shape constant a of prior proposed for 80 should be chosen moderately large for smaller values of te. The values of scale constant ft of prior proposed for the total number of failures i.e. 80 should be chosen

Rajesh Singh, Kailash R. Kale, Pritee Singh татя тчт пял ESTIMATION OF PARAMETERS OF BINOMIAL TYPE Kl&A, No 2 (/8) EXPONENTIAL CLASS SRGM USING BAYESIAN TECHNIQUE_Volume 19, June, 2024

smaller when values of te are small. The values of prior parameters rç and v should be chosen smaller

for smaller values of t„.

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