On Warranty Cost Analysis For a Software Reliability Model Via Phase Type Distribution Текст научной статьи по специальности «Медицинские технологии»

WARRANTY COST ANALYSIS VIA PHASE TYPE SOFTWARE Volume 14, June 2019 RELIABILITY MODEL

On Warranty Cost Analysis For a Software Reliability Model Via Phase Type Distribution

Y. Sarada and R. Shenbagam

CEGC, Anna University, Chennai, Tamil Nadu, India - 600 025 sarada@ annauniv.edu, [email protected]

Abstract

This research work investigates an optimal software release problem via phase type distribution, warranty and risk cost analysis. The inter arrival time of software failure is assumed to be a phase type distribution. The PH-SRM is one of the most flexible models, which overarches the existing non-homogeneous Poisson process (NHPP) models, and can approximate any type of NHPP-based models with high accuracy. Based on the phase type Non- homogeneous Poisson Process (PHNHPP) formulation and using the renewal reward theorem, the long run average cost rate is obtained. As model parameter estimation is an important issue in developing software reliability models, the software failure parameter has been estimated by the moment matching method. Finally, a numerical example is provided to illustrate the theoretical results therein.

Keywords: phase type distribution, software reliability, method of moments, renewal reward theorem, optimal software release policy

I. Introduction

The most significant feature of commercial software is Reliability, since it quantifies software failures during the growth procedure. Software is examined from an assortment of judgment in the testing procedure. For example, functionality, reliability, usability and the defects located in the procedure should be defined before it is released to the society. Decreasing the development cost and improving the quality of software are notable facts in software testing management. Software managers are confronted with many complicated problems in software testing. The plan for software maintenance (patch schedule) and the decision for the software release (Okumoto and Goel (1980), Pham (1996)) are based on reliability. Therefore, software reliability is an excellent tool to estimate the number of bugs (Musa et al. (1987), Musa (1999), Pham (2000)). Previous efforts on software reliability models (SRMs) mainly focused on NHPPs owing to their mathematical tractability and varied modeling situations such as imperfect debugging, change points, testing efforts and fault detection process (Xie et al. (2007), Okamura et al. (2013)). For example, the debugging process is modelled as a counting process which follows Poisson distribution with a time dependent hazard rate function (Goel and Okumoto (1979); Littlewood (1981), Yamada et al. (1983), Goel (1985), Langberg and Singpurwalla (1985), Laprie et al. (1991), Gokhale and Trivedi (1998)).

Y. Sarada, R. Shenbagam RT&A, No 2 (53) WARRANTY COST ANALYSIS VIA PHASE TYPE SOFTWARE Volume 14, June 2019 RELIABILITY MODEL_

One of the major issues addressed in NHPP based software reliability is that of determining the best model whose solution lies in the statistical methods encompassing fitting of the observed bug data and to decide the model parameters (Langberg and Singpurwalla (1985)). Working in this direction, Okamura and Dohi (2006, 2008) introduced the phase type software reliability model (PHSRM) wherein the fault detection time follows a phase type distribution with the underlying counting process following NHPP.

A warranty is an agreement between a buyer and a seller at the time of product sale. It is a detailed study of the reimbursement type for a given product at the time of occurrence of failures. Also, it plays a significant role to safeguard the customer's interest particularly in the case of complicated products such as automobiles or electric devices. Recently attention has been directed towards warranty policies and warranty cost modelling (Nguyen and Murthy (1984), Blischke and Murthy(1995)). In today's market, many goods like mobile phones, electronic items and home devices are sold with extended warranty policy, in which a few choices are available for the consumer at the expiry time of the free warranty period. Extended warranties present extra security in the event of expensive failures after the initial warranty period and thereby safeguard the buyer against inflation. Also, extended warranty has attracted significant attention among practitioners. Lam and Lam (2001) proposed an extended warranty model with options open to customers to obtain an optimal policy for the consumers.

Furthermore, in the software management scheme the most significant calculation is to find an optimum software release time, referred to as an optimal software release problem. An optimal release problem with warranty cost and reliability requirement was studied by Yamada (1994). Also, Jain and Handa (2001) developed a software reliability model by employing a Hybrid warranty policy. Zhang and Pham (1998) studied a software cost model under warranty with a risk cost due to software failure and a cost to remove each error detected in the software. Prince Williams (2007) derived optimal release time policies to predict the optimal release time of software using imperfect debugging phenomena and warranty. The optimal release problem with simulated cost and reliability requirements was further implemented by Okumoto et al. (2013).

From a thorough review of the existing literature done, a combination of Phase type distribution and warranty (fixed / extended) has not been employed anywhere in the literature, in the analysis of software reliability. Motivated by this and in order to fill the gap in the literature, following Okamura and Dohi (2006,2008), the two new features attempted in this research article are the parameter estimation through moment matching method and the long run average cost rate analysis under the combination of phase type distribution and extended warranty in the context of software reliability models.

The structure of the paper is organized as follows: Section II furnishes the basics for the related work. Section III gives a detailed problem description and model assumptions. An explicit expression for the long run average cost rate is obtained in Section IV, while parameter estimation is discussed in Section V. Further, Section VI provides the numerical illustrations. Finally Section VII presents the concluding remarks.

II. Basics

Software reliability model based on Non-Homogeneous Poisson process:

The NHPP modelling in the SRMs essentially treats a counting process of software failures / faults / bugs in software system testing. It is virtually similar to a functioning profile of released software (Musa 1999), which provides information regarding the number of software failures in the system testing vis-à-vis the software reliability in the working phase. Further, existing NHPP-based SRMs are categorized into finite and infinite models. In the finite models, the detection

Y. Sarada, R. Shenbagam RT&A, No 2 (53) WARRANTY COST ANALYSIS VIA PHASE TYPE SOFTWARE Volume 14, June 2019 RELIABILITY MODEL_

slowly reduces with testing time and ultimately becomes zero, while in the infinite model, it does

not become zero, that is, the number of software faults infinitely increases with time.

Specifically, if M(t) represents the number of software faults by time t with F(t) as the

cumulative distribution function(c.d.f) of the detection times of software faults while the random

variable N is the total number of software faults with mean m, then the probability mass function

(p.m.f) of M(t) is given by

P{M (t) = k} - Pk (t) = WWt e -m F (t k - 0,1,2,... (1)

k!

(Refer Langberg and Singpurwalla (1985)). 2.1 PH-SRM

Okamura and Dohi (2006, 2008) introduced phase type distribution in SRMs in which the fault detection time is a PH distribution in the NHPP- based model, referred to as PH-SRM.

A PH distribution is defined as the time to absorption in a continuous-time Markov chain (CTMC) with one absorbing state. Let Q denote the infinitesimal generator matrix of the CTMC with one absorbing state. Without loss of generality, Q is assumed to be partitioned as follows:

e-U u0' 10 0 /

where U and U0 correspond to transition rates of transient states and exit rates from transient states to the absorbing state, respectively and a probability vector (a,am+i) exists such that

Fph(x) -1 - a. exp(U.x).e for x>0. Here e denotes a column vector of ones with an appropriate dimension (Neuts (1981)). Note that the exit vector U0 is given by U0 = -Ue. The transient states of PH distribution are often called phases. PH distribution is proved to be dense, so that it can approximate any probability distribution with any precision as the size of U (the number of phases) increases (Asmussen and Koole (1993)).

By substituting the c.d.f. of PH distribution into (1), the p.m.f. of PH-SRM is given by

\k

-e mFPH (t),k - 0,1,2,... (2)

P{M(t) = k} = Pk (t) = (m FpH(t)} e"m FPH« k = 0,1,2,...

k!

Let

A(t) = mFPH (t) = (1 -n exp(Ut )e)m (3)

so that

P{M(t) = k} = Pk(t) = (A(t)) e"A(t),k = 0,1,2,... (4)

k!

Thus PH-SRM exactly comprises of NHPP-based SRMs whose fault detection time distributions are exponential (Goel & Okumoto 1979), fc-stage Erlang (Yamada et al. 1983; Khoshgoftaar 1988; Zhao & Xie 1996), hyperexponential (Laprie et al. 1991) and hypoexponential distributions (Fujiwara & Yamada 2001).

WARRANTY COST ANALYSIS VIA PHASE TYPE SOFTWARE Volume 14, June 2019 RELIABILITY MODEL

III. Model assumptions

We make the following assumptions about software reliability model for a phase type distribution in the context of warranty (fixed and an extended) modelling.

i. C1: The set up cost of software development process is a constant.

ii. C2: The cost to remove errors during debugging period is proportional to the total time of removing all errors detected during this period.

iii. C3: The cost to remove errors during fixed warranty period (TW) is proportional to the total time of removing all errors detected in the time interval [ T, T+Tw].

iv. C4: The cost to remove errors during extended warranty period (TE) is proportional to the total time of removing all errors detected in the time interval [T+Tw, T+ Tw+Te].

v. C5: Risk cost due to the software failure after its release.

vi. It takes random time to remove errors and hence it is assumed that the time to remove each error follows a phase type distribution.

IV. Cost analysis

No efforts were made previously to carry out the cost analysis under warranty (fixed / extended) and phase type modeling in SRMs. Thus the goal of cost analysis is to obtain the long-run average cost rate for the proposed warranty model. By applying the standard result based on the renewal reward theorem, the long-run average cost per unit time is given by, (Ross, 1996):

C(T) =

Expected cost incurred in a cycle _ E(C) Expected length of a cycle E(L)

(5)

(6) (7)

Here, the expected length of the cycle E(L) is given by,

E(L) = T + TW + Te

Next, the expected cost in a cycle E(C) can be expressed as,

E(C) = E(Q) + E(C2) + E(C3) + E{C4) + EC5)

In what follows, the calculations pertaining to the cost analysis are presented. > E(C2):

y = y0 + (i - 1)Y, i = 1,2,3,....

Let yi be the cost of fixing ith software fault. It consists of a deterministic part y0 and an incremental random part (i-1) Y, where Y is a phase type random variable with mean

jUy = -%S~^e2 . Note that the cost of fixing a fault is increasing as the number of faults removed is increasing. This is reasonable because it may become difficult to identify and fix a fault that occurs in the later testing phases.

Let N(T) denote the number of detected errors removed by time T, so that:

E(C2) = C21E

N (T ) S (J0 + (i - 1)Y) i=1

i x

= C2 \ S E

\n=l

N (T )

S (J0 + (i - 1)Y | N(T) = n)P[N(T) = n]

i=1

= C21 S n[2^0 + MY P (T) + S n2MyPn (T) jj

Employing (3) we have,

E(C2) = C2 ^

2 yQA(T) + j {At)?

> E(Cs): E(C3) = C3 |e

'n (T+TW)

N (T+TW) 2 Wi N (T)

Using E

2 Wi N (T)

= juW {A(T + TW) - A(T)}, we have,

E(C3) = C3J [A(T + Tw ) -A(T)]}

(8)

(9)

> E(C4):

E(C4) = C4J [A(T + Tw + TE ) -A(T + TW )]}

(10)

> E(Cs): E(Cs) = C5{1 - R( x / T)}

Here R(x/T) is the software reliability expressed as:

R(X/T) = e-[A(T + x)-A(T)] = e-m[aeUTei -aeU(T+x)ei ],

so that

E(C5) = C5 |l - e-m[aeUTei-aeU (T+x)

ei]

(11)

Now, employing (8) - (11), the expected total software cost E[C] can be expressed as in (12).

E(C) = Ci + C2

2 yo A(T) + jy {A(t))2 + C3 J {A(T + TW )-A(T) ]

+C4 [je {A(T+tw + te )-A(T+tw )] + C5

i - e

aeUTei-aeU (T + x)ei

(12)

The optimal T* can be determined from (5), by using numerical or analytical methods. Optimization:

This section is devoted to obtain the optimal software release time T* using the concept of pseudo-convexity of a function.

Lemma:

Using the convexity property of C(T), the optimal software release time T* is determined by solving the following equation.

[t + Tw + TE ]

C2 (yo A(T) + UY A(T)A(T}) + C3vw A'(T + Tw ) - A(T)) + C4 [ue (A(T + Tw + Te ) - A(T+Tw ))]

+ C5mi(eeUTUel-aeU (T+^e-™^ e1-aeU (T+X)e1]

C

2 yo AT ) + UY A(t))2 + C3 [uw (A(T + Tw ) - A(T )]

-C4 [ue (a(t + Tw + Te )-A(T+Tw )]

+ C

1 - e

aeUTe1 -oeU (T+^

= Ci + C5

(13)

-m

Proof

Consider the derivative of C(T) with respect to T as given below:

[t + Tw + Te ]

C2 (yA'(T ) + UY A(T )A'(T )) + C3Uw (A(T + Tw ) - A(T )) + C4 [ME (A(T + Tw + TE ) - A(T + Tw ))]

+ cJ(eeUTUe1-œU (T+^ )e"™[«eUTei -aeU (T+x)q] '

C2

1 [2 yo A(T ) + uy (A(t ))2 + C3 [Mw (A(T + Tw ) - A(T )] + C4 [ue (A(T + Tw + Te ) - A(T + Tw )]

+ C5

i - e

aeUTe1 -aeU (T+%

= C1 + C5

If the cost function given in (5) is pseudo-convex, then it has only one local minimum and thus there will exist a unique global minimum. Now consider (12):

—E(C) = dT

C2 (yoA'(T) + Uy A(T )A'(T )) + C3Uw (A'(T + Tw ) - A'(T)) + C4 [ue (A'(T + Tw + TE ) - A'(T + Tw ))] + Qm((aeUTUe - œU(T+U) e^^a<T+")

> 0

d2

—r-E(C) = dT2 v 7

C2 (yoA"(T ) + UY (A (T ))2 + UY A(T )A"(T ))+Cu (A"(T + Tr ) - A"(T )) + C4 [ue (A'' (T + Tw + TE )-A" (T + Tw ))]

r

+ Qm

(ae

U2e -aeU(T+x)U2e -m(aeUTUel-aeU(T+x)Ue,))

-m[ae e

^ -,»U (T + x)

dT:

-E(C) > 0

Hence E(C) is positive and convex. Since C(T) is linear in T and positive, C(T) is pseudo- convex in E(C). Hence the lemma.

-m

2

d

V. Parameter estimation

In this section statistical estimation of the model parameters is to be performed which will take the modelling closer to the realm of applicability. Methods based on the estimation of the parameters for a phase type distribution, can be classified into three types such as (i) moment matching method (MM), (ii) maximum likelihood (ML) estimation and (iii) Bayes estimation. In this section, the method of moments (MM) is mainly used for the PH distribution. The concept of MM method is to find PH parameters so that the population moments can fit the moments derived from samples or probability density function of the true distribution. As mentioned before, the accuracy of MM method depends on the number of moments used.

Consider the Hyper - exponential distribution of order 2, that can be expressed as, A(t) = mF (t) = {i - n exp(Ut )e)m, so that i

* = [p i - p], U=

--0

Ai

0 -J-

Az.

e = [i 1]\

Therefore, the

f (t ) = A'(t) = m

:).d.f and the first four moments are given by:

JL~Ai + il-p) e_A2 Ai A2

t > 0, 0 < p < l.

mi = m[ pA + (l - p)A2]

m2 = m[2pA + 2(1 - p)a2 ]

(14)

(15)

m3 = m[6pA3 + 6(1 - p)A3 ]

m4 = m[24 pAJ + 24(1 - p)AJ ]

(16) (17)

From (14) we have,

A2 =

mi - pwAi m(1 - p)

Substituting (18) in (15), and after some simple calculations,

Ai =

mi

w

i ±

Ï

2

(m2m - 2m^ )(i - p)

2 pmj

(18)

(19)

We have two cases:

Case (i)

When

Ai =

mi m

i -

1

2

(m2m - 2m ^ )(i - p)

2 pmj

Substituting (20) in (18) we have,

t

t

WARRANTY COST ANALYSIS VIA PHASE TYPE SOFTWARE Volume 14, June 2019 RELIABILITY MODEL

h = m

m

1 +

1

(OT2OT - 2m, )(p)

2(1 - p)mf

(21)

Since (20) and (21) involve ^ and % expressed in terms of the unknown parameters p and m only. Using (16) and (17) the following system of equations are obtained:

m3-

6m

m

3m2m

- 2 +

2m,2

m2m - 2m 2m2

2 ^

2 p -1

Vp(I - p)

= 0

(22)

3

2

24m4

m4

m2m3

-1 +

3m3

m2m - 2m

2 ^ 1

2m

2

2(2p -1) Vp(1 - P)

m2m - 2m

2 1

2m

2

2

3 p 2 - 3 p +1 p(1 - p)

(23)

= 0

Thus, (22) and (23) are to be solved to get the parameters m and p. Subsequently the remaining parameters are obtained by the method of back substitution.

Case (ii)

Proceeding similarly as in case (i), we have,

h =

m1 m

1 +

]

2

(m2m - 2m^ )(1 - p)

2 pm^

(24)

i mi

h2 =-<

m

1 -

1

2

(m2 m - 2m^ )p

2(1 - p)m2

(25)

m3

6m3

m

3m2m

- 2 +

2m2

m2m - 2m 2m2

2 ^

1 - 2 p

>(1 - p)

= 0

(26)

m4

24m4

m

m2m3

-1 +

3m3

m2m - 2m 2m2

2 ^

2(1 - 2 p) Vp(1 - p)

m2m - 2m 2m}

2 W

2

3 p 2 - 3 p +1 p(1 - p)

= 0

(27)

3

2

1

+

3

m

3

2

1

3

2

1

+

It is easily seen that both cases lead to the same result, but with the roles of p and 1-p interchanged. Hence, the parameter estimation carried out may help the practitioners in making flexible decisions for the proposed software reliability model via phase type distribution.

Additionally a numerical example is provided to demonstrate the applicability of the proposed software reliability model with warranty in Section III.

VI. Numerical illustration and sensitivity analysis

In this section, a numerical example is given to illustrate the impact of combining the phase type distribution and an extended warranty in the field of software reliability. The first four moments are assumed to be:

mi = 2.0i25 x i04, m2 = i.62i25 xi07, m3 = i.9607 xi0i0, m4 = 3.i644 xi0i3.

The inter arrival times of software faults are assumed to be hyper exponential. Employing the moment matching method and using the above moments, the estimated values are obtained as:

WARRANTY COST ANALYSIS VIA PHASE TYPE SOFTWARE Volume 14, June 2019 RELIABILITY MODEL

w = 50, n = [0.95 0.05], U =

1

400 0

0

1

450

, e = [1 1]'

Also, assume that,

C1 = 5000, C2 = 50, y0 = 0.5, jy = 0.9, jW = 0.95, jE = 0.85,TW = 500, TE = 400, x = 1.5, C3 = 360, C4 = 200, C5 = 500

Utilizing the above parameter values in (5), the result has been presented in Table 1. The optimal software release time T*=67 and the corresponding cost 10.0249 is depicted in Table 1. Additionally, the sensitivity analysis of a proposed software reliability model using C(T) is analyzed. Tables 2 and 3 illustrate the sensitiveness of the long run average cost rate. Table 2 shows that as C1, C2 , C3 and C5 increase respectively, the long run average cost rate increases while an increase in C4 results in a decrease in C(T). In a similar manner, Table 3 indicates that the long run average cost rate increases as x, ^y, ^w and y0 increase and decreases as the parameter ^e increases. Further, Tables 2 and 3 illustrate that the optimal software release time T* increases with an increase in the parameters C1, C3, x and ^w while it decreases as the parameters C2, C4, ^y, ^e and y0 increase. Also, the optimal software release time T* is unchanged as the parameter C5 increases. Thus, the sensitivity analysis of such parameters may aid the software system manager in making decisions to model the software system testing.

Table 1: C(T) versus T

T C(T) T C(T) T C(T) T C(T) T C(T)

20 10.6718 45 10.1554 62 10.0316 67 10.0249 80 10.0631

25 10.5325 50 10.1018 63 10.0292 68 10.0250 85 10.0975

30 10.4121 55 10.0629 64 10.0274 69 10.0257 90 10.1420

35 10.3097 60 10.0379 65 10.0261 70 10.0268 95 10.1959

40 10.2244 61 10.0345 66 10.0252 75 10.0393 100 10.2587

Table 2 : Sensitivity analysis for the parameters Ci, C2, Cs, C4 and C5 in C(T)

Ci t* C(T*) C2 t* C(T*) C3 t* C(T*) C4 t* C(T*) C5 t* C(T*)

4000 65 8.9898 40 88 9.6337 360 67 10.024 150 69 11.882 400 67 10.0098

5000 67 10.024 50 67 10.024 400 76 11.189 200 67 10.024 500 67 10.0240

6000 69 11.057 60 54 10.288 450 87 12.595 250 65 8.1645 600 67 10.0400

7000 72 12.088 70 45 10.478 500 98 13.947 300 63 6.3018 700 67 10.0550

Table 3: Sensitivity analysis for the parameters x, ¡y, ¡w, ¡e and yo in C(T)

X t* C(T*) t* C(T*) t* C(T*) t* C(T*) yo t* C(T*)

0.5 67 9.9760 0.9 67 10.024 .75 52 7.7208 .75 68 10.899 .5 67 10.024

1.5 67 10.024 1.1 54 10.276 .85 59 8.8888 .85 67 10.024 1.5 57 10.3964

2.5 67 10.068 1.3 45 10.451 .95 67 10.024 .95 66 9.1497 2.5 47 10.7144

3.5 68 10.108 1.5 39 10.581 1.05 76 11.129 1.05 65 8.2740 3.5 37 10.9770

WARRANTY COST ANALYSIS VIA PHASE TYPE SOFTWARE Volume 14, June 2019 RELIABILITY MODEL

VII. Conclusion

In this study, a phase type Non - homogeneous Poisson process software reliability model that incorporates warranty (fixed and an extended) via long run average cost rate in order to determine the optimal software release time is developed. Hence, PH-SRM is a promising tool to reduce the effort to select the best models in software reliability assessment. Also, the parameter estimation was carried out using moment matching method. The graphical illustrations conform to the theoretical observations made earlier. Additionally, a sensitivity analysis has been carried out for all the parameters, to exemplify the optimal software release policy (T*) and the corresponding long run average cost rate C(T*). To the best of authors' knowledge, it is observed that, phase type distribution in the cost analysis has not been studied from the view point of software reliability systems with fixed and extended warranty model. As a final remark, the extended warranty model enables the software manger to decide on whether the software is sufficiently tested to allow its release or unrestricted use. Such predictions provide a quantitative basis for achieving reliability, risk and cost goals.

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