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Statistics & Probability Letters 72 (2005) 187–194

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A mixture and self-exciting model for software reliability

R. Wang

Department of Statistics, Tunghai University, 181, Sec. 3, Taichung-Kan Rd., Taichung 407, Taiwan

Received 2 October 2003; received in revised form 14 November 2004; accepted 28 November 2004

Abstract

This paper develops a model, based on both properties of mixture and self-exciting, that generalizesexisting software reliability models. We also offer a model classification method. Details are proven withsome specific cases and examples illustrating the results are given.r 2005 Elsevier B.V. All rights reserved.

Keywords: Software reliability; Intensity function; Concatenated failure rate; Self-exciting; Mixture; Martingale

1. Introduction

A large number of statistical software reliability models have been proposed over the pastdecade for evaluating the reliability of computer software. One of the main approaches of thosemodels is to obtain results from assumptions about the stochastic behavior of how failures occurduring software’s running time. In this approach, the methods deal with the failure process bycharacterizing the time pattern of failure occurrences. The use of stochastic models was proposedby Jelinski and Moranda (1972), one of earliest model in the field. It is of great interest to unify orclassify the large class of models. Several generalizations of software reliability models have beenintroduced, for example, Langberg and Singpurwalla (1985), Miller (1986), Musa et al. (1987), Al-Mutairi et al. (1998), and Singpurwalla and Wilson (1999). Among them, Chen and Singpurwalla(1997) have introduced a generalization to unify software reliability models through self-excitingprocesses.

see front matter r 2005 Elsevier B.V. All rights reserved.

spl.2004.11.027

ress: rtwang@mail.thu.edu.tw.

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The purpose of this paper is to introduce and study a model for software reliability, extendedfrom the one proposed by Chen and Singpurwalla (1997), by adding the consideration of amixture. In the process of developing this model, we also point out a method for the classificationof models. Assumptions and notations are given in Section 2 where a close relation between theintensity function and the concatenated failure rate function is shown. The result indicates aclassification for some existing software reliability models according to the specified functionalforms. In Section 3, an intensity function defined on both properties of mixture and self-exciting isgiven and details are proven with some specific cases. Finally, examples which illustrate the resultsare given in Section 4.

2. Assumptions and notations

Let fX ðtÞ; tX0g be a counting process, which characterizes failure occurrences of a computersoftware, defined on a filtered probability space ðO;F; ðFtÞtX0;PÞ with fX ðtÞ; tX0g adapted to thefiltration ðFtÞtX0; where

Ft ¼ sfX ðsÞ; 0psptg.

We assume that the filtration satisfies the ‘usual conditions’ of right-continuity and completeness(see Meyer, 1966, p. 71).A counting process is said to be self-exciting if it depends on the entire or some fraction of its

history ðFtÞtX0 that affects or excites the intensity function LðtjFtÞ of the process defined as

LðtjFtÞ ¼ limdt#0

1

dtE½X ðt þ dtÞ X ðtÞjFt�.

Thus, a self-exciting process is not necessarily equipped with the property of independentincrements. Assume that the process does not have failures which occur simultaneously. In otherwords, the process satisfies the conditional orderliness property (Cox and Isham, 1980, p. 25): forany Gt � Ft and tX0; as d # 0;

PðX ðt þ dÞ X ðtÞX2jGtÞ ¼ oðdÞPðX ðt þ dÞ X ðtÞ ¼ 1jGtÞ.

When such is the case, the intensity function becomes

LðtjFtÞ ¼ limdt#0

1

dtPðX ðt þ dtÞ X ðtÞ ¼ 1jFtÞ; SX ðtÞptoSX ðtÞþ1,

where Si is the time of occurrence of the ith failure with S0 0: Let Ti ¼ Si Si1 be the inter-occurrence time, for i ¼ 1; 2; . . . :A self-exciting process takes account of the points generated by the failure times so that, for

example, Ft ¼ fX ðtÞ;S1; . . . ;SX ðtÞg: A point process is said to have no-memory if LðtjFtÞ ¼ LðtÞ;it has 0-memory if LðtjFtÞ ¼ LðtjX ðtÞÞ; and it has m-memory if LðtjFtÞ ¼

LðtjX ðtÞ;SX ðtÞ; . . . ;SX ðtÞmþ1Þ; for m ¼ 1; 2; . . . : It has been shown that if fX ðtÞ; tX0g has m2-memory then it is m1-memory, for 0pm1pm2o1; and a counting process has 0-memory if andonly if it is a Markov process (Snyder and Miller, 1991, p. 306). Note that the intensity functionLðtjFtÞ is continuous from the left, and we assume for tX0 that E½LðtjFtÞ�o1:

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Another function which is used in reliability modelling (to include software reliability), is theconcatenated failure rate function of the failure process. Given Ft; it is defined (see Chen andSingpurwalla, 1997),

HðtjFtÞ ¼ limdt#0

1

dtPðtoTX ðtÞþ1ptþ dtjFtÞ; tX0.

Let us consider a system that consists of only one component whose lifetime is T, and defineX ðtÞ ¼ IfTptg; thus, for a one-component system LðtjFtÞ is equivalent to HðtjFtÞ: As might beexpected, there is a close relation between LðtjFtÞ and HðtjFtÞ as is shown in the following:

LðtjFtÞ ¼ Hðt SX ðtÞjFtÞ; SX ðtÞptoSX ðtÞþ1,

HðtjFtÞ ¼ Lðtþ SX ðtÞjFtÞ; 0ptoSX ðtÞþ1 SX ðtÞ.

Accordingly, this relation enables us to indicate a method to classify software reliability models.The implication is that if we can specify LðtjFtÞ then we have HðtjFtÞ; and vice versa. Here wegive five cases as examples:

ðaÞ LðtjFtÞ ¼ ðN X ðtÞÞcðtÞ¼)HðtjFtÞ ¼ ðN X ðtÞÞcðtþ SX ðtÞÞ;

ðbÞ LðtjFtÞ ¼ ðN X ðtÞÞcX ðtÞþ1ðtÞ¼)HðtjFtÞ ¼ ðN X ðtÞÞcX ðtÞþ1ðtþ SX ðtÞÞ;

ðcÞ LðtjFtÞ ¼ lðtÞ¼)HðtjFtÞ ¼ lðtþ SX ðtÞÞ;

ðdÞ HðtjFtÞ ¼ ðN X ðtÞÞcðtÞ¼)LðtjFtÞ ¼ ðN X ðtÞÞcðt SX ðtÞÞ;

ðeÞ HðtjFtÞ ¼ cX ðtÞþ1ðtÞ¼)LðtjFtÞ ¼ cX ðtÞþ1ðt SX ðtÞÞ;

where lð�Þ and cð�Þ are continuous functions. Obviously, LðtjFtÞ is no-memory in case (c),0-memory in cases (a) and (b), and 1-memory in cases (d) and (e). Except for case (b), the givencases correspond to some categories of software reliability models in the context of Chen andSingpurwalla (1997). Thus, case (a) corresponds to the class of i.i.d. order statistic models, case (c)corresponds the class of non-homogeneous Poisson process models, and cases (d) and (e)correspond to the class of concatenated failure rate models.Apart from the self-exciting, another property for the model in our discussion is mixture which

is assumed that individual faults come with i.i.d. random failure rates l and that failure times haveintensity function fðtjl;FtÞ: Let X iðtÞ be the indicator function of failure due to fault i upto timetX0: Therefore, the cumulative number of software failures upto time tX0 is given by

X ðtÞ ¼XN0

i¼1

X iðtÞ,

where N0 is the unknown initial number of faults in the software.Now we can discuss an intensity function which have both properties of self-exciting and

mixture. Assumptions are given in the following:

(i)

The X i are mutually independent, given N0; fligiX0; and Ft; (ii) fðtjli;FtÞ ¼ limdt#0 ð1=dtÞPðX iðt þ dtÞ X iðtÞ ¼ 1jN0; fligiX0;FtÞ; (iii) The failure process satisfies the conditional orderliness property,

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where li is the random failure rate of faults i. The intensity function of the software can bedetermined and

LðtjN0; fligiX0;FtÞ ¼ limdt#0

1

dtPðX ðt þ dtÞ X ðtÞ ¼ 1jN0; fligiX0;FtÞ. (1)

3. Main results

In this section, we assume that once a fault in the software is detected then it is removedperfectly with no new faults generated.

Theorem 3.1. Under the conditions (i), (ii) and (iii), if the li have a mixing distribution function G,then the intensity function of fX ðtÞ; tX0g; given N0 and Ft; is

LðtjN0;FtÞ ¼ ðN0 X ðtÞÞ

R10 fðtjl;FtÞ exp½Fðtjl;FtÞ�dGðlÞR1

0 exp½Fðtjl;FtÞ�dGðlÞ,

where

FðtjlÞ ¼Z t

0

fðsjl;FsÞds.

Proof. From (1) and the given conditions, we have

LðtjN0; fligiX0;FtÞ ¼XN0

i¼1

limdt#0

1

dtPðX iðt þ dtÞ X iðtÞ ¼ 1jN0; fligiX0;FtÞ

¼ ðN0 X ðtÞÞfðtjfligiX0;FtÞ. ð2Þ

Let dX ðsÞ ¼ X ðs þ dsÞ X ðsÞ; and it is not so difficult to see that ðdX ðsÞjN0; fligiX0;FsÞ follows abinomial distribution

ðdX ðsÞjN0; fligiX0;FsÞ � Binom ðN0 X ðsÞ;fðsjfligiX0;FsÞdsÞ.

So that

E½dX ðsÞjN0; fligiX0;Fs� ¼ ðN0 X ðsÞÞfðsjfligiX0;FsÞds.

Next, let us write

dMðsÞ ¼ dX ðsÞ E½dX ðsÞjN0; fligiX0;Fs�.

This is equivalent toZ t

0

dMðsÞ ¼

Z t

0

dX ðsÞ

Z t

0

E½dX ðsÞjN0; fligiX0;Fs�.

It follows that

MðtÞ ¼ X ðtÞ

Z t

0

ðN0 X ðsÞÞfðsjfligiX0;FsÞds. (3)

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The MðtÞ in (3) is a martingale (see Bremaud, 1981). Because the integration part of this Doob-decomposition is a predictable process, therefore, for fixed N0;

X ðtÞ E

Z t

0

ðN0 X ðsÞÞfðsjfligiX0;FsÞdsjFt

� �(4)

is also a martingale. Eq. (4) is equivalent to

X ðtÞ

Z t

0

ðN0 X ðsÞÞE½fðsjfligiX0;FsÞjFt�ds

� �or equivalently

X ðtÞ

Z t

0

ðN0 X ðsÞÞE½fðsjli;FsÞjX iðsÞ ¼ 0�ds

� �. (5)

Following the assumption that the li have a mixing distribution function G, the conditionalexpectation in (5) is given by

E½fðsjli;FsÞjX iðsÞ ¼ 0� ¼

R10 fðsjl;FsÞ exp½FðsjlÞ�dGðlÞR1

0 exp½Fðsjl;FsÞ�dGðlÞ,

where

Fðsjl;FsÞ ¼

Z s

0

fðujl;FuÞdu.

Hence (5) becomes

X ðtÞ

Z t

0

ðN0 X ðsÞÞ

R10 fðsjl;FsÞ exp½Fðsjl;FsÞ�dGðlÞR1

0 exp½Fðsjl;FsÞ�dGðlÞds

( ). (6)

By comparing (6) with (3) and (2), we have

LðtjN0;FtÞ ¼ ðN0 X ðtÞÞ

R10 fðtjl;FtÞ exp½Fðtjl;FtÞ�dGðlÞR1

0 exp½Fðtjl;FtÞ�dGðlÞ.

This completes the proof. &

The following results are immediate.

Corollary 3.2. Under the conditions of Theorem 3.1, if the li degenerate to a constant l0; then

LðtjN0;FtÞ ¼ ðN0 X ðtÞÞfðtjl0;FtÞ.

Furthermore, if

fðtjl0;FtÞ ¼ fðtjl0;X ðtÞÞ,

then LðtjN0;FtÞ is the intensity function of a 0-memory self-exciting process and

LðtjN0;FtÞ ¼ ðN0 X ðtÞÞfðtjl0;X ðtÞÞ.

Also, if

fðtjl0;FtÞ ¼ fðtjl0;X ðtÞ;SX ðtÞÞ,

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then LðtjN0;FtÞ is the intensity function of a 1-memory self-exciting process and

LðtjN0;FtÞ ¼ ðN0 X ðtÞÞfðtjl0;X ðtÞ;SX ðtÞÞ.

Corollary 3.3. Under the conditions of Theorem 3.1, if fðtjl;FtÞ l; then

LðtjN0Þ ¼ ðN0 X ðtÞÞ

R10 lelt dGðlÞR10 elt dGðlÞ

.

The following result is obtained by considering that the unknown initial number of faults in thesoftware has a Poisson prior.

Corollary 3.4. Under the conditions of Theorem 3.1, if N0 � PoissonðyÞ; then

LðtjFtÞ ¼d

dty 1 exp

Z t

0

xðujFuÞdu

� �� �,

where

xðujFuÞ ¼

R10 fðujl;FuÞ exp½Fðujl;FuÞ�dGðlÞR1

0 exp½Fðujl;FuÞ�dGðlÞ.

Furthermore, if fðtjl;FtÞ l; then

LðtÞ ¼d

dty 1 exp

Z t

0

R10 lelu dGðlÞR10 elu dGðlÞ

du

!" #( ).

Proof. Let us begin with

ðX ðtÞ X ðsÞjN0; fligiX0;FsÞ � Binom N0 X ðsÞ; 1W ðtÞ

W ðsÞ

� ,

where

W ðtÞ ¼ exp

Z t

0

xðujFuÞdu

� .

In particular,

ðX ðtÞjN0; fligiX0Þ � BinomðN0; 1 W ðtÞÞ.

If N0 has a Poisson prior with parameter y; then, given fligiX0;

X ðtÞ � Poissonðyð1 W ðtÞÞ.

In fact, given fligiX0; it can be further observed that fX ðtÞ; tX0g is a non-homogeneous Poissonprocess with the mean value function yð1 W ðtÞÞ:Due to the property of independent increments,the intensity function is the derivative of the mean value function. This completes the proof. &

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4. Examples

Example 4.1 (Littlewood, 1981). Assume that the mixing distribution G is gamma distributedwith the scale parameter b and the shape parameter a: If fðtjl;FtÞ l; from Corollary 3.3,R1

0 lelt dGðlÞR10 elt dGðlÞ

¼ a=ðbþ tÞ

and the intensity function is

LðtjN0Þ ¼ ðN0 X ðtÞÞa

bþ t.

This is an example that corresponds to case (a) in Section 2.

Example 4.2 (Goel and Okumoto, 1979; Miller, 1986). Assume that N0 � PoissonðyÞ andfðtjl;FtÞ l: If the li degenerate to a constant l0 then, referring to Corollary 3.4, the intensityfunction is

LðtÞ ¼ yl0el0t.

Furthermore, if the mixing distribution G is gamma distributed with the scale parameter b and theshape parameter a then the intensity function is

LðtÞ ¼yaba

ðbþ tÞaþ1.

These are examples that corresponds to case (c) in Section 2.

Example 4.3 (Schick and Wolverton, 1973). Assume that the li degenerate to a constant l0 and

fðtjl0;FtÞ ¼ l0ðt SX ðtÞÞ,

then, referring to Corollary 3.2, the intensity function is

LðtjN0;FtÞ ¼ ðN0 X ðtÞÞl0ðt SX ðtÞÞ.

The concatenated failure rate function is

HðtjN0;FtÞ ¼ ðN0 X ðtÞÞl0t; tX0.

This is an example that corresponds to case (d) in Section 2.

This paper has introduced an intensity function on both properties of mixture and self-exciting,and has offered a method for classifying existing software reliability models. The examples givenin this paper have demonstrated both results.

Acknowledgements

The author is grateful to the referee for helpful comments. This work is supported by theNational Science Council of Taiwan NSC92-2118-M-029-006.

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