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We study a class of software reliability models using quantile function. Various distributional properties of the class of distributions are studied. We also discuss the reliability characteristics of the class of distributions. Inference procedures on parameters of the model based on L-moments are studied. We apply the proposed model to a real data set.

Software reliability models play an important role in developing software systems and enhancing the performance of computer software. In general, software reliability model can be classified into two types, depending on the operating domain. The most popular category of models depends on time, which uses the concepts such as the mean time between failures and the failure intensity function. The second category of software models measures reliability as the ratio of successful runs to the total number of runs. The intensity (or failure rate) function plays a pivotal role for modelling software failure time data. Throughout the literature on failure time of software systems, certain parametric models have been used repeatedly such as the Rayleigh model by Schick and Wolverton [

The models described above are based on distribution function of failure time and reliability measures derived from it. An alternative and equivalent approach for modelling statistical data is to use quantile function. Even though both functions convey the same information about the distribution, the methodologies and concepts based on distribution function are more popular in practice. However, quantile functions have several properties that are not shared by distributions, which makes it more convenient for analysis. There are explicit general distribution forms for the quantile function of order statistics. Also random numbers from any distribution can be generated using appropriate quantile functions, a purpose for which lambda distributions were originally conceived. There are many simple quantile functions which are very good in empirical model building where distribution functions are not effective. In such situations, conventional methods of analysis using distribution functions are not appropriate.

For various properties and applications of quantile functions, we refer to Parzen [

The rest of the paper is organized as follows. In Section

Let

For every

If

Quantile function for the class of distributions in (

The derivative of (

Plots of quantile density function at different values of parameters.

The quantile function defined in (

The distributional characteristics such as location, dispersion, skewness, and kurtosis can be expressed through quantile terms. For the class of distributions in (

In the case of extreme positive skewness,

The

The third

The

The

The

One of the important concepts in reliability analysis is the hazard function which is defined as

The shape of the hazard function is determined by the derivative of

Since

Plots of

Mean residual function is a well-known measure, which has been widely used in various fields of reliability and survival analysis. In quantile setup, the mean residual quantile function is expressed as

The above identity can also be expressed as

Since the class of distributions (

Probability density function of inverse Gaussian is given by

Probability density functions of inverse Gaussian and its approximation.

The probability density function of Weibull distribution is given by

Probability density functions of Weibull and its approximation.

For estimating the parameter of the distributions, which are expressed in terms of quantile function, there are different methods available (see Gilchrist, [

Hosking (1990) has studied asymptotic properties of

Let

where

Now, we apply the model (

Hazard for the given data set.

A probability distribution can be specified either in terms of the distribution function or by quantile function. Although both convey the same information about the distribution, with different interpretations, the concepts and methodologies based on distribution functions are traditionally employed in most forms of statistical theory and practice. One reason for this is that quantile based studies were carried out mostly when the traditional approach fails to provide results of desired quality. Except in a few isolated areas, there have been no systematic parallel developments aimed at replacing distribution functions in modelling and analysis by quantile functions. However, the feeling that through an appropriate choice of the domain of observations a better understanding of a chance phenomenon can be achieved is fast gaining acceptance.

Motivated by this fact, in the present work, we have introduced a class of quantile function models, useful in software reliability analysis. The proposed class has several desirable properties and several existing well-known distributions that are members of the class of distributions as special cases or through approximations. Various reliability characteristics were discussed. The parameters of the model were estimated using

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors thank the referee and the editor for their helpful and constructive comments.