This paper consider an MMLE (Modified Maximum Likelihood Estimation) based scheme to estimate software reliability using exponential distribution. The MMLE is one of the generalized frameworks of software reliability models of Non Homogeneous Poisson Processes (NHPPs). The MMLE gives analytical estimators rather than an iterative approximation to estimate the parameters. In this paper we proposed SPC (Statistical Process Control) Charts mechanism to determine the software quality using inter failure times data. The Control charts can be used to measure whether the software process is statistically under control or not.
The software reliability is one of the most significant attributes for measuring software quality. The software reliability can be quantitatively defined as the probability of failure free operation of a software in a specified environment during specified duration. [1]. Thus, probabilistic models are applied to estimate software reliability with the field data. Various NHPP software reliability models are available to estimate the software reliability. The MMLE is one of such NHPP based software reliability model. (2). The software reliability models can be used quantitative management of quality (3). This is achieved by employing SPC techniques to the quality control activities that determines whether a process is stable or not. The objective of SPC is to establish and maintain statistical control over a random process. To achieve this objective, it is necessary to detect assignable causes of variation that contaminate the random process. The SPC had proven useful for detecting assignable causes(4).
This section presents the theory that underlies exponential distribution and maximum likelihood estimation for complete data. If ’t’ is a continuous random variable with
. Where, the mathematical relationship between the pdf and cdf is given by:
. Let ‘a’ denote the expected number of faults that would be detected given infinite testing time in case of finite failure NHPP models. Then, the mean value function of the finite failure NHPP models can be written as:
When the data is in the form of inter failure times also called Time between failures, we will try to estimate the parameters of an NHPP model based on exponential distribution [6]. Let N(t) be an NHPP defined as ,
is the mean value function of the process of an NHPP given by ) a>0, b>0,t>=0 (2.1.1)
The intensity function of the process is given by
The constants ‘a’, ‘b’ which appear in the mean value function and hence in NHPP, in intensity function (error detection rate) and various other expressions are called parameters of the model. In order to have an assessment of the software reliability ‘a’,’ b’ are to be known or they are to be estimated from a software failure data. Suppose we have ’n’ time instants at which the first, second, third…, n th failures of a software are experienced. In other words if is the total time to the k th failure, is an observation of random variable
Inverse of the above matrix is the asymptotic variance covariance matrix of the MLEs of ‘a’,’ b’. Generally the above partial derivatives evaluated at the MLEs of ‘a’, ‘b’ are used to get consistent estimator of the asymptotic variance covariance matrix.
However in order to overcome the numerical iterative way of solving the log likelihood equations and to get analytical estimators rather than iterative, some approximations in estimating the equations can be adopted from [2] [8] and the references there in. We use two such approximations here to get modified MLEs of ‘a’ and ‘b’. (2.2.12) It can be seen that the evaluation of , C are based on only a specified natural number ’n’ and can be computed free from any data. Given the data observations and sample size using these values along with the sample data in equation (2.1.12)(2.2..7) we get an approximate MLE of ‘b’. Equation (2.2.2) gives approximate MLE of ‘a’.
Based on the time between failures data give in Table-1, we compute the software failure process through mean value control chart. We use cumulative time between failures data for software reliability monitoring through SPC. The parameters obtained from Goel-Okumoto model applied on the given time domain data are as follows: a = 33.396342, b = 0.003962 ’ ’ and ’ ’ are Modified Maximum Likelihood Estimates (MMLEs) of parameters and the values can be computed using analytical method for the given time between failures data shown in Table 1. Using values of ‘a’ and ‘b’ we can compute They are used to find whether the software process is in control or not by placing the points in Mean value chart shown in figure-1. A point below the control limit indicates an alarming signal. A point above the control limit indicates better quality. If the points are falling within the control limits it indicates the software process is in stable [9]. The values of control limits are as shown in Table-2.
No.
No.
No.
No.
No.
This Mean value chart (Fig 1) exemplifies that, the first out -of -control and second our-ofcontrol situation is noticed at the 10 th failure and 25 th failure with the corresponding successive difference of m(t) falling below the LCL. It results in an earlier and hence preferable out -ofcontrol for the product. The assignable cause for this is to be investigated and promoted. The out of control signals in and the model suggested in Satya Prasad at el [2011] [ 13 ] are the same. We therefore conclude that adopting a modification to the likelihood method doesn’t alter the situation, but simplified the procedure of getting the estimates of the parameter
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