We briefly discuss the aspect of the discretized NHPP model by showing a discretized exponential software reliability growth model [5,6], which has the simplest mathematical structure. Now we define a discrete counting process 
representing the cumulative number of faults detected up to n-th testing-period. And we can say that the discrete counting process 
follows a discrete-time NHPP [
which is derived based on a continuous-time NHPP. In Equation (1), 
means the proba of event
. 
is a mean value function of the discrete-time NHPP. The mean value function, 
, represents the expected cumulative number of faults detected up to nth testing-period.
The discretized exponential software reliability growth model is a discrete analog of the original (continuous-time) exponential software reliability growth model. Let 
denote the mean value function following the discretized exponential software reliability growth model. The discretized exponential software reliability growth model is given as
from the basic assumptions of the original exponential software reliability growth model. In Equation (2), 
represents the constant time-interval, 
the expected total number of potential faults to be detected in an infinitely long duration or the expected initial fault content, and 
the fault detection rate per fault. Regarding the discretization method, we use the Hirota’s bilinearization methods [
as
As
, Equation (3) converges to an exact solution of the original continuous-time exponential software reliability growth model, which is derived by the differential equation.
The discretized exponential software reliability growth model in Equation (3) has two parameters, 
and
, which have to be estimated by using actual data. The parameter estimations of 
and
, 
and
, can be obtained by the following procedure using the method of least-squares. First of all, if we observed fault counting data
, where 
represents the cumulative number of faults detected up to nth testing-period, we derive the following regression equation from Equation (2):
where
Based on the regression analysis, we can estimate 
and
, which are the estimations of 
and 
in Equation (4). Then, the parameter estimations, 
and
, can be obtained as
in Equation (4) is independent of 
because 
is not used in calculating 
as showing Equation (5). Hence, we can obtain the same parameter estimates 
and
, respectively, when we choose any value of 
[5,6,15,16].
Software reliability assessment measures are useful in quantitative software reliability assessment. This paper discusses the expected number of remaining faults and the software reliability function, which are well-known software reliability assessment measures. The expected number of remaining faults, 
, represents the expected number of undetected faults in the software system at arbitrary testing-period. Then, we have
if we assume that 
follows a discrete-time NHPP with mean 
in Equation (3). The software reliability function, 
, is defined as the probability that a software failure does not occur in the time-interval 
given that the testing has been going Up to the nth testing-priod. Then, we have
and 
in Equation (4) based on the linear regression scheme by using fault counting data. We indicate 
and 
as 
and
, respectively.
at each observation point by
by assuming the residual errors 
follows the independent and identically probability distribution and putting mass 
at each ordered point
.
and let 
be the iteration count.
.
and 
from the bootstrap sample
.
and go back to Step 5 if. b < B Step 11: We have 
samples for
, 
and a software reliability assessment measures.
be parameter of interest.
, where 
is the bootstrap statistic. We can approximate the 
and 
quantile denoting va and va-1, respectivelyof the distribution of 
by 
and
. Then,
basic bootstrap confidence interval is given by
can be approximated by the distribution of 
and
. That is,
.
standard normal bootstrap confidence interval as
is
, which is the 
quantile of the standard normal distribution. For example,
.
. Then, the 
percentile bootstrap confidence interval is calculated by
represents the 
quantile of the empirical cumulative probability distribution function.
by deriving
, where 
is the variance of
. Letting 
and 
are the 
and 
quantile of
, we have
and
, which are 
and 
quantile of
, into 
and
, which are the 
and 
quantile of
. Then, the 
bootstrap-t confidence interval is derived as
is the standard deviation of 
and 
is estimated by the bootstrap-t statistics
.
.
, in which 
is the bootstrap distribution for the estimator. And 
is the acceleration constant derived as
where 
is a jackknife iteration value, which is estimated by using the data, removed ith data and
.
[
.
and 
. Then, we obtain bootstrap samples for 
and
. Figures 1 and 2 show histograms for the bootstrap samples 
and 
to see bootstrap distributions of 
and
, respectively. And we have bootstrap samples for the software reliability assessment measures, such as the expected number of remaining fault at the termination time of the testing, 
, and the software reliability, 
, respectively.
and 
are calculated by
and
, respectively. Further, 
, 
, 
, 
, 
, and
, respectively. And, 
, in which we use the means of the bootstrap samples of 
and 
as the point estimations, respectively. The means of 
and
, which are denoted by 
and
, are calculated by
. From 
are estimated in the basic, standard normal, and bootstrap-t confidence intervals because the number of re-
are notably different each other. These results are caused by assuming the symmetric distributions to derive these bootstrap confidence intervals. However, the probability distribution of an estimator follows an asymmetric distribution and the approximate accuracy is influenced by the bias and the skewness of the probability distribution for the estimator generally. As we show in 
follows an asymmetric distribution. On the other hand, the percentile bootstrap confidence interval give us an appropriate interval estimations on 
because the interval estimation based on the percentile method is estimated based on only the bootstrap distribution, not assumed a symmetric distribution. Of course, the interval estimations based on the BCa method can be thought that we have more appropriate interval estimation on 
because the BCa method is the improved estimation method for the percentile bootstrap confidence interval.