Computing the Distribution Function of the Number of Renewals

The method of Laplace transforms is used to find the distribution function, mean, and variance of the number of renewals of a renewal process whose inter-arrival time distribution has a rational Laplace transform. Where the Laplace transform is not rational, we use the Padé approximation method. We apply our method to certain examples and the results are compared to those reported by other researchers.


Introduction
Renewal and reliability theories are powerful modeling tools in many applications, considering, for example, a series of renewals during a time interval   0,t with the inter-renewal times having certain distributions.The interest, usually, is to find the distribution or the moments of the number of renewals during the interval   0,t , , X X   .Although the theoretical aspects of renewal theory have been discussed in various books such as Cox [1] and Feller [2,3], not much seem to have been done in applying the theoretical results to practice for lack of availability of easily computable results.Using the cubic splining algorithm to compute the recursively-defined convolution integrals that appear in renewal theory, Baxter et al. [4] is able to construct some tables for the mean and variance of the number of renewals for different inter-renewal time distributions with varying parameters.A straightforward method to compute the convolution integral is to use the Laplace transform (LT) method.Using the root-finding method, Chaudhry [5] develops a unified method to compute the mean and variance of the number of renewals.Though the means and variances are useful for many applications, this paper goes a step further and deals with computing the distribution of the number of renewals from which one can get more information than from the mean and variance.
There are several methods that can be used for the numerical inversion of generating functions (GFs); see the excellent reviews on the numerical inversion (of GFs as well as of LTs) by Abate et al. [6][7][8][9].Some researchers have been critical of using the method of roots, see, e.g.Cox and Smith [10] and Kleinrock [11].In view of this, this paper serves another useful purpose and shows that with the availability of high precision in current software packages, the roots can be found successfully and used to invert the transforms.Further, using the roots method, the results can be first given in an analytically explicit form and then used to find the final results.

Problem Description and Method of Laplace Transforms
Let 1 2 be a governing sequence of independently identically distributed (i.i.d.) inter-renewal times for the renewal process

and probability density function (pdf)
Define the probability mass function (pmf) , n represents the time when the n-th renewal occurs.Define


F t with itself, and (see Chaudhry and Templeton [12]) Taking the LST on both sides of Equation ( 1), we get and hence which is a rational function.Without loss of generality, we assume that the equation have the same constant terms, . And Equation ( 4) can be expressed in partial fractions as where the constant coefficient is given by The final inversion can be written as The case when

2.2.2.
We use the Padé approximation method.Assume  0 D s  has repeated roots can be dealt with similarly.
is the n-th moment of the inter-renewal time.We can find a rational approximation function N s and D s are polynomials of degree where K and L , respectively with undetermined coefficients and n , such that the first . We denote the above Padé approximation as   K L a (Baker et al. [13]).In practice, 0 is set to one and K and L are chosen by trial and error.Equating the moments and formulating the simultaneous equations, the coefficients b n and a n are uniquely determined (Baker et al. [13], Harris [14]).
Use of continued fractions is another way to obtain an approximate rational function   . The method is, in fact, a special case of the Padé method (Baker et al. [13]).
There are times when using Padé method directly is not possible or does not give the desired results.For example, the Pareto distribution has an infinite second moment, and directly equating the moments cannot be done.For the lognormal distribution, the waveform of the Padé approximated distribution function shows quite large errors.The solution is to have a two-step approximation.The first step is to use line segments to approximate the distribution function.In the second step, the Padé method is used to generate a rational LT.By adjusting the parameters in these two steps, f s   can be obtained with the desired properties.We will illustrate this technique in the examples.

   N t in
The expected number of renewals of 0,1 is given by and its variance by , and are checked against the results obtained by (Chaudhry [5]) using In addition, we also match the resulting mean and variance with those of Baxter et al. [4].

Erlang Distribution
We first consider the simple case where  f t  follows the Erlang distribution with the shape parameter equal to 2, and we can get exact analytical results.
The pdf and CDF of are given, respectively, by 2 16 It may be remarked that the analytic expression for the mean has been incorrectly reported by Parzen [15, p. 177].

Mixed Generalized Erlang Distribution
The pdf and LST of a Mixed Generalized (MGE) distribution are given, respectively, by

5, c c c
We consider the case of MGE with The t t , mean and variance are obtained by the LT method discussed in Section 2.2.Table 1 shows the results for several values of .They are checked against the known results obtained without using roots by Chaudhry [5].

 
The graph in Figure 1 shows that the E N t       calculated from the obtained distribution functions asymptotically converges to the line

 
. As expected, one can see from this graph that the results given in Equation ( 6) are good only for large t.

Matrix Exponential Distribution
Consider the matrix exponential and non-phase-type dis-tribution (see Bladt [17]) given below.The pdf and its LT are, respectively,


Using partial fractions for transform inversion, we obtain for some selected values.The results are listed in Table 2.

Gamma Distribution
The pdf and LST of the Gamma distribution are given, respectively, by where  and  are the shape and scale parameters, respectively.In this example, we set 1

   
and α = 0.55.The Padé [4/5] The   n P t are obtained and some selected pmf's, means, and variances are tabulated together with those from Bax- ter et al. [4] in Table 3.

Weibull Distribution
The pdf of the Weibull distribution do not have the closed-form LT.However, the moments of the Weibull distribution can be obtained from its CDF to use in the Padé method.For   and 1   , we equate moments up to the 8-th moment in the Padé [2/6]  Some selected results are listed in Table 4 together with Baxter's (see Baxter et al. [4]).It is noted that when t becomes large, our resulting means and variances match those of Chaudhry [5].

Truncated Normal Distribution
The pdf of the truncated normal distribution is given by  n P we are able to obtain t .Some results and compari- sons are shown in Table 5.

Inverse Gaussian Distribution
The pdf for the inverse Guassian distribution is Some results are shown in Table 6 along with the results of Baxter et al. [4].

Lognormal Distribution
The pdf of the lognormal distribution have no closed-form LT.
Using the Padé approximation method directly on the lognormal distribution does not lead to the desired results.
Our solution is a two-step approximation.In the first step, we sample uniformly N points of

F t N
, and then connect the adjacent points to form line segments as the first approximation of the lognormal distribution.The  

Conclusions
Using Laplace transforms, it is shown that computing the distribution of the number of renewals is straightforward when the LT of the inter-renewal time distribution is rational.For inter-renewal time distributions having nonrational LTs, the Padé method provides good approxima-tions.For the case where the inter-renewal time distributions do not have Laplace transforms, we provide the line segment approximation for the distribution function and apply the Padé method thereafter.
The proposed numerical method is not limited to compute the distribution of number of renewals.It can be applied to other similar processes such as alternating renewal process, the superposition of renewal processes, and cumulative processes, for which Laplace transforms are used to solve such problems.Further, the method discussed here can be applied to more complex models such as bulk-renewal processes both in discrete-and continuous-times once the analytic results are known. 0

Table 2 . t for the non-phase-type distribution discussed in Section 3.3.
n P

Table 6 . t for the Inverse Gaussian distribution with
t  

Table 7 .
N tis close to 1.In addition, the last line segment is modified to reach 1 at t N .The resulting first approximation function can be written as

Table 7
[4]the second step, the LST of the first approximation function is obtained and expanded as a Taylor series.Using the Padé [7/8] method, we have the second approximation of   lists selected t , means and variances a- long with the results of Baxter et al.[4].
N n P