An asymptotic solution of the integral equation for the second moment function in geometric processes.

In this study, we derive an asymptotic solution of the integral equation satisfied by the second moment function M 2 t , a . We first find the Laplace transform M 2 L s , a and then obtain M 2 t , a asymptotically by inversion. Further, we have derived the asymptotic expressions of M 2 t , a for some special lifetime distributions such as exponential, gamma, Weibull, lognormal and truncated normal. Finally, the asymptotic solution is compared with the numerical solution to evaluate its performance.

Introduction

Geometric process (GP) is first introduced by [1] to facilitate the modelling of the deteriorating systems in which the consecutive working times of the system are stochastically decreasing and the successive repair times after failure are stochastically increasing. This process is defined in the following way.

Let be a counting process and be the interarrival time between th and th event of this process for . The counting process is said to be a GP with the ratio parameter if there exists a real number such that are independent and identically distributed random variables with a cumulative distribution function (cdf) .

The GP is also called quasi-renewal process with ratio parameter by [2].

Let be a GP with the ratio parameter and be the distribution function of . Then, it is obvious that for . If , then is stochastically increasing. If , then is stochastically decreasing. When , the GP reduces to a renewal process (RP).

Some important characteristics of a GP are the mean value function and the second moment function . These are defined as follows.

The mean value function of a GP, which is also called the geometric function, is given by where denotes Stieltjes convolution.

It is well known that the geometric function satisfies the following integral equation, see [3].

If , the geometric function is finite for all . Furthermore, if is continuous, then the integral equation (1.1) has a unique solution although cannot be obtained in an analytical form. If , is infinite for all ([4] and [3]).

The second moment function of a GP is given by

[5] shows that the second moment function of a GP satisfies the following integral equation.

For , it is easily seen that a GP with ratio has finite moments of all orders. Hence for , the function is finite for all . If and for all , is infinite for all ([4] and [3]).

GP is widely used as a stochastic monotone model in many practical applications since its introduction. A considerable amount of the research on the GP has been published. For example, the GP has been applied in software reliability [6], maintenance [7], warranty cost analysis [8] and [9], modelling of an epidemic disease [10]. Numerical examples also show that the GP is of importance. In [3], model performance of the GP is validated by 14 real-world data sets according to criteria of the mean square error (MSE) and maximum percentage error. After validating the model performance of the GP, some characteristics such as mean value function and the second moment function are needed in most applications. Following applications of the GP can be given in order to illustrate how these functions are actually used.

[8] carries out the warranty cost analysis by using the GP model to compute the expected warranty cost and the variability of the warranty cost. In this model, denotes the number of the product’s failure in where is the length of the warranty period. They conduct the analysis depending on the cost where is a constant. Then, the expected warranty cost and the variability of the warranty cost are and , respectively. See [8] for the details.

[6] proposes a GP model to investigate software reliability and testing costs. In this model, denotes the number of software faults in and the cost of fixing software fault is given by where is a constant and is an incremental cost variable with mean . Then, the expected total debugging cost in is In addition to above cost model, [6] also considers the cost of testing per unit time and assumes that it is a random variable with mean . Then, the total expected testing and debugging cost in is given as See [6] for the details.

[9] constructs the cost analysis of the repair-limit risk-free policy by using the GP. In this model, and denote the number of repairs and the number of replacement under the warranty in , respectively, where is the length of the warranty period. They carry out the analysis depending on the cost where and are the repair cost per failure and the replacement cost per unit, respectively. They investigate the measures and which include the functions and . See [9] for the details.

The analytical forms of the geometric function and the second moment function do not exist. In the literature, many studies on their computations have been done. Some of the known studies on the geometric function are as follows. When the distribution of the first interarrival time has an exponential distribution, upper and lower bounds for the geometric function are obtained by [4], and [11] obtains power series expansion for by using the integral equation (1.1). [12] proposes a numerical solution to the integral equation given in (1.1) and applies this method to four common lifetime distributions, namely, exponential, gamma, Weibull and lognormal. These proposed methods for the geometric function have the considerable amount of computation needed to obtain the function with a desired accuracy. In practice, an asymptotic formula for should be useful for applications requiring the geometric function . [3] gives an asymptotic expression for the geometric function by using its Laplace transform in the following theorem.

Let be a GP with the ratio parameter . Assume that the first interarrival time has a distribution function and probability density function (pdf) with and .

[3]

Let . If then

To the best of our knowledge, some studies on the second moment function are as follows. [13] proposes a numerical approximation and Monte Carlo estimation for the second moment function depending on the convolutions of the distribution functions. [5] adapts Tang and Lam’s method to compute the integral equation (1.3) and applies this method to exponential, gamma, Weibull and lognormal distributions. [5] also derives a power series expansion for the second moment function by the help of the integral equation (1.3) under the assumption of the first interarrival time having an exponential distribution. They give some computational procedures for the variance function after the calculation of .

The aim of this study is to derive an asymptotic solution of the integral equation (1.3). For this purpose, we first find the Laplace transform and then obtain asymptotically by inversion. By the obtained asymptotic solution, the second moment function is calculated rapidly for an arbitrary distribution . This approximation also works well for even small values. For some special lifetime distributions such as exponential, gamma, Weibull, lognormal and truncated normal, we have given the asymptotic expressions of . Then, the asymptotic solution is compared with the numerical solution proposed by [5] to evaluate its performance.

An asymptotic solution to the integral equation (1.3)

In this section, we give an asymptotic solution of the integral equation (1.3) for by using its Laplace transform.

Let be a GP with the ratio parameter . Assume that the first interarrival time has a distribution function and pdf with and .

Let . If , then the asymptotic solution of (1.3) is obtained as

The proof is completed in the following way. The Laplace transform of the second moment function is obtained by taking the Laplace transform of both sides of the integral equation (1.3). Then, we expand into a Taylor series with respect to about At the end, an asymptotic expression for will be found by taking the inverse Laplace transform of .

Since , By substituting , we obtain can be expanded into a Taylor series with respect to about as The first three terms in (2.3) can be expressed based on the function and its derivatives as follows.

and are the mean value and second moment functions of the renewal process with the interarrival time distribution function . It is easily seen that Then, the Laplace transform of is found as and so this function has been written as a function of .

It is easily obtained that and Further, [3] gives and By substituting and differentiating both sides of (2.2) with respect to , we see that Since by [3], taking in (2.5) we find Then, and so this function has been expressed as a function of and .

By differentiating the both sides of Eq. (2.5) with respect to , it is found that Hence, we have The partial derivation of (2.6) with respect to can be given as Then, we obtain the third term in (2.3) as a function of and by It can be written that as Hence, we obtain and By using the above expressions in (2.4), (2.6), (2.7), it follows that and Using Eqs. (2.8), (2.9), (2.10) in Eq. (2.3), we see that Thus, inverting (2.11) yields By using the Tauberian theorem, we have that is, . This completes the proof of Theorem 2.1. ■

Comments on some extensions of the GP

This section presents two extensions of the GP and discusses whether the asymptotic expressions for their mean value and second moment functions can be obtained or not by the method used in this paper. We begin with the definitions and some properties of them are as follows.

Threshold Geometric Process (TGP):

[10] extends the GP to TGP in modelling of the Severe Acute Respiratory Syndrome (SARS) data set which has multiple trends as follows.

A stochastic process is said to be a TGP if there exists real numbers and integers such that for each forms a RP with a common cdf.

The monotonicity property of the TGP varies according to the value of ratio parameter . If , then is stochastically increasing. If , then is stochastically decreasing. When , forms a RP.

Let us consider a TGP proposed in [10] to model the SARS data, where is the number of infected people with SARS on the th day. Then, a corresponding counting process can be defined as where denotes the greatest integer which is not more than . denotes the total number of infected cases in days. In principle, there exists an such that for the TGP. Then, can be written as where and . Depending on the definition of , this counting process has finite moments of all orders. As an example, we give the mean value function of this process as follows. Let for . It is clear from the definition of the TGP that for . Then is obtained as

Doubly Geometric Process (DGP):

Let be the sequence of independent and the positive-valued random variables. The counting process is said to be a DGP with the parameter if there exists a real number such that are independent and identically distributed with a common cdf where is a function of with for  [14].

[14] gives some properties of the DGP by choosing where is a real number. The reason that [14] chooses is: [14] fits the DGP with different , which are and , on ten real data sets and finds that the DGP with performs better than the other three ’s. Let be a DGP with the parameters and , and be the distribution function of . Then for .

[14] gives the monotonicity property of the DGP as follows: If and or if and , then is stochastically increasing. If and or if and then is stochastically decreasing. If varies between negative and positive values, then is not stochastically monotonous over ’s, where represents all possible values on .

Let be a DGP with the ratio parameter and . Suppose that the sequence of interarrival times follows this DGP. Denote with . Now, consider a RP with the interarrival times having a common cdf which is the same as the cdf of . Denote with . If is stochastically decreasing, then . Thus, it can be shown that for each fixed by using the fact that . If is stochastically increasing, then and so, for each fixed . It is well known that the RP has finite moments of all orders. Thus, the DGP possesses moments of when is stochastically increasing. Then, we make following comments on the moment functions of the DGP .

(i) If and then the DGP has finite moments of all orders.

(ii) If and then the DGP has finite moments of all orders.

In addition, by Proposition 5(ii) in [14],

(iii) If and then the mean value function of the DGP is infinite and so the second moment function is also infinite.

However, for the other cases in terms of and the distribution , it has not been determined yet whether the mean value and second moment functions of the DGP are finite.

Although the alternative processes provide more flexible models for wider application than the GP, they may create difficulties in mathematical derivations. For example, the integral equations (1.1), (1.3) can be obtained from the fact that for the GP. However, such equality cannot be written for DGP. Therefore, it does not seem possible to obtain an integral equation for the DGP. Since the integral equations do not exist, Tang and Lam’s method given in [12] or power series expansion given in [5] cannot be adapted to compute its mean value and second moment functions. In addition, the asymptotic expressions for the mean value and second moment functions of the DGP cannot be obtained by using the similar argument of this study since we use the Laplace transform of the integral equations (1.1), (1.3). To the best of our knowledge, the only computational procedure for the mean value and second moment functions of the DGP is to obtain Monte Carlo estimation of these functions depending on the convolutions of the distribution functions proposed by [13].

Numerical examples

In this section, the performance of the proposed asymptotic solution is evaluated by comparing with the power series expansion and the numerical solution. To do this, we consider five numerical examples each with exponential distribution, gamma distribution, Weibull distribution, lognormal distribution and truncated normal distribution. In the first example, the approximate values are compared with the values computed by the power series expansion given in [5]. In other examples, the approximate values are compared with the values computed by the numerical method based on the trapezoidal integration rule with the step length given in [5]. In all five examples, the ratio parameter is chosen as since it satisfies the condition for many real data sets fitted by the GP [3]. The solutions are calculated up to time . After computation, the approximate and numerical solutions are plotted together in the same figure for comparison. The figures are presented for only . In these figures, we use solid and dash-dotted line for the asymptotic solution and numerical solution (power series expansion for only exponential distribution), respectively. For easy comparison, we also plot the 95% and 105% of the values of the numerical method as the lower and upper bounds of the second moment function by using dotted lines.

Consider a GP with ratio parameter and assume that the first interarrival time has an exponential distribution with the pdf . Then and . From (2.1), an asymptotic solution is given by

The approximate values of calculated by (4.1) and the power series expansion with an absolute error not exceeding are plotted in Fig. 1. The solutions are calculated up to time .

Fig. 1

Exponential distribution and .

Fig. 1 shows that the results obtained by the asymptotic solution and power series expansion are very close to each other. Asymptotic solution lies inside the lower and upper bounds. As expected, solutions are getting closer when t increases. In general, the absolute relative errors are not more than 1%. Note that for , similar comments can be made for the asymptotic solution. When , asymptotic solution lies inside the lower and upper bounds for the interval (1.5, 12] and it is very close to solution of the power series expansion. For the intervals (0, 1.5] and (12, 20], the deviation of the asymptotic solution is a bit larger. However, in general, the absolute relative errors are not more than 5%.

Consider a GP with ratio parameter and assume that the first interarrival time has a gamma distribution with the pdf . Then and . From (2.1), an asymptotic solution is given by

The solutions are calculated up to time .

It is seen from Fig. 2 that the asymptotic solution lies inside the lower and upper bounds in the interval [1.5, 20], the deviation of this solution in the interval (0, 1.5) is a bit larger. However, in general, the absolute relative errors are not more than 3%. Note that for , the results are similar for this approximation. When , although asymptotic solution lies inside the lower and upper bounds for the interval (0.5,15], the deviation of the asymptotic solution in the intervals (0,0.5] and (15,20] is a bit larger and generally, the absolute relative errors are not more than 5%.

Fig. 2

Gamma distribution and .

Consider a GP with ratio parameter and assume that the first interarrival time has a Weibull distribution with the pdf . Then and . From (2.1), an asymptotic solution is given by

The solutions are calculated up to time since .

We can see from Fig. 3 that the results obtained by the asymptotic and numerical solution are very close to each other. The approximate solution lies inside the bounds. The absolute relative errors are generally smaller than 3%. Note that similar results can be yielded for . When , the values lie inside the bounds for the interval . Out of this interval, i.e. in the interval (], the absolute relative errors are more than 5% but generally less than 10%.

Fig. 3

Weibull distribution and .

Consider a GP with ratio parameter and assume that the first interarrival time has a lognormal distribution with the pdf . Then and . From (2.1), an asymptotic solution is given by

The solutions are calculated up to time since .

From Fig. 4, the deviation of asymptotic solution in the interval (0, 7) is a bit larger. In general, the absolute relative errors are not more than 10%. Out of this interval, i.e. in the interval [7, 18], asymptotic solution is very close to the solution of the numerical method. The absolute relative errors are not more than 3%. Note that for and 0.95, the deviations of the asymptotic solution from the numerical solution are larger. To obtain an accurate approximation, it is not sufficient that is large enough. At the same time, should be close to 1.

Fig. 4

Lognormal distribution and .

Consider a GP with ratio parameter and assume that the first interarrival time has a truncated normal distribution with the pdf and . Then, the first five moments of this truncated normal distribution are , , , and . From (2.1), an asymptotic solution is given by

The solutions are calculated up to time since .

It is seen from Fig. 5 that the asymptotic solution lies inside the lower and upper bounds and it is very close to solution of the numerical method. In general, the absolute relative errors are not more than 1%. As expected, the solutions are getting closer when increases. Note that for , the results are similar. When , the deviation of the asymptotic solution in the interval (0, 2] is a bit larger. Out of this interval, the asymptotic solution is very close to the solution of the numerical method. In general, the absolute relative errors are not more than 5%.

Fig. 5

Truncated normal distribution and .

Conclusions

In this study, the asymptotic solution of the integral equation given for the second moment function of a GP is obtained by using Laplace transform of the integral equation. Further, we have derived the asymptotic expressions of for some special lifetime distributions such as exponential, gamma, Weibull, lognormal and truncated normal. Then, we consider five numerical examples to evaluate the performance of the solution given. According to the numerical examples, the asymptotic solution is generally accurate on the interval at least for the distributions considered here. It is getting closer to the numerical solution when increases and approaches 1. Also, it can be concluded that the ratio parameter is more effective than . That is, the asymptotic solution gets more closer to the numerical solution as approaches 1 for fixed .