A Moment Based Approximation for Expected Number of Renewals for Non-Negligible Repair

This paper focuses on the renewal function which is simply the mathematical expectation of number of renewals in a stochastic process. Renewal functions are important, and they have various applications in many fields. However, obtaining an analytical expression for the renewal function may be very complicated and even impossible. Therefore, researchers focused on developing approximation methods for them. The purpose of this paper is to explore the renewal functions for non-negligible repair for the most common reliability underlying distributions using the first four raw moments of the failure and repair distributions. This article gives the approximate number of cycles, number of failures and the resulting availability for particular distributions assuming Mean Time to Repair is not negligible and that Time to Restore, or repair has a probability density function denoted as . When Mean Time to Repair is not negligible and Time to Restore has a probability density function denoted as , the expected number of failures, cycles and the resulting availability were obtained by taking the Laplace transforms of corresponding renewal functions. An approximation method for obtaining the expected number of cycles, number of failures and availability using raw moments of failure and repair distributions are provided. Results show that the method produces very accurate results for especially large values of time t .


Introduction
Renewal functions give the expected number of failures of a system or a component during a time interval [1]. They have wide variety of applications in decision making such as supply chain planning [2,3], inventory theory [4], continuous sampling plans [5,6], insurance application and sequential analysis [7,8].
Since they have wide variety of applications, obtaining closed form analytical expression for renewal functions have certain advantages including carrying out parametric studies of the functions [9]. However, for most distributions, obtaining the renewal function analytically is complicated and even impossible [2,11]. Therefore, development of computational techniques and approximations for renewal functions has attracted researchers [10,11].
Our previous paper [11] explores the renewal function for minimal repair using time discretizing method. In this paper, we explore the renewal functions for non-negligible repair for commonly used reliability distributions using the first four raw moments of the failure and repair distributions.
Let the variates X 1 , X 2 , X 3 ,… represent time to failure (TTF i ) be iid (independent and identically distributed) with the underlying failure density f(x) having means MTBF( mean time between failures) = and variance ; further, let Y 1 , Y 2 , Y 3 , … represent the i th Time-to-Repair (TTRi),+ i = 1, 2, 3, 4,… with the pdf r(y) having means MTTR = and variance . Then, T i = X i 774 A Moment Based Approximation for Expected Number of Renewals for Non-Negligible Repair + Y i represents the time between cycles (TBCs) which are also iid whose density is given by the convolution , and whose Laplace transform is given by g(s) f (s) r (s) . Clearly the mean and variance of the cycle-times T i 's are and . As described by U. N. Bhat [12] there will be two types of renewals: (1) A transition from a Y-state (i.e., when system is under repair) to an X-state (at which the system is operating reliably), (2) A transition from an X-state (or operating-reliably-state) to a Y-state (where system will go under repair).
Let M 1 (t) represent the expected number of renewals of type 1, and M 2 (t) represent the expected number of failures (or renewals of type 2). Then, as stated by Bhat [12] and E. A. Elsayed [1], the Laplace transforms of the two renewal functions, respectively, are given by General expressions can be obtained for the renewal functions M 1 and M 2 by inverting eq. (1a) and eq. (1b). Eq. (1a) shows that where is the corresponding cdf of cycle time T. Eq. (1b) shows that Therefore, in general the expected number of type 1 is given by while the expected number of failures is

Exponential TTF and Exponential TTR
which gives the expected number of transitions from a repair-state to an operational-state. Similarly, which upon inversion yields representing the expected number of failures during an interval of length t. Note that the limit of both renewal functions M 1 (t) and M 2 (t) as r   is equal to t as expected. Further, a comparison of M 2 (t) with M 1 (t) reveals that M 2 (t) > M 1 (t) for all t > 0, which is intuitively meaningful because the expected number of failures must exceed the expected number of cycles for all t > 0. As an example, if  = 0.0005/hour and repair-rate , then , while .

System Availability
Availability is the probability that a system or component is performing its required function at a given point in time or over a stated period of time when operated and maintained in a prescribed manner [13].
Because we are assuming that a system can be either in an operational-state, or under repair, then the reliability function must be replaced by the instantaneous (or point) availability function at time t, denoted A(t), which represents the probability that a repairable unit or system is functioning reliably at time t. Thus, if there is no repair, the availability function is simply A(t) = R(t), the reliability function. However, if a component (or system) is repairable, then there are two mutually exclusive possibilities: (1) The system is reliable at t, in which case A 1 (t) = R(t), (2) The system fails at time x, 0 < x < t, gets renewed (or restored to almost as-good-as-new) in the interval (x, x+x) with probability element (x) dx, and then is reliable from time x to time t [14]. This second availability is given by Because the above two cases are mutually exclusive, then Taking Laplace transform of eq. (3) [and observing that where r(t) is the density of repair-time.
The inverse Laplace transform of Eq. (4) results in the point availability A(t). If the underlying distributions are not exponential, problems arise in inverting the Laplace transform [15]. Therefore, numerical solutions and approximations become the only alternatives for obtaining A(t) [16]. There are numerous approximation techniques in the literature such as, Sarkar & Chaudhuri [15] uses Fourier transform technique to determine the availability of a maintained system under continuous monitoring and with perfect repair policy. They also obtain closed-form expressions when the system has gamma life distribution and exponential repair time. Ananda and Gamage [17] consider statistical inference for the steady state availability of a system when repair distribution is two-parameter lognormal and failure distributions are Weibull, gamma and lognormal. There are also other articles in the literature that work on confidence limits for steady state availability of a system such as [18,19] etc.
In order to approximate availability and renewal functions we used moment-based approximation, which only requires knowing the first four raw moments of failure and repair distributions. "There are a number of cases where the moments of a distribution are easily obtained, but theoretical distributions are not available in closed form [20]." And also, efficient estimators for the various moments of the underlying distribution could be calculated from the observed sample data [21]. Kambo et. al. [21], uses first three moments of failure distribution in order to approximate the renewal function for negligible repair and they conclude that the method produces exact results of the renewal function for certain important distributions like mixture of two exponential and Coxian-2.
In this paper, we propose an approximation for the evaluation of expected number of cycles, number of failures and availability based on first four raw moments of failure and repair distributions where convolution of f(t) and r(t) is intractable. We conclude that the method produces very accurate results for especially large values of time t.

Intractable Convolutions of F(T) with R(T)
Obtaining the convolutions of f(t) with r(t) for the general classes of failure and repair distributions is not always tractable, such is the case of both TTF and TTR being Weibull, then g(t) cannot be obtained. Therefore, below we will develop an approximate method based on raw moments that will yield approximations for the three functions M 1 (t), M 2 (t), and the resulting A(t) for any failure and repair distributions.
It has been well documented, since Pierre Laplace, that the Laplace transform of any density function is given by Note that the inclusion of higher exponents s 8 , s 9 , etc. in the brackets inside eq. (6a) will require the 5 th , 6 th , etc. raw moments in the above D(s) which we will not consider. Thus, the 4 th -order approximation for D(s) is given by The comparison of s 3 coefficients will give rise to the second equation in the seven unknowns c 3 , c 4 , …, c 9 . Letting i j5 ij R r    , j = 3, 4… 9 be the sum of products of any five distinct roots out of seven, excluding the j th root, and hence it will have exactly 6 C 5 = 6 terms. Using these notations, the second equation by comparing the coefficients of s 3 will be as follows:

Summary and Conclusions
The moment-based approximation for expected number of cycles M 1 (t), number of failures M 2 (t) and availability A(t) were obtained for the three parameter Weibull, normal, lognormal, exponential, logistic, loglogistic and gamma distribution as failure and repair distributions. As we discussed in Section 2, the exact results of M 1 (t), M 2 (t) and A(t) when TTF and TTR are exponentially distributed have been known. So, we used those results at λ= 0.001 and r = 0.05 to compare the approximation method that we have developed. Based on these results relative errors were calculated and concluded that the method produces very accurate results for especially large values of t versus small values of t. The figures and tables below explain the results better. Table 1 shows the error of approximation relative to the exact value. As it is seen from the table relative error is almost 95% when time is 20 units, but as the time increases relative error decreases dramatically. Moreover, when time is 5000 units and relative error is zero on the six decimals.  Table 2 shows the percent error of approximation of M 2 (t) relative to the exact value. As it is seen from the table relative error is almost 0.698% when time is 20 units, but as the time increases relative error decreases dramatically. When time is 5000 units and relative error is almost zero. Further, relative error is much higher for M 1 (t) then M 2 (t) for smaller values of t.  Table 3 shows the percent error of approximation of A(t) relative to the exact A(t). Same conclusion can be made for availability also.  Figure 1 is the graphical representation of Table 1, 2 and 3. It also shows that the approximation method works well for large values of t.
In this article, we explored the renewal functions for non-negligible repair for the most common reliability underlying distributions using the first four raw moments of the failure and repair distributions when the convolution of f(t) and r(t) is intractable. We conclude that the method produces very accurate results for especially large values of time t.

Abbreviations
MTTR, Mean Time to Repair; TTR, Time to Restore, or repair; pdf, probability density function; MTBF, Mean Time Between Failure.