[Editor's Note: In the online version of this article, we have corrected "Table 1: Analysis Results Comparison" used in the printed edition of Volume 6, Issue 1. In the "Proposed Model" column, the values for Beta and Lambda were switched.] In repairable system maintainability and availability analysis using the BlockSim software, the user is given the option to specify a restoration factor (repair effectiveness) that describes the percentage to which a component will be restored after the performance of the maintenance action. This provides the ability to model maintenance using "used parts" or imperfect maintenance. The commonly used models for analyzing repairable systems data are perfect renewal processes (PRP), corresponding to perfect repairs, and nonhomogeneous Poisson processes (NHPP), corresponding to minimal repairs. However, most repair activities may realistically not result in such two extreme situations but in a complicated intermediate one (general repair or imperfect repair/maintenance). In this article, we explore a model that can analyze complex repairable systems with various degrees of repair. A general likelihood function formulation for single and multiple repairable systems is presented for estimation of the new model parameters. The practical use of the proposed statistical inference is demonstrated by two examples, and the results show that the proposed method is a promising and efficient approach with the potential to become useful in industry. The proposed model will be available in upcoming ReliaSoft products.
Introduction Repairable systems receive repair/maintenance actions that restore system components when they fail. These actions change the overall makeup of the system and affect the system behavior differently due to different maintenance approaches. Basically, there are two major categories: corrective maintenance or preventive maintenance. Each can be classified in terms of the degree to which the operating condition of an item is restored by maintenance in the following way [1][2]:
In this article, a model is proposed to analyze complex repairable systems and to estimate the restoration factor. A general likelihood function formulation for single and multiple systems with time truncated data and failure truncated data is presented for the estimation of the parameters. Assumptions
Proposed Model and
Maximum Likelihood Estimates
where for convenience we define t_{0} = 0. The sequence t_{1}, t_{2}, ... of failure times and the sequence x_{1}, x_{2}, ... of interarrival times thus contain exactly the same information about a particular realization of the process.
Previous research assumes that the nth repair can remove the damage incurred only during the time between the (n1)th and nth failures. In practice, not only does the nth repair depend on the (n1)th repair, but it also depends on all previous repairs. We assume that the repair action could remove all damage accumulated up to the nth failure; accordingly, the virtual age after the nth repair becomes:
where q is the degree of the nth repair (restoration factor) where 0 <= q <= 1, thus:
If the system has the virtual age V_{n1} = y immediately after the (n  1)th repair, the nth failure time X is distributed according to the following cumulative distribution function (cdf ):
In this model, q = 0 corresponds to a perfect repair (PRP, as good as new) while q = 1 indicates a minimal repair (NHPP, as bad as old). The case of 0 < q < 1 corresponds to an imperfect repair (better than old but worse than new) while q > 1 indicates worse repair (worse than old). The case of q < 0 suggests a system restored to a condition of better than new. Physically speaking, therefore, q can be an index for repair effectiveness.
Maximum Likelihood Estimation of the
Parameters
Single Repairable System
Thus the conditional probability density function (pdf ) of t_{i} is:
Where t_{i} > t_{i}1. The corresponding likelihood is:
Taking the natural log on both sides gives:
Where v_{i} can be obtained by Eqn. (1).
Multiple Systems
Taking the natural log on both sides gives:
There are three parameters (q, λ and β) that need to be estimated. However, there is no closed form mathematical solution. A numerical algorithm has been developed to solve both the single repairable system and multiple repairable systems.
Grouped Data The likelihood function is as follows:
Taking the natural log on both sides gives:
In order to estimate the unknown parameters, we use a numerical method to maximize the log likelihood function like the single repairable system and the multiple repairable systems.
Example 1
This data set is failure truncated. Based on this data set, different ML estimates of λ, β and repair degree q can be calculated corresponding to different model assumptions. The results of the ML estimates are shown in Table 2. For the PRP column, we assume the repair activities are perfect repairs and the failure intensity is as good as new. Thus, we can obtain the λ, β estimates and LKV using the Weibull++ 6 software. For the NHPP column, we assume that the repair actions restore the system operating state to be as bad as old. λ and β can be estimated using the RGA 6 software, which provides an excellent tool to analyze this type of repairable system. For the Proposed Model column, we use MLE to obtain λ, β and q estimates.
Table 2 shows that, based on LKV, the proposed model is the best fit for this data set. The q parameter is estimated to be 0.93156903. This value can be used as the restoration factor in BlockSim. Figure 1 shows the cumulative number of failures. Cumulative number of failures depends on the virtual time. This plot can also be used to estimate the cumulative number of failures for future times.
Example 2
This failure data set is from multiple repairable systems. We can estimate λ, β, q and the cumulative number of failures utilizing the data from all six systems in Table 3. Table 4 shows the results of ML estimates based on different models.
From Table 4, based on LKV, the proposed model is the best fit for this data set. The estimated q value, 0.55215907, can be used as the restoration factor in BlockSim. Figure 2 shows the cumulative number of failures and can be used to estimate the cumulative number of failures for future times. We can see that the proposed model fits these multiple repairable systems very well and provides promising results.
Conclusion
References 2. Wang, H. and Pham, H. "Some maintenance models and availability with imperfect maintenance in production systems." Annals of Operations Research, 91: 305318, 1999. 3. Kijima, M. and Sumita, N. "A useful generalization of renewal theory: counting process governed by nonnegative Markovian increments." Journal of Applied Probability, 23, 7188, 1986. 4. Kijima, M. "Some results for repairable systems with general repair," Journal of Applied Probability, 20, 851859, 1989. 5. Kaminskiy, M. and Krivtsov, V. "A Monte Carlo approach to repairable system reliability analysis." Probabilistic Safety Assessment and Management, New York: Springer; p. 10631068, 1998. 6. ReliaSoft, Reliability Growth and Repairable System Data Analysis Reference, 2004.
