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Two approximations of renewal function for any arbitrary lifetime distribution

  • S.I. : Reliability Modeling with Applications Based on Big Data
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Abstract

The renewal functions (RFs) of most distribution functions do not have closed-form expressions while such expressions are desired for the optimization problems involved RF. Many efforts have been made to develop approximations of RF. However, it seems that no RF approximation is accurate enough in the entire time range. In this paper, we propose two RF approximations. The first approximation is obtained through smoothly connecting two limiting relations and fairly accurate in the entire time range. The second approximation has the same function form as the first part of the first approximation but the model parameter is determined in a different way so as to achieve higher accuracy for small to moderate time range. The expressions of the proposed approximations are simple and applicable for any arbitrary lifetime distribution. Their accuracy is analyzed and, the appropriateness and usefulness are illustrated by a numerical example.

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References

  • Ayhan, H., Limon-Robles, J., & Wortman, M. A. (1999). Approach for computing tight numerical bounds on renewal functions. IEEE Transactions on Reliability, 48(2), 82–188.

    Article  Google Scholar 

  • Blischke, W. P., & Murthy, D. N. P. (2003). Case studies in reliability and maintenance. New Jersey: Wiley.

    Google Scholar 

  • Brezavšček, A. (2016). Stochastic approach to planning of spares for complex deteriorating industrial system. Quality Technology and Quantitative Management, 12(4), 465–480.

    Article  Google Scholar 

  • Çetinkaya, S., Tekin, E., & Lee, C. (2008). A stochastic model for joint inventory and outbound shipment decisions. IIE Transactions, 40(3), 324–340.

    Article  Google Scholar 

  • Chaudhry, M. L. (1995). On computations of the mean and variance of the number of renewals: A unified approach. Journal of the Operational Research Society, 46(11), 1352–1364.

    Article  Google Scholar 

  • Constantine, A. G., & Robinson, N. I. (1997). Weibull renewal function for moderate to large arguments. Computational Statistics & Data Analysis, 24(1), 9–27.

    Article  Google Scholar 

  • Cui, L., & Xie, M. (2003). Some normal approximations for renewal function of large Weibull shape parameter. Communications in Statistics-Simulation and Computation, 32(1), 1–16.

    Article  Google Scholar 

  • Daley, D. J., & Vesilo, R. (1997). Long range dependence of point processes, with queueing examples. Stochastic Processes and their Applications, 70(2), 265–282.

    Article  Google Scholar 

  • Dohi, T., Kaio, N., & Osaki, S. (2002). Renewal processes and their computational aspects. In Stochastic models in reliability and maintenance (pp. 1–30). Berlin, Springer.

  • From, S. G. (2001). Some new approximations for the renewal function. Communications in Statistics-Simulation and Computation, 30(1), 113–128.

    Article  Google Scholar 

  • Garg, A., & Kalagnanam, J. R. (1998). Approximations for the renewal function. IEEE Transactions on Reliability, 47(1), 66–72.

    Article  Google Scholar 

  • Gertsbakh, I., & Shpungin, Y. (2004). Renewal function and interval availability: A numerical Monte–Carlo study. Communications in Statistics. Theory and Methods, 33(3), 639–649.

    Article  Google Scholar 

  • Ghodrati, B. (2011). Efficient product support—Optimum and realistic spare parts forecasting. In L. Tadj, et al. (Eds.), Replacement models with minimal repair (pp. 225–269). London: Springer-Verlag London Limited.

    Chapter  Google Scholar 

  • Jiang, R. (2008). A Gamma–normal series truncation approximation for computing the Weibull renewal function. Reliability Engineering and System Safety, 93(4), 616–626.

    Article  Google Scholar 

  • Jiang, R. (2010). A simple approximation for the renewal function with an increasing failure rate. Reliability Engineering and System Safety, 95(9), 963–969.

    Article  Google Scholar 

  • Jiang, R. (2015). Product warranty. Introduction to quality and reliability engineering (pp. 283–301). Berlin: Springer.

    Chapter  Google Scholar 

  • Jiang, R. (2018). Performance evaluation of seven optimization models of age replacement policy. Reliability Engineering and System Safety, 180, 302–311.

    Article  Google Scholar 

  • Jiang, R., & Murthy, D. N. P. (2008). Maintenance: Decision models for management. Beijing: Science Press.

    Google Scholar 

  • Jin, T., & Gonigunta, L. (2009). Weibull and gamma renewal approximation using generalized exponential functions. Communications in Statistics: Simulation and Computation, 38(1), 154–171.

    Article  Google Scholar 

  • Kumar, U. D., Crocker, J., Knezevic, J., & El-Haram, M. (2000). Reliability, maintenance and logistic support: A life cycle approach. Dordrecht: Kluwer Academic Publishers.

    Book  Google Scholar 

  • Lin, P. C., & Shue, L. Y. (2005). Application of optimal control theory to product pricing and warranty with free replacement under the influence of basic lifetime distributions. Computers & Industrial Engineering, 48(1), 69–82.

    Article  Google Scholar 

  • Parsa, H., & Jin, M. (2013). An improved approximation for the renewal function and its integral with an application in two-echelon inventory management. International Journal of Production Economics, 146(1), 142–152.

    Article  Google Scholar 

  • Rao, B. M. (1995). Algorithms for the free replacement warranty with phase-type Iifetime distributions. IIE Transactions, 27(3), 348–357.

    Article  Google Scholar 

  • Riascos-Ocho, J., Sanchez-Silva, M., & Akhavan-Tabatabaei, R. (2014). Reliability analysis of shock-based deterioration using phase-type distributions. Probabilistic Engineering Mechanics, 38(38), 88–101.

    Article  Google Scholar 

  • Samanta, S. K., Chaudhry, M. L., Pacheco, A., & Gupta, U. C. (2015). Analytic and computational analysis of the discrete-time GI/D-MSP/1 queue using roots. Computers & Operations Research, 56, 33–40.

    Article  Google Scholar 

  • Smeitink, E., & Dekker, R. (1990). A simple approximation to the renewal function. IEEE Transactions on Reliability, 39(1), 71–75.

    Article  Google Scholar 

  • Spearman, M. L. (1989). A simple approximation for IFR Weibull renewal functions. Microelectronics Reliability, 29(1), 73–80.

    Article  Google Scholar 

  • Xie, M. (1989). On the solution of renewal-type integral equations. Communications in Statistics-Simulation and Computation, 18(1), 281–293.

    Article  Google Scholar 

  • Zacks, S. (2010). The availability and hazard of a system under a cumulative damage process with replacements. Methodology and Computing in Applied Probability, 12(4), 555–565.

    Article  Google Scholar 

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Acknowledgements

The research was supported by the National Natural Science Foundation of China (No. 71771029).

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  1. School of Automotive and Mechanical Engineering, Changsha University of Science and Technology, Changsha, 410114, Hunan, People’s Republic of China

    R. Jiang

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Appendix A: Renewal function of the Weibull distribution

Appendix A: Renewal function of the Weibull distribution

The exact values of RF of the Weibull distribution with \( \eta = 1 \) and \( \beta = 1.5(0.5)3.5 \) are shown in Table 4.

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Jiang, R. Two approximations of renewal function for any arbitrary lifetime distribution. Ann Oper Res 311, 151–165 (2022). https://doi.org/10.1007/s10479-019-03356-2

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