A long-time asymptotic solution to the g-renewal equation for underlying distributions with nondecreasing hazard functions


The Kijima’s type 1 maintenance model, representing the general renewal process, is one of the most important in the reliability theory. The g-renewal equation is central in Kijima’s theory and it is a Volterra integral equation of the second kind. Although these equations are well-studied, a closed-form solution to the g-renewal equation has not yet been obtained. Despite the fact that several semi-empirical techniques to approximate the g-renewal function have been previously developed, analytical approaches to solve this equation for a wide class of underlying distributions is still of current interest. In this paper, a long-time asymptotic for the g-renewal rate is obtained for distributions with nondecreasing hazard functions and for all values of the restoration factor \(q\in [0,1]\). The obtained analytical result is compared with the numerical solutions for two types of underlying distributions, showing a good asymptotic match. The obtained approximate g-renewal rate is employed for maintenance optimization, considering the repair cost as a function of the restoration factor. Several numerical examples are performed in order to show the efficiency of our results.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13


  1. 1.

    The existence of the second derivative of function H(x) follows from Theorem 1, which is necessary for existence of the derivative \(\partial _{q}w(x,q)\) and, therefore, for representing the g-renewal equation in the form (33).


  1. Barlow RE, Proschan F (1965) Mathematical theory of reliability. Wiley, New York

    Google Scholar 

  2. Bartholomew-Biggs M, Zuo MJ, Li X (2009) Modelling and optimizing sequential imperfect preventive maintenance. Reliab Eng Syst Saf 94:53–62

    Article  Google Scholar 

  3. Cassady CR, Iyoob IM, Schneider K, Pohl EA (2005) A generic model of equipment availability under imperfect maintenance. IEEE Trans Reliab 54:564–571

    Article  Google Scholar 

  4. Finkelstein M (2008) Failure rate modedling for reliability and risk. Springer, London

    Google Scholar 

  5. Gavrilov LA, Gavrilova NS (2011) Mortality measurement at advanced ages: a study of the social security administration death master file. N Am Actuar J 15(3):432–447

    MathSciNet  Article  Google Scholar 

  6. Kahle W (2007) Optimal maintenance policies in incomplete repair models. Reliab Eng Syst Saf 92:563–565

    Article  Google Scholar 

  7. Kaminskiy MP, Krivtsov VV (2006) A monte carlo approach to estimation of g-renewal process in warranty data analysis. Reliab Theory Appl 1:29–31

    Google Scholar 

  8. Kijima M, Sumita M (1986) A useful generalization of renewal theory: counting processes governed by nonnegative Markovian increments. J Appl Prob 23:71–88

    MathSciNet  Article  Google Scholar 

  9. Kijima M, Morimura H, Susuki Y (1988) Periodical replacement problem without assuming minimal repair. Eur J Oper Res 37:194–203

    MathSciNet  Article  Google Scholar 

  10. Krivtsov V, Yevkin O (2013) Estimation of g-renewal process parameters as an ill-posed inverse problem. Reliab Eng Syst Saf 115:10–18

    Article  Google Scholar 

  11. Murthy DNP (1991) A note on minimal repair. IEEE Trans Reliab 40:245–246

    Article  Google Scholar 

  12. Nakagawa T (1979) Optimum policies when preventive maintenance is imperfect. IEEE Trans Reliab R–28:331–332

    Article  Google Scholar 

  13. Nakagawa T, Kowada M (1983) Analysis of a system with minimal repair and its application to replacement policy. Eur J Oper Res 12:176–182

    Article  Google Scholar 

  14. Tanwar M, Rai RN, Bolia N (2014) Imperfect repair modeling using Kijima type generalized renewal process. Reliab Eng Syst Saf 124:24–31

    Article  Google Scholar 

  15. Vladimirov VS, Zharinov VV (2004) Equations of mathematical physics. Phys. Mat. Lit., Moscow

    Google Scholar 

  16. Wong JSW, Wong R (1976) On asymptotic solutions of the renewal equation. J Math Anal Appl 53:243–250

    MathSciNet  Article  Google Scholar 

  17. Yevkin O, Krivtsov V (2012) Approximate solution to g-renewal equation with underlying Weibull distribution. IEEE Trans Reliab 61(1):68–73

    Article  Google Scholar 

  18. Yevkin O, Krivtsov V (2013) Comparative analysis of optimal maintenance policies under general repair with underlying Weibull distributions. IEEE Trans Reliab 62(1):82–91

    Article  Google Scholar 

Download references


The authors are grateful for financial support to the following Mexican Institutions: CONACYT, SNI and PROMEP.

Author information



Corresponding author

Correspondence to Serguei Maximov.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Maximov, S., Cortes-Penagos, C.J. A long-time asymptotic solution to the g-renewal equation for underlying distributions with nondecreasing hazard functions. Math Meth Oper Res (2020). https://doi.org/10.1007/s00186-020-00715-9

Download citation


  • G-renewal process
  • Optimal maintenance
  • Volterra integral equation
  • Asymptotic solution

Mathematics Subject Classification

  • 62N05
  • 45D05
  • 34M30