Renewal theory

Yijun Xie

doi:10.54254/3029-0880/2025.28965

Advances in Operation Research and Production ManagementOpen access

Renewal theory

Research Article

Open Access

Renewal theory

Yijun Xie ^1*

¹ Department of Mathematical Sciences, Durham University

^*Corresponding author: leooooooxie@outlook.com

Published on 4 November 2025

AORPM Vol.4 Issue 3

ISSN (Print): 3029-0899

ISSN (Online): 3029-0880

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Abstract

Renewal theory originated from research on component failure and replacement. It has since developed into a key framework for analysing systems of repeated events within applied probability. This paper reviews the key concepts and principal findings in this field, while demonstrating several of its applications. The paper first introduces the Poisson process, highlighting the inter-arrival intervals of the exponential distribution and its 'memorylessness’, thereby introducing the general renewal process. The renewal function, elementary renewal theorem, renewal equation, and key renewal theorem are discussed, with attention to their assumptions, interpretations, and asymptotic conclusions, showing how they can be applied. The paper also presents several practical extensions of the renewal processes, including the delayed renewal process, renewal reward process, alternating renewal process, and age-dependent branching processes. Finally, concise examples illustrate the computation of long-run replacement and success rates, as well as the use of demographic renewal equations. Applications in reliability, service operations, and demography show that renewal models provide transparent asymptotic rates and availability with modest modelling complexity.

Keywords:

poisson process, renewal theory, renewal equation, key renewal theorem, stochastic process

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References

[1]. Grimmett, G.R., & Stirzaker, D.R. (1992). Probability and Random Processes (Third Edition). Oxford University Press, New York, NY.

[2]. D. R. Cox. (1962). Renewal Theory. Methuen Ltd., London.

[3]. Ross, S.M. Stochastic Processes (Second Editions). Wiley.

[4]. Pinsky, M.A., & Karlin, S. (2011). An Introduction to Stochastic Modeling (Fourth Edition). Academic Press.

[5]. Lindvall, T. (1977). A probabilistic proof of Blackwell’s renewal theorem.Ann. Probab., 5(3), 482-485.

[6]. Zamparo, M. (2023). Large deviation principles for renewal–reward processes. Stoch. Proc. Appl., 156, 226–245.

[7]. Johnson, R.A., & Taylor, J.R. (2008). Preservation of some life length classes for age distributions associated with age-dependent branching processes.Stat. Probab. Lett., 78(17), 2981–2987.

References

[1]. Grimmett, G.R., & Stirzaker, D.R. (1992). Probability and Random Processes (Third Edition). Oxford University Press, New York, NY.

[2]. D. R. Cox. (1962). Renewal Theory. Methuen Ltd., London.

[3]. Ross, S.M. Stochastic Processes (Second Editions). Wiley.

[4]. Pinsky, M.A., & Karlin, S. (2011). An Introduction to Stochastic Modeling (Fourth Edition). Academic Press.

[5]. Lindvall, T. (1977). A probabilistic proof of Blackwell’s renewal theorem.Ann. Probab., 5(3), 482-485.

[6]. Zamparo, M. (2023). Large deviation principles for renewal–reward processes. Stoch. Proc. Appl., 156, 226–245.

[7]. Johnson, R.A., & Taylor, J.R. (2008). Preservation of some life length classes for age distributions associated with age-dependent branching processes.Stat. Probab. Lett., 78(17), 2981–2987.

Cite this article

Xie,Y. (2025). Renewal theory. Advances in Operation Research and Production Management,4(3),53-67.

Data availability

The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.

About volume

Journal: Advances in Operation Research and Production Management

Volume number: Vol.4

Issue number: Issue 3

ISSN: 3029-0880(Print) / 3029-0899(Online)

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