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Numerical Treatment of Homogeneous Semi-Markov Processes in Transient Case–a Straightforward Approach

Author

Listed:
  • Gianfranco Corradi

    (Universitá “La Sapienza”)

  • Jacques Janssen

    (CESIAF)

  • Raimondo Manca

    (Universitá “La Sapienza”)

Abstract

This paper presents the numerical solution of the process evolution equation of a homogeneous semi-Markov process (HSMP) with a general quadrature method. Furthermore, results that justify this approach proving that the numerical solution tends to the evolution equation of the continuous time HSMP are given. The results obtained generalize classical results on integral equation numerical solutions applying them to particular kinds of integral equation systems. A method for obtaining the discrete time HSMP is shown by applying a very particular quadrature formula for the discretization. Following that, the problem of obtaining the continuous time HSMP from the discrete one is considered. In addition, the discrete time HSMP in matrix form is presented and the fact that the solution of the evolution equation of this process always exists is proved. Afterwards, an algorithm for solving the discrete time HSMP is given. Finally, a simple application of the HSMP is given for a real data social security example.

Suggested Citation

  • Gianfranco Corradi & Jacques Janssen & Raimondo Manca, 2004. "Numerical Treatment of Homogeneous Semi-Markov Processes in Transient Case–a Straightforward Approach," Methodology and Computing in Applied Probability, Springer, vol. 6(2), pages 233-246, June.
    • Handle: RePEc:spr:metcap:v:6:y:2004:i:2:d:10.1023_b:mcap.0000017715.28371.85
    DOI: 10.1023/B:MCAP.0000017715.28371.85
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    References listed on IDEAS

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    1. D. A. Elkins & M. A. Wortman, 2001. "On Numerical Solution of the Markov Renewal Equation: Tight Upper and Lower Kernel Bounds," Methodology and Computing in Applied Probability, Springer, vol. 3(3), pages 239-253, September.
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    Cited by:

    1. Andrey Launov & Klaus Wälde, 2013. "Estimating Incentive And Welfare Effects Of Nonstationary Unemployment Benefits," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 54(4), pages 1159-1198, November.
    2. Vlad Stefan Barbu & Guglielmo D’Amico & Andreas Makrides, 2022. "A Continuous-Time Semi-Markov System Governed by Stepwise Transitions," Mathematics, MDPI, vol. 10(15), pages 1-12, August.
    3. Marta O. Soares & Luísa Canto e Castro, 2012. "Continuous Time Simulation and Discretized Models for Cost-Effectiveness Analysis," PharmacoEconomics, Springer, vol. 30(12), pages 1101-1117, December.
    4. Yu Fang & Lijun Sun, 2019. "Developing A Semi-Markov Process Model for Bridge Deterioration Prediction in Shanghai," Sustainability, MDPI, vol. 11(19), pages 1-15, October.
    5. Vlad Stefan Barbu & Guglielmo D’Amico & Thomas Gkelsinis, 2021. "Sequential Interval Reliability for Discrete-Time Homogeneous Semi-Markov Repairable Systems," Mathematics, MDPI, vol. 9(16), pages 1-18, August.
    6. Márcio das Chagas Moura & Enrique López Droguett, 2010. "Numerical Approach for Assessing System Dynamic Availability Via Continuous Time Homogeneous Semi-Markov Processes," Methodology and Computing in Applied Probability, Springer, vol. 12(3), pages 431-449, September.
    7. Yunhui Hou & Nikolaos Limnios & Walter Schön, 2017. "On the Existence and Uniqueness of Solution of MRE and Applications," Methodology and Computing in Applied Probability, Springer, vol. 19(4), pages 1241-1250, December.
    8. Moura, Márcio das Chagas & Droguett, Enrique López, 2009. "Mathematical formulation and numerical treatment based on transition frequency densities and quadrature methods for non-homogeneous semi-Markov processes," Reliability Engineering and System Safety, Elsevier, vol. 94(2), pages 342-349.
    9. Andrey Launov & Irene Schumm & Klaus Walde, 2008. "Estimating insurance and incentive effects of labour market reforms," CDMA Conference Paper Series 0813, Centre for Dynamic Macroeconomic Analysis.
    10. Sophie Mercier, 2008. "Numerical Bounds for Semi-Markovian Quantities and Application to Reliability," Methodology and Computing in Applied Probability, Springer, vol. 10(2), pages 179-198, June.
    11. Bei Wu & Brenda Ivette Garcia Maya & Nikolaos Limnios, 2021. "Using Semi-Markov Chains to Solve Semi-Markov Processes," Methodology and Computing in Applied Probability, Springer, vol. 23(4), pages 1419-1431, December.
    12. Marta Soares & Luísa Canto e Castro, 2012. "Continuous Time Simulation and Discretized Models for Cost-Effectiveness Analysis," PharmacoEconomics, Springer, vol. 30(12), pages 1101-1117, December.
    13. Marta O Soares & L Canto e Castro, 2010. "Simulation or cohort models? Continuous time simulation and discretized Markov models to estimate cost-effectiveness," Working Papers 056cherp, Centre for Health Economics, University of York.
    14. Bo, Yimin & Bao, Minglei & Ding, Yi & Hu, Yishuang, 2024. "A DNN-based reliability evaluation method for multi-state series-parallel systems considering semi-Markov process," Reliability Engineering and System Safety, Elsevier, vol. 242(C).

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