Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

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Rusilko T. V. Application of queueing network models in insurance. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2022, vol. 22, iss. 3, pp. 315-321. DOI: 10.18500/1816-9791-2022-22-3-315-321, EDN: ONZHCB

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Application of queueing network models in insurance

Rusilko Tatiana V., Yanka Kupala State University of Grodno

The purpose of this paper is to study the issues of the functioning of insurance companies using the methods of the queueing networks theory. The introduction provides a brief overview of scientific publications in this area. In particular, research based on the use of Markov stochastic processes and queueing systems are considered. In the first section of the article, a closed exponential queueing network is proposed as a model for the process of processing insurance claims. A detailed description of the corresponding network model is given. The stay of each job at a specific network node and its routing between the nodes correspond to the customer claim status in the insurance company and the process of its routing between claims adjusters of different types of risks. The process of changing the number of jobs at the nodes was studied under the asymptotic assumption of a large number of jobs in the second section of the article. In this case, its probability density function  satisfies the Fokker – Planck – Kolmogorov equation. The system of differential equations for the first-order and second-order moments of the state vector was substantiated in the third section of the article. The solution of this system allows for predicting the dynamics of the expected number of insurance claims in the model nodes in both transient and steady states. Second-order moments can be used to calculate the variability of the number of insurance claims at the model nodes and to study the correlation between the number of claims at different nodes with time. The areas of implementation were considered.

  1. Cramer H. Historical review of Filip Lundberg’s works on risk theory. Scandinavian Actuarial Journal, 1969, vol. 52, pp. 6–12. https://doi.org/10.1080/03461238.1969.10404602
  2. Gerber H. U. Life Insurance Mathematics. Berlin, Springer-Verlag, 1995. 220 p. https://doi.org/10.1007/978-3-662-03153-7
  3. Grandell J. Aspects of Risk Theory. New York, Springer-Verlang, 1991. 175 p. http://dx.doi.org/10.1007/978-1-4613-9058-9
  4. Bening V. E., Korolev V. Yu. Nonparametric estimation of the ruin probability for generalized risk processes. Theory of Probability and its Applications, 2003, vol. 47, iss. 1, pp. 1–16. https://doi.org/10.4213/tvp2954
  5. Rotar V. I., Bening V. E. An introduction to the mathematical theory of insurance. Surveys on Applied and in Industrial Mathematics, 1994, vol. 1, iss. 5, pp. 698–779 (in Russian).
  6. Abdyusheva S. R., Spivak S. I. Markov models in actuarial analysis. Middle Volga Mathematical Society Journal, 2003, vol. 5, iss. 1, pp. 224–232 (in Russian). EDN: XWOCZN
  7. Glukhova E. V., Zmeyev O. A., Livshits K. I. Mathematical Models of Insurance. Tomsk, Tomsk State University Publ., 2004. 180 p. (in Russian).
  8. Akhmedova D. D., Zmeyev O. A., Terpugov A. F. Optimization of activity of insurance company with account of expenses for advertising. Tomsk State University Journal, 2002, iss. 275, pp. 181–184 (in Russian). EDN: OYCDID
  9. Asmussen S. Applied Probability and Queues. New York, Springer-Verlag, 2003. 438 p. https://doi.org/10.1007/b97236
  10. V. M., Livshits K. I., Nazarov A. A. Investigation of nonstationary interminable linear queueing system and their application for mathematical model of insurance company. Tomsk State University Journal, 2002, iss. 275, pp. 189–192 (in Russian). EDN: OYCDIX
  11. Matalytski M., Rusilko T., Pankov A. Asymptotic analysis of the closed queueing structure with time-dependent service parameters and single-type messages. Journal of Applied Mathematics and Computational Mechanics, 2013, vol. 12, iss. 2, pp. 73–80. https://doi.org/10.17512/jamcm.2013.2.09
  12. Matalytskiy M. A., Romanyuk T. V. Mathematical analysis of stochastic models of processing claims of various types in insurance companies. Doklady of the National Academy of Sciences of Belarus, 2005, vol. 49, iss. 1, pp. 18–23 (in Russian).
  13. Medvedev G. A. About optimization of closed queueing systems. Proceedings of the USSR Academy of Sciences. Engineering Cybernetis, 1975, no. 6, pp. 65–73 (in Russian). EDN: YHFVDR
  14. Tikhonov V. I., Mironov M. A. Markov Processes. Moscow, Sovetskoe Radio, 1977. 488 p. (in Russian).
  15. Rusilko T. V., Matalytski M. A. Queuing Network Models of Claims Processing in Insurance Companies. Saarbrucken, LAP Lambert Academic Publishing, 2012. 336 p. (in Russian).
  16. Rusilko T. V. The first two orders moments determination method for the state vector of the queueing network in the asymptotic case. Vesnik of Yanka Kupala State University of Grodno. Series 2, 2021, vol. 11, iss. 2, pp. 152–161 (in Russian). EDN: EZQQOM