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.