I.A. Turenova1, M.K. Kiryachek2, S.P. Moiseeva3, E.V. Pankratova4

1–3 National Research Tomsk State University (Tomsk, Russia)
4 V.A. Trapeznikov Institute of Control Sciences of the RAS (Moscow, Russia)
1 irenaturena@yandex.ru, 2 radkovaaa@yandex.ru, 3 smoiseeva@mail.ru, 4 pankratova@ipu.ru

With the development of communication systems, the emergence of complex and multi-level information and computing systems, and the creation of automated control systems in almost all areas of industry and daily life, interest in queuing theory as a mathematical tool for studying random processes in various systems has significantly increased in recent decades. Ensuring high performance and reliability of web servers has become critically important, as they play a key role in providing access to website content, processing user requests, and interacting with databases.

Consider the process of users accessing an Internet resource (e.g., watching video content). Each user may recommend content to their acquaintances, creating an "implicit advertising" effect, resulting in the total intensity of resource access depending on the number of users currently on the site, while each user's time spent on the site (e.g., watching a video) is random. One of the characteristics of content demand is the number of "clicks" (number of pages viewed) over a certain period, leading to the problem of studying event streams that do not possess stationarity and lack of after-effect properties. Considering these features justifies the relevance of constructing probabilistic models and their study, which will allow predicting content load and optimizing costs.

The aim of this work is to construct and study a mathematical model of Internet resource traffic in the form of a system with an unlimited number of servers with variable arrivals intensity and repeated service. Within this work, a Markov process describing the functioning of a queuing system with an unlimited number of service units, variable incoming flow intensity, and feedback was investigated. A distinctive feature is the analysis of the system in non-stationary mode. As a result of the study, one-dimensional generating functions were obtained, characterizing the number of users on the website, as well as the streams of repeated access to content. The presence of these functions allows finding the main numerical characteristics of the corresponding random processes, such as mathematical expectations, variances, and other probabilistic characteristics, which significantly expands the possibilities for analyzing system behavior over time.

The proposed model allows quantitatively and qualitatively describe the dynamics of Internet resource traffic, make predictions about future traffic, and explains the mechanism of traffic growth, which will allow assessing the expediency of investing in any Internet resource. The research results can also serve as a basis for evaluating the probabilistic-temporal characteristics of designed information and telecommunications networks and distributed computer systems, which provide for the presence of processes for repeated or additional processing of data sets and requests.

Turenova I.A., Kiryachek M.K., Moiseeva S.P., Pankratova E.V. Mathematical model of web resource access requests as an infinite-server queueing system with repeated requests and variable intensity // Dynamics of complex systems. 2026. V. 20. № 4. P. 63−73. DOI: 10.18127/j19997493-202604-05

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