## 1 Introduction  
Ensuring high quality of service (QoS) was and remains the most important task entrusted to modern communication networks. In the increasing network load, a growing variety of contemporary traffic, and QoS requirements, the problem involves improving the efficiency of using available network resources, such as the bandwidth of communication links, queue buffers, and processing time of routers [1, 2]. Quality of service is a key characteristic of modern networks, which can be evaluated using a variety of indicators (metrics). Traditional QoS indicators in solving OSI network layer problems are network performance indicators: bandwidth, network delay, jitter, and packet loss probability [1, 2]. The nature of the calculated routes largely determines the values of end-to-end QoS indicators and the characteristics of the communication links included in the routes: bandwidth and its utilization. Thus, according to the listed indicators, the current research developed an approach where a balanced communication link utilization improves the QoS level. As the analysis [3, 4, 5, 6, 7, 8, 9] showed, an effective mechanism for solving the given task is the implementation of traffic engineering (TE) principles to ensure the balanced loading of available network resources. For example, routing protocols (OSPF-TE, IS-IS-TE), signaling, and resource reservation, as well as RSVP-TE, modified for traffic engineering requirements, have found their use in MPLS networks [2]. Great attention is also being paid to the development of TE solutions by scientists working on optimizing network solutions [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. Their proposed findings apply to load balancing in software-defined networks (SDNs), cloud environments, mobile networks, queue management on routers and switches, etc. Another group of TE solutions involves developments based on using artificial intelligence to make network decisions [23; 24; 25; 26; 27]. The shortcomings of known solutions include implementing a heuristic approach to solving traffic limitation problems based on the token bucket and leaky bucket algorithms [1; 2]. In addition, these algorithms do not consider the results of solving routing problems. Although these solutions' goal is common--to combat congestion and ensure the quality of service--new solutions aimed at the coordinated solution of routing and traffic limitation problems are subject to the following requirements: optimality and adaptability to changes in the network state (its structure, load, etc.) and packet flow characteristics (intensity, priority, etc. ), as well as ensuring load balancing based on TE principles to guarantee a given QoS level. Nevertheless, modern communication networks often work in overload conditions when the amount of available network resources is insufficient, which negatively affects the QoS level, and is accompanied by an increase in service failures and traffic limitation at the network edge. However, the delays and packet loss increases are unacceptable for specific flows. Therefore, it is essential to ensure the predictability of the QoS level and controllability of limiting traffic entering the network, which should be implemented through maximum coordination by routing tools and traffic limitations. The desired effect is possible using unified optimization mathematical models that describe the relation of such critical network processes. Specific solutions in this direction were proposed in studies [21; 22] and modified, for example, under the problem of fault-tolerant routing. However, already-known mathematical models need further improvement to increase the adaptability of load balancing solutions to changes in the network state, flow parameters, requirements for the level of differentiated quality of service, etc. Therefore, in the current study, a mathematical optimization model is proposed, which received two modifications depending on the chosen type of optimality criterion.