## 1 Introduction  
Queueing theory provides a powerful tool for the optimization of sharing the restricted resources in many telecommunication, manufacturing, logistic, social, and other systems and networks. In particular, it has wide applications for optimization of routing and energy saving in modern networks where heterogeneous information flows have to be delivered from one node to another one with minimal delay and energy consumption; see, e.g., [1; 2; 3; 4; 5; 6; 7]. The flows are heterogeneous with respect to the required bandwidth (e.g., elephant, dog, and mice flows) and CPU (computationally sparse and computationally dense flows). Very often, the requests receiving service in such systems are inhomogeneous with respect to the indicators of the quality of service (e.g., delay sensitive or insensitive customers, customers tolerant or intolerant to partial loss, streaming or elastic customers, perishable and non-perishable goods, etc. ), as well as with respect to their economic or social value. Therefore, certain mechanisms for providing preferences for some types of customers are offered and widely analyzed in the literature. Among these mechanisms, we can mention various polling disciplines with suitably chosen round tables and the maximum attendance times and the generalized processor sharing disciplines. In polling disciplines, which assume a cyclic connection of the common server to the buffers designed for storing different types of customers, priority can be provided to some type of customers via more frequent connection to the buffer for storing the priority customers and via a longer duration of maintaining this connection. The generalized processor sharing discipline is the generalization of the usual processor sharing discipline, which assumes that all customers present in the system receive service simultaneously with the rate inversely proportional to the number of these customers. This generalization assumes that various types of customers can use not equal, but different shares of the capacity of the server. As a simpler mechanism, which does not permanently require the sharing of time or processor and may be much easier implemented in real-world systems, customer prioritization can be used. Static priorities define the discipline of the choice of the next customer for service based only on the types of customers present in the system. A non-preemptive static priority does not suggest an interruption of non-priority customer service when the priority customer arrives. The preemptive static priority assumes such an interruption. A non-priority customer in this situation may be lost or return to the buffer and later try to receive service again (there exist many variants of the duration of the repeated service). The evident shortcoming of the static priorities is their evident unfairness with respect to the low priority customers, especially in the situation when the real values of the priority and non-priority customers for the system are not essentially different. A low priority customer can have a very long waiting time and succeed to start service essentially later than the priority customer that arrived only recently; see, e.g., [8]. In this respect, the dynamical priorities, which take into account not only the types of customers present in the system, but also the lengths of the corresponding queues, are much more flexible. However, the problems of proving the optimality of some intuitively reasonable strategies (e.g., the threshold or hysteresis) in the class of all available strategies and making the optimal choice of the parameters of the control strategy are, as a rule, quite complicated. Furthermore, what is even more essential is that the practical realization of such a strategy (requiring, in particular, the permanent monitoring of the lengths of queues of all types) can be very difficult or costly.