## 2 Related Work
Typically, the most convenient models for the analytical modeling of traffic characteristics of multi-service networks with traffic overflow are the so-called _multi-rate models_. The basic multi-rate models were initially developed for wired systems (networks) with integrated (multi-rate) services--ISDN (Integrated Services Digital Network) and B-ISDN (Broadband ISDN) [46; 47; 48; 49]. Currently, they are also used to model wireless multi-service telecommunications systems [50; 51; 52]. The initial assumption in multi-rate models is that the resources demanded by calls of particular traffic classes are a multiple of the so-called _allocation unit_ (AU), which can be defined as the greatest common divisor of the resources demanded by calls of all traffic classes offered to the system. Depending on the multi-service multi-rate system (multi-service systems, systems with integrated traffic) under investigation, the AU can refer to both bitrate and other physical quantities that limit the resources of a system. In research on modeling multi-service overflow systems, models based on an analysis of the call admission processes are predominant (the methods based on the analysis of the call admission processes are characterized by exponential computational complexity, since they require--to determine moments--solutions to state equations related to the multi-dimensional Markov process that takes place in the system under consideration). The interrupted Poisson process [IPP], the Markov-modulated Poisson process [MMPP]) [31], and methods based on an analysis of the state space of systems are under consideration. These methods are in fact extensions of existing solutions for single-service systems and take into consideration the influence of traffic streams with different parameters (call intensity, service time, the volume of demanded resources) and their mutual correlation. The first effective methods to model and dimension multi-service overflow networks were proposed between 2007 and 2008, in [28] and concurrently in [34]. In [28; 34], the primary resources are described by a model of a full-availability resource, that is, a state-independent model, in which a call will be admitted for service if the system has enough free resources to service this call. The solutions proposed in [28; 34] make it possible to determine in a simple way the variance and intensity of traffic that overflows to alternative resources. To determine the traffic characteristics of the alternative resources that service overflow traffic, in [28; 34] a modified full-availability group model, modified according to the approach proposed by [PERSON] [5] for single-service systems, is used. In this model, both the capacity of the full-availability group and the values of offered traffic intensities for individual classes are divided by appropriate peakedness coefficients. An alternative analytical model, one that also assumes full availability of the primary resources, is proposed in [53]. This model is based on an appropriately defined, bi-dimensional convolution operation and allows overflow systems with any type of traffic offered to the primary resources to be modeled. The models in [54; 55; 56] consider overflow systems in which overflow traffic changes the service parameters, such as service time and bitrate, in the secondary resources. The assumed changes in the parameters were not, however, linked to any network traffic shaping mechanisms. Over time, the model in [28] for multi-service overflow systems, based on the Fredericks-Hayward approach, was successively expanded. Subsequent studies considered, among others, (i) limited availability of the secondary resources [57], (ii) thresholdless compression (elastic traffic) [58], (iii) threshold compression (adaptive traffic) [35], and (iv) multi-service Erlang, Engset, and Pascal traffic streams [35; 59].