## 8 Conclusions  
This article proposes an analytical model of a multi-service hierarchical system with overflow traffic. The initial assumption is that the primary resources of the system are offered mixtures of Erlang, Engset, and Pascal multi-service traffic streams. Existing methods for modeling such systems are reviewed and analyzed, while the reasons for their insufficient accuracy are presented and discussed. The results of the analysis allowed us to develop a new method for calculating the parameters of overflow traffic that has the advantage of obtaining higher accuracy in modeling systems with multi-service overflow traffic.  
To the best of our knowledge, the proposed method is also the first method that enables the analysis of systems with traffic overflow for any range of values of the offered traffic. The analysis of the previously developed solutions showed that for Pascal streams, they do not allow for the analysis of systems in which the traffic intensity offered by Pascal streams is equal or close to 1 Erl per system AU. Owing to its application versatility (Erlang, Engset and Pascal traffic streams) and high accuracy, the developed method can be employed in analyzing and dimensioning telecommunications networks at the flow (call) level. The use of Erlang, Engset, and Pascal streams allows taking into account the diverse nature of the traffic sources. Erlang traffic can be used to characterize the behavior of many flows whose average traffic volume is constant over time. Engset traffic allows us to take into account the impact of a limited (compared with the system capacity) number of traffic sources on the change (reduction) in the intensity of the offered traffic along with the number of serviced traffic sources. Engset traffic is often used to approximate flows with a long service time and a limited number of traffic sources. Pascal traffic can be used to model traffic with large values of the peakedness coefficient (variance-to-mean ratio).  
The great interest in using traffic overflow to optimize 4G and 5G networks [15; 16; 17; 43] indicates that the proposed method, by taking into account the different sizes of the requested resources and types of traffic offered as well as the high accuracy of calculations, can be widely used in the analysis and dimensioning of 4G and 5G access networks. The proposed method can also be used to develop call admission control mechanisms for multi-service telecommunications networks.  
Conceptualization, D.K. and M.G.; methodology, D.K.; software, D.K.; validation, D.K. and M.G.; formal analysis, D.K., M.G., and M.S.; investigation, D.K.; resources, D.K.; data curation, D.K.; writing--original draft preparation, D.K. and M.G.; writing--review and editing, D.K., M.G., and M.S.; visualization, D.K.; supervision, M.G.; project administration, D.K.; funding acquisition, M.S. All authors have read and agreed to the published version of the manuscript.  
This research is in part funded by the Polish National Science Centre (supervised by the Ministry of Science and Higher Education), Grant 2016/23/B/ST7/03925, and by the Ministry of Education and Science, Grant 0313/SBAD/1301.  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.