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The article proposes an analytical method for determining occupancy distribution and blocking probability in systems which are offered overflow traffic composed of multi-service traffic streams. The described analytical model enables determination of parameters of traffic overflowed from primary groups in hierarchically constructed telecommunication networks. The proposed method is based on an appropriate modification of the Kaufman-Roberts recursion for the full-availability group with multi-rate traffic and uses the modified Fredericks & Hayward’s approximation. Additionally, an approximate method for dimensioning systems with multi-service overflow traffic is also presented. The analytical results of the blocking probability and the results obtained in the dimensioning processes calculated using the presented methodology are compared with the data obtained from the system simulation process.