In the paper, we consider a stochastic game model of data transmission with three asymmetric players (i.e. network nodes), in which the network is given and does not change over time. The players aim to transmit as many packages as possible to the corresponding terminal nodes through the common node whose capacity is two. We assume that each player has a finite capacity buffer for storing data packages. The dynamic process of data transmission is modeled as a stochastic game with finite set of states. Existence of the Nash equilibrium and a cooperative solution is proved. We find the cooperative strategy profile and Nash equilibrium in pure strategies. The estimation of the price of anarchy is calculated for a numerical example.