We consider the multiplexing of several variable bit rate (VBR) connections over one variable bit rate connection where the multiplexing uses a multiplexing buffer of size B. The VBR trunk is itself a connection and has a multidimensional connection descriptor, reflecting peak and sustainable rates. Given a cost function for the VBR trunk and a connection admission control (CAC) method for the input connections, we focus on the problem of finding the VBR trunk connection descriptor that minimizes the cost function and is able to accept set given set of VBR input connections. First, we show that, under reasonable assumptions on the cost function, the optimization problem can be reduced to a simpler one. Then we consider the homogeneous, loss-free case, for which we give an explicit CAC method. In this case, we find that, for all reasonable cost functions, the optimal VBR trunk is either of the CBR type, or is truly VBR, with a burst duration equal to the burst duration of the input connections. We show that the optimal peak cell rate is fixed for a given B (thus for a CBR trunk), and a VBR choice can only be an improvement. Lastly, we take as an example of the cost function the equivalent capacity of the VBR trunk. These results are expected to form the basis for a general method for a connection manager at a multiplexing node in an integrated services packet network.