A new network connection setup procedure for network computers to set up data transmission connections over a network that implements traffic prioritization (e.g., according to IEEE 802.1p) has the effect of providing admission control and implicit bandwidth reservation. A sending network device (the "sender") that wants to initiate a new transmission over the network first enters a probing mode in which it sends data packets at a priority level that is lower than the target priority level and at a transmission rate up to a peak probing transmission rate that may be the same or slightly higher than the desired transmission rate. If the sender determines based on feedback regarding the probing transmission that the network can support a transmission at the peak probing transmission rate and the probing priority level, it switches to a steady mode in which it sends data packets at the desired transmission rate and at the target priority level.