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We consider a single link accessed by a single source which responds to congestion signals from the network. The design of controllers for such sources in the presence of feedback delay has received much attention recently. Here, we present conditions for the global, asymptotic stability and semiglobal exponential stability of congestion controllers. These conditions are natural extensions of conditions obtained through linearized analysis of such systems. Our result on exponential stability provides the missing link in the proof of how one obtains a single deterministic congestion control equation from a system with many congestion-controlled sources and random disturbances. Using numerical examples, we compare the conditions on the congestion-control parameters obtained using local and global stability analysis.