We perform the necessary calculations to determine the stability and asymptotic forms of solutions bifurcating from steady state in a nonlinear delay differential equation with a single discrete delay. The results are used to examine the loss of local stability in a selection of congestion control algorithms employed over a single link. In particular, we analyze the fair and the delay-based dual algorithms. Explicit conditions are derived to ensure the onset of stable limit cycles as these algorithms just lose local stability. Further, we are able to quantify the effect parameters of the system have on the amplitude of the bifurcating periodic solutions.