Internet Congestion Control

and hence is stable. Note that the fact that PM>0 implies that $T(j\overline{\omega })<1$, where $\overline{\omega }$ is the frequency at which $\text{arg}(T(j\overline{\omega }))=\pi$, that is, the gain margin is also positive.

From Equation 67 it follows that

$\text{arg}(T(j{\omega }_c))={\omega }_c{T}^{+}+{\tan }^{-1}\frac{{\omega }_c}{K}+{\tan }^{-1}\frac{{\omega }_c}{2N^{-}/C{(T^{+})}^2}+{\tan }^{-1}\frac{{\omega }_c}{1/T^{+}}$ (73)

(73)

Since

${\omega }_c=0.1\min \left\{\frac{2N^{-}}{{(T^{+})}^{2}C},\frac{1}{T^{+}}\right\}$

it follows that

${\omega }_c{T}^{+}=0.1\min \left\{\frac{2N^{-}}{T^{+}C},1\right\}$

so that ...