Global stability of nonlinear optimization flow control with time delay

It was shown that an optimization-based algorithm converges in a delay-free network condition. We present the sufficient conditions under which the stability can be global focusing on the scenario of a single bottleneck link with multiple users under heterogeneous delays. Using Lyapunov Razumikhin theory we achieve the globally asymptotic stability with heterogeneous delays by scaling down the stepsize. Then we provide a result that even with delays, the queue length increment at the router is bounded. The upper bound grows with increase in the number of flows as well as the maximum source sending rate and the maximum round trip delay. The upper bound decreases as the link departing rate and stepsize increase.