Global stability with time delay in optimization flow control

In this paper we consider a dual-gradient optimization flow control scheme. In an earlier work it was shown that such algorithms converge in a delay free case. We present the sufficient condition under which the stability can be global focusing on the scenario of a single flow and bottleneck link with delay. We first show that this synchronous algorithm is convergent in general network topology without delay. Then we provide a result that even with delays, the queue length increasing 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.