Message passing delay in network congestion management

Network Utility Maximization (NUM) framework has been extensively studied. Generally, existing solutions for NUM require message exchange between network and users to regulate the flow of network traffic, and information is frequently assumed to be available instantaneously and the traffic flow adjustment is accomplished immediately. However, realistically, there is delay in message exchange because time is required for the messages to reach the designated destinations and for the traffic flow adjustment in network to take effect. Consequently, without proper synchronization, transmission rate and network pricing may oscillate, resulting in the algorithm's failure to converge, even when there is a solution that converges to an optimal solution. Here, we propose a synchronization methodology to prevent the algorithm from oscillating.