Optimal transmission of messages in computer networks – an optimal control problem involving control-dependent time-delayed arguments

In this paper, we find the optimal transmission of messages in computer networks. This problem has been formulated as a nondelayed optimal control problem in several recent papers on TCP (transmission control protocol). Since the actual transmission of messages from origins to destinations should consist of both forward transmission delays of the buffers and latency of the links, we remodel the problem as a time-delayed optimal control problem consisting of both control-dependent time-delayed arguments and discrete time-delayed arguments. We then develop a modified control parameterization method for solving this time-delayed optimal control problem. The gradients of the new objective function and constraint functions generated by this modified control parametrization method are derived. A numerical example is solved by using the time-delayed version of the problem that we formulate, as well as the nondelayed version of the problem in the literature. Numerical results clearly illustrate the efficiency of the modified control parameterization method for solving both versions of this optimal transmission problem. Comparison of results of the two versions concerning the optimal transmission rates at the origins, the optimal output flow rates at the destination, and the queue sizes at the buffers are obtained. These comparison results clearly reflect how the optimal transmission of messages in computer networks in real life can be affected by both the forward transmission delays of the buffers and the latency of the links.