We adopt the optimization framework for the rate allocation problem proposed by Kelly and investigate the stability of the system with arbitrary communication delays between network elements with time-varying queue dynamics. We first present the conditions for the existence of a solution of the system. Second, we establish the conditions for the system stability with arbitrary delays for a family of popular utility and price functions, and then extend the results to more general utility and price functions. We demonstrate that the stability of such a system can be studied by considering a discrete time system derived from a simpler homogeneous delay case where all users have the same fixed delay. Numerical examples are provided to validate our analyses.