We adopt the optimization framework for rate allocation problem proposed by Kelly and characterize the stability condition with an arbitrary communication delay in the case of single resource. We demonstrate the existence of a fundamental trade-off between users price elasticity of demand and the responsiveness of resource through a choice of price function as well as between system stability and resource utilization. We investigate the effects of non-responsive traffic on system stability and show that the presence of non-responsive traffic enhances the stability of system. We also investigate the system behavior after the system loses its stability.