This paper analyzes influences of global and local cues on the synchronization rate of inter-connected oscillator networks subject to time delays. The coupling between oscillators is assumed to be directed, which includes the symmetric bi-directional coupling as a special case. In order to study the rate of synchronization, we first give a synchronization condition, which guarantees synchronization of oscillator networks independent of delays. Then we prove that the rate of synchronization always benefits from a stronger global cue, namely a stronger global cue leads to a faster rate of synchronization. The role of the local cue in the rate of synchronization is proven to be more complex: when the delays are below a certain value, the local cue has no influence on the rate of synchronization, but when the delays are above the value, a stronger local cue lowers the rate of synchronization. Finally, simulation results are given to illustrate the theoretical results.