For domestic award travel on United, I call the Premier Executive desk and almost never wait. The phone doesn't even ring, the agent just picks up.

For international travel, I call the Premier Executive desk anyway, tell the agent that I want to ticket an international award, and they transfer me immediately to an international desk agent. Again, very little wait in between. The one time I called the international awards desk directly, I was on hold so long that I forgot I was on hold. The other thing that helps is that I call at some pretty strange hours sometimes (1 AM EDT, for example).

I dread the day that I might no longer have elite status with United. Let's just hope I hit Million Miler first...