Calling relative's cell phone in Mexico?

When you're calling the cell phone of someone who's in Mexico(an AT&T phone that was purchased in the US)and there's a longer-than-usual pause,two audible "clicks" sounds,then the air-like sound that always starts before the phone beeps,but it doesn't beep,instead just a long airy-echo-like pause before going to voicemail?I know the phone is turned on,because when off it goes straight to voicemail without the long pause and the echo-chamber sound,only it doesn't beep.Does it mean there's no service or what?