Does an imessage recipient get the message instantly?

I'm considering to buy an iPhone 5s for imessaging and facetiming with my girlfriend.
Some things are not clear to me about iMessage. My girlfriend thought iMessage works even without internet, but I read that it needs data or wifi, right?
In case cellular data is turned off, but wifi is on, with android, after some time of inactivity the wi-fi is disconnected. Is this the same with iPhone?
If it is the same, what happens when I try to iMessage my girlfriend who has only wi-fi on, but her wi-fi connection is disconnected?
In all the different cases, would my girlfriend get my messages always instantly, like text messages?
Thank you!

My girlfriend thought iMessage works even without internet, but I read that it needs data or wifi, right?
iMessage requires a data connection - WiFI or cellular.
In case cellular data is turned off, but wifi is on, with android, after some time of inactivity the wi-fi is disconnected. Is this the same with iPhone?
When cellular data is disabled in iPhone settings, WiFi remains active indefinitely.
If it is the same, what happens when I try to iMessage my girlfriend who has only wi-fi on, but her wi-fi connection is disconnected?
Not sure what you mean.  If her WiFi is disconnected, how can it be "on"?  If she has no WiFi or cell data connection, no iMessages can be received.

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