How much time does it take from user pressing one of the 4 Internet Button buttons, Particle cloud receiving the button# (by cellular or WiFi), and finally the iOS device getting notification from cloud which button pressed?
We have chosen Particle as the platform for demonstrating our exciting product at CES 2017.
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I would say in the milliseconds range.
@jonlogan can we do something to supoort them? 
That’s a rather broad question.
Cellular and wifi will each have different latency due to the different technologies involved.
(Furthermore, the Internet button doesn’t accommodate an electron, so cellular would probably not be an option, unless you make a PCB of your own).
Receiving a notification on an iOS device can be done in many ways. Are we talking native notifications, or third party ones? That’ll make a difference.
Depending on the answers above, the total latency will be a sum of all the components above, and will be subjected to the ‘quality’ of Internet inbetween. There’s no one right answer to that question, as it depends on a great many deal of factors.
All that in mind, assuming you’re in a ‘normal’ environment, I’d say… Less than a second?