I know the Inquirer is about getting the scoop first, but this article is pure misinformation.

Apple was promising push notifications as a work-around for background processing, NOT background processing itself. Apple's hardware/software is capable of background processing, but they won't allow 3rd-party developers to run applications in the background for their various (not necessarily good) reasons.

My good old trusty Psion series 3 organiser had background processing capabilities back in the 1990s. 

There were system calls that would wake the application, and if necessary, turn the PDA back on so the app could do its processing and then go back to sleep.

Yes, this required hardware that had the ability to turn the unit on when a timer expired. 

One would expect any PDA to have the ability to wake up/turn on when it is time to ring an alarm.