Question

What happens if an application is waiting for a push notification and the internet connection is unavailable?

For example, I tell my server to do task “A”, and notify me via APNS when the task is ready. But, after I send the task to the server, the application loses internet connectivity.

Answer 1

Notification will be delivered when internet connection on iPhone becomes available again.

Apple Push Notification Service includes a default Quality of Service
(QoS) component that performs a store-and-forward function. If APNs
attempts to deliver a notification but the device is offline, the QoS
stores the notification. It retains only one notification per
application on a device: the last notification received from a
provider for that application. When the offline device later
reconnects, the QoS forwards the stored notification to the device.
The QoS retains a notification for a limited period before deleting
it.

See “Quality of Service” section of Local and Push Notification Programming Guide.