You can check all invited customers by editing the survey and then going to Customers tab within the survey. You should find a list of customers there. The status of the single invitation will be updated (invited, started, etc).
For more information about the questionnaire statuses please see here.
Invites are sent only when the survey status is set to active. The automated import of invitations is usually scheduled from 10:00 am to 8:00 pm. Manually started imports will be executed immediately.