Understanding topology of wireless POS terminal card authorization

When a wireless POS terminal is requesting authorization for a
credit/debit card...

Does the wireless terminal communicate to a local/on-site "hub/server"
which in-turn acts as a client to communicate with the EFT network's
authorization host? (ie, the hub/server acts as the authorization
client).

OR does the wireless POS terminal communicate DIRECTLY to the
authorization host, via a wireless access point and then across a
public/private IP network (ie, the wireless terminal acts as the
authorization client).

OR is it sometimes done one way and sometimes the other? And, if so,
what are some examples of vendors that use the two topologies?

Mainly, I'm trying to understand where the authorization "client"
software is running... on a hub/server on directly or the wireless POS
terminal(s). And what are the associated implications on which
run-time environments that the client must be able to operate in...

Thanks!

David T.