Is your device accessible through network?

Using serial console (from inside)

Login as root and ping some IP in your local network.
# ping -c 3 This command will send ping packets to for three times.

If previous command run successfully and you're using DHCP with DNS enabled
configured your TPS to use some DNS servers
try to ping some intranet/internet hostname
# ping -c serv.localnet
in case of private DNS or
# ping -c
if global network is accessible from your intranet
to see if name resolution works or not.

Traditional Linux CLI utilities
# ifconfig -a ,
# route -n ,
# ip addr show and
# ip route show may give you exhaustive information about current network interfaces and routing status.

Using network tools (from outside)

In order to make this tests you need to know the IP address of your Linux TPS box.

TPS always have it's own MAC address and in most cases it gets some IP from DHCP or static IP system configuration.

If you know MAC address of your TPS box and it gets IP by DHCP, you may find your device IP in DHCP server listing.

If you know the IP or your box, run:
# ping your_box_ip to determine if your base system is alive.

Successfull ping does not guarantee that your TPS is functional, but it definitely means that your Linux kernel and network stack is running and mostly available.

Next you may try to access your device via secure shell (ssh in Linux or PuTTY in Windows) and HTTP client (via browser).
If you Linux TPS asks for login and password for both SSH and HTTP sessions it means general network services is running and system is mostly functional.