There are two ways to acces the LSP. If you are connected to the network, and there is no network issue preventing you from accessing the site where the LSP is located, you can telnet, use ASA, or web browse into it over the LAN/WAN.
If network connectivity is down, there is a modem you can use to dial into the S8300 LSP. Then use a PPP connection to telnet, use ASA, or web browse into the S8300 LSP.
The "display lsp" command will list all LSPs adminstered on the system and show the service state. Properly connected LSPs will be "in-service idle" or something like that. If they are in "out-of-service" state, either network connectivity was lost between the S8700 and the LSP and the remote site is in LSP mode, or there is a problem with the LSP communicating to the S8700 (system shut down, network issue, license error, etc.)
A Local Survivable Processor (LSP) is actually an S8300 processor in either a G700 or G350 Media Gateway and needs to have an S8700/S8710 or S8500 primary Media Server. The S8300 will run for 144 hours in LSP/license error mode if the site looses connection to the primary Media Server.