A business phone system is often referred to as a PBX, which stands for Private Branch Exchange. A PBX is responsible for connecting phones and voice applications such as an auto attendant, voicemail box, etc. Today, a PBX is telephony software that runs on a server in your office or is hosted by a third party. Most IP PBXs and the phones that work with them use the SIP protocol, a communications protocol for signaling and controlling multimedia communication sessions.
Whether or not an IP PBX is hosted or in your office, it will need to be connected to the Public Switched Telephone Network (PSTN) for outbound dialing. The PSTN is not all VoIP-based (or SIP-based), so one must use a SIP trunking service to connect the PBX to the PSTN. With hosted PBX service, the PBX and SIP trunking service are kept offsite and maintained by the provider. With SIP trunking, the PBX is installed and managed on-site by your own IT staff, and you are also responsible for purchasing and maintaining a SIP trunking service.
So in short: a hosted PBX service is a complete phone system solution that connects to the PSTN and is maintained by a third party. SIP trunking is a gateway for your on-premise PBX to an Internet Telephony Service Provider (ITSP).
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