Splunk doesn't use IIS or any other packaged web services, it has it's own, self contained web-service that doesn't rely on any external programs/resources.
To provide access on a public URL, you need to map that URL to the public IP address and port of the SplunkWeb service running on your machine. When Splunk is running, it can usually be accessed at http://localhost:8000 from the local machine, or http://servername:8000 and http://10.0.0.1:8000 from a remote machine, assuming that 10.0.0.1 is your machines IP address - obviously your real IP will be different, this is just an example.
The steps to map the public URL to your machine will depend on your ISP, DNS provider, firewalls etc, so I can't give you those here. However, if you're familiar with how to do this for a regular IIS website, then the process shouldn't be very different. Bear in mind that the default port for an IIS service is 80, while Splunk uses 8000.
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