To answer your question: you may not have to restart SQL Server. You can connect on the admin connection. You do that by prefixing the server\instance with ADMIN:, for instance
By default the Admin connection is only reachable on the machine itself. It is likely that you will get an error message when you connect, because SSMS wants to make a second connection for Intellisense, but the Admin connection is one single scheduler, and there can only be one connection this way.
Once you are connected, you can start to kill processes, if the problem is a deluge of connections.
That said, many experienced DBAs, including people who monitor this forum, would prefer to take the shortcut to just restart SQL Server.