I think the OP was looking for their server name or IP address to log in for the transfer, etc, but I suppose that's good information to have anyway.
Personally, I think the actual server specifications are less relevant on shared hosting because more powerful servers would just have more customers loaded on them. In fact, I'd bet that within Dreamhost itself this is true between the older, less powerful servers and the newer, shinier ones.
Even simple metrics like "how many shared users are on the server" aren't really very good. I'd guess that the 80/20 rule applies and that 80% of the resources are being used by 20% of the users. Actually, I suspect the ratio is more like 98%/2%.
In some ways, the shared hosting is a red herring in that the server itself can be looked at as a giant web server and managed that way. You manage to peak load and try to ensure that the server can handle the load at those times. In some ways, shared hosting can be more efficient because the peaks and valleys should even out across websites with varying usage profiles - though I suppose there's still demand-side peaks that apply across all sites.
Web hosting is pretty demanding from the perspective of acceptable load. By this I mean that there's little difference between a slowdown and an unacceptable slowdown. What kinds of slowdown did you experience? Chances are that slowdowns are the result of a runaway neighbor and a support call should be in order to help save everyone else on the server.
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