Do not know if CenturyLink is imposing any limits or otherwise throttling your services. Or if your internet/network setup just cannot handle the load.
Here is the CenturyLink "manual" for your modem:
http://internethelp.centurylink.com/internethelp/modem-pk5001z-setup-basic.html
I do not see any tabs or selections that would lead to the logs (if any - no indications of there being logs). However I cannot see into all of the tabs.
Have you changed the admin login and passwords? Just to keep your network secure and keep others from stealing your services....
Keep looking about via the modem's tab. E.g., "Modem Status" - monitor that via a minimized window and check when things go bad.
Another thing to check is the splitter/filter. You should have a wire coming in somewhere to that device. The wire is split into two wires: one to the telephone and one to the modem/router. No at all uncommon for those wires to be cheap or otherwise sub-standard for the intended use. Likewise for the splitter itself. Check the connections and swap out any wires that are flimsy and/or have loose connectors. Loose either connector on the wire or loose connector into a port. Could be worth trying another splitter if you can get one via Century Link.
Just need to see if the problem "area" can be narrowed down. Cabling, splitter, modem/router, configuration, or PC....