I don't like inquiries that lead nowhere so, although I had satisfied my curiosity, I posted my comments so that others might benefit from what I had learned. Over the years, from time-to-time, my computer has been telling me to use Network Diagnostics, but I didn't know where to find it, so I didn't pursue the matter. This time, I decided to make a greater effort. What I found, I posted, although I had expected more. Maybe Network Utilities is the app to use, but since I don't know how to use it, so I can't comment further on that app; however, I should have mentioned the following:
(1) When you click on the OPTION button together with the Wirelesss icon in the menu bar, the extended dropdown menu you get contains a link entitled, "Create Diagnostics Report.".
(2) I was troubled by the first entry in the above dropdown menu because it kept alternating between "Wi-Fi:On" and "Wi-Fi: Looking for Networks." My thought was, "Make up your mind. If it's on, why are you looking for Networks"? The technical advisor at Linksys told me that this is normal and that nothing was amiss. I still don't quite understand but, apparently, even though the wireless is on, it still keeps looking for Networks. I don't understand, but I'll just accept that.
My new router is working fine now, although that's not due in any way to any of the Diagnostics apps mentioned above.