Trying to Find Out What's Overloading My Modem/Router

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Of course you can start looking through a microscope to see where the firehose is (assuming there is one), this approach sounds to me as a step to far. 1) we don't know what to look for, since we do not know the inner workings of the home gateway proprietary firmware, only the exhibited behaviour, and we don't know the baseline traffic. that is the traffic patterns before the issues started to pop up. 2) having no experience with Wireshark (and network packet capture in general I assume), there's a learning curve for this power tool.

If an expert were to be looking at successful capture (made preferably on the home gateway itself) (s)he would have to look at every protocol in there to see what stands out, if anything. OTOH, I've seen devices go nuts because of in itself normal, but for the device unexpected protocols.

Even though it's not bad to learn using Wireshark for this (but our view may be biased :) ), I would look at taking a different step first. Since you don't know which of your network devices causes this, take each one offline in turn and see how the home gateway behaves. I know this can be tedious and annoying, but so is the acting up home gateway. Once this source is isolated it can be investigated further, depending on what type of network device it is.