ISP Failed, Internet Connection Intermittent

For a long time, my internet connection has gone through phases in which pages, images, videos, etc. stop loading something like every 1-2 minutes. I have seen gifs start loading incredibly quickly, in real time almost, then stop dead. Refreshing in this case yields an endless period of loading. I am fairly certain that this problem extends to all devices on my wireless network, but I rarely use two computers simultaneously, so I'm not certain of this. The problem is usually present for a few weeks at a time, then is stops happening for a month or two, if I remember correctly.

Whenever pages fail to load, I have found that, by opening Network Diagnostics (System Preferences>Network>Assist me…>Diagnostics…), I can temporarily restore internet connectivity. The Network Diagnostics window usually has orange lights for ISP, Internet, and Server when I first open it, though ISP's light is sometimes red. As I proceed through the program, the lights usually change after I select my network from the list. Network Settings will sometimes turn red, at which time all the lights below it will turn red as well. Whether or not this happens, though, the lights end up as green, and I am told that my internet connection seems to be working correctly. Most of the time, my internet connection will then work for some time (the length of which may vary). I have pored over all of Apple's network troubleshooting articles, none of which have resolved my issue. I have restarted my router and modem in the past, though not recently. I do not believe that the problem lies in either my iMac or wireless router (Time Capsule, 2 terabytes, 4th generation or earlier), as my modem (or possibly cable splitter) seems the liklier culprit, but my landline phone service and cable television service has not seen any service disruptions (all 3 services operate over the same cable, and are provided by the same company).

During a recent interruption of service, I used the ping command in Terminal to try to connect to the 8.8.8.8 server. The full text of the command and its output is as follows:

Rays-iMac-8:~ lorenzocastoldi$ ping 8.8.8.8

PING 8.8.8.8 (8.8.8.8): 56 data bytes

ping: sendto: No route to host

ping: sendto: No route to host

Request timeout for icmp_seq 0

ping: sendto: No route to host

Request timeout for icmp_seq 1

ping: sendto: No route to host

Request timeout for icmp_seq 2

ping: sendto: No route to host

Request timeout for icmp_seq 3

ping: sendto: No route to host

Request timeout for icmp_seq 4

ping: sendto: No route to host

Request timeout for icmp_seq 5

ping: sendto: No route to host

Request timeout for icmp_seq 6

ping: sendto: No route to host

Request timeout for icmp_seq 7

Request timeout for icmp_seq 8

Request timeout for icmp_seq 9

Request timeout for icmp_seq 10

Request timeout for icmp_seq 11

Request timeout for icmp_seq 12

Request timeout for icmp_seq 13

Request timeout for icmp_seq 14

Request timeout for icmp_seq 15

Request timeout for icmp_seq 16

^C

--- 8.8.8.8 ping statistics ---

18 packets transmitted, 0 packets received, 100.0% packet loss

I followed this with a traceroute command to the same server.

Rays-iMac-8:~ lorenzocastoldi$ traceroute 8.8.8.8

traceroute to 8.8.8.8 (8.8.8.8), 64 hops max, 52 byte packets

1 * * *

2 *traceroute: sendto: No route to host

traceroute: wrote 8.8.8.8 52 chars, ret=-1

*traceroute: sendto: No route to host

traceroute: wrote 8.8.8.8 52 chars, ret=-1

^C

I can't find much on this "sendto: No route to host" message, but the few forum posts I was able to find seemed to be made by users of the Linux operating system, or variants thereof, making me even more certain that the issue is with my modem, cable, or ISP. After running Network Diagnostics again, I got this

Rays-iMac-8:~ lorenzocastoldi$ traceroute 8.8.8.8

traceroute to 8.8.8.8 (8.8.8.8), 64 hops max, 52 byte packets

1 10.0.1.1 (10.0.1.1) 2.198 ms 1.216 ms 2.369 ms

2 * * *

3 67.59.227.133 (67.59.227.133) 31.147 ms 8.546 ms 10.267 ms

4 rtr2-ge1-18.mhe.whplny.cv.net (67.83.231.5) 15.358 ms 22.261 ms 12.545 ms

5 65.19.121.217 (65.19.121.217) 14.326 ms

65.19.121.209 (65.19.121.209) 16.525 ms

r1-ge6-1.cst.nrwlct.cv.net (65.19.121.161) 14.141 ms

6 451be0d6.cst.lightpath.net (65.19.120.214) 12.331 ms 17.755 ms

451be0ca.cst.lightpath.net (65.19.120.202) 14.891 ms

^C

for traceroute, and this

PING 10.0.1.1 (10.0.1.1): 56 data bytes

64 bytes from 10.0.1.1: icmp_seq=0 ttl=255 time=2.135 ms

64 bytes from 10.0.1.1: icmp_seq=1 ttl=255 time=2.478 ms

64 bytes from 10.0.1.1: icmp_seq=2 ttl=255 time=2.179 ms

64 bytes from 10.0.1.1: icmp_seq=3 ttl=255 time=1.058 ms

^C

--- 10.0.1.1 ping statistics ---

4 packets transmitted, 4 packets received, 0.0% packet loss

round-trip min/avg/max/stddev = 1.058/1.962/2.478/0.539 ms

for ping, in that order. I believe I've heard that the second entry in traceroute, with the three asterisks, is my router. Whenever I run traceroute, entry 2 is always like this, and I think this is the case for other people as well. This would mean, if I am not mistaken, that the first IP address, 10.0.1.1, is the weak link. However, I do not know how to proceed from here. When I traceroute other websites, it always appears as IP address number one, and it is contacted before my router, if #2 is in fact my router. I have read that 10.0.1.1 is also a private IP address, though I do not know exactly what that would mean for me. I am tempted to think that this IP address may be my computer itself, but that wouldn't make sense. It may be my router, again, but I do not know this for certain.

Because this is an intermittent problem, it can be difficult to examine it in any detail, but I should be able to answer most questions about it. If anyone has any potential solutions, or even hypotheses as to the nature of the problem, please let me know. I cannot solve this problem on my own.

tl;dr: What is the 10.0.1.1