I was wondering if anyone could help me diagnose a problem. I am an IT professional with a math degree, so I know there are large gaps in my knowledge.
Here is the scenario: we are a small international school with a basic network configuration using all Apple routers. The base router is an Airport Extreme, and there are 11 access points configured to broadcast the same network schoolwide. In addition, I have two Airport Express routers functioning as extenders, because there was no hardline available where they were needed. My problem is with the extender at the front office.
There is no ethernet cable running near the admin offices, so I needed to setup an extension from the nearest hardline access point. In the beginning this has worked fine, but now I am increasingly running into LOS problems. The extension continues to broadcast to end users, but loses its connection with the hardline router it was extending. This event is irregular, but gaining in frequency.
It should be noted that this extension is doing the most work from among the access points. It is the first access point that every device connects to the network through, because its just inside the front door. In addition, it supports 5 offices , and a wifi-enabled printer which recieves jobs from all the other access points. There is an immense amount of traffic passing through this device (although I have no idea if the load is greater than normal).
Where should I begin looking for a solution? Is there more information I should collect? My temporary solution has been to powercycle the router, but this is becoming increasingly more necessary (had to cycle it twice today). If you have any advice for me, I would love to have it.

I suspect that your hardware, specifically the extender for the front office which is giving the problems, is starting to fail. To verify that, then swap it with one of the others that is not exhibiting these issues. If the problems move with the extender, voila! Time for a visit to the Apple Store for replacement or repairs. If as I suspect you don't have an Apple Store nearby, then you may need to order a new one online and send the bad one in for repairs. This is where I recommend that you keep the old one after it is fixed as a backup.

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