Thanks for the feedback, everyone.
We're pretty sure that the slowdown, when it occurs, has very little to do with any normal activities or operations of the board. With our current configuration, when things are working, they're working just fine.
Without too much detail ... The Chiffboard runs on 2 different servers: 1. A front-end http server for phpBB, implemented as a virtual private server (VPS) hosted on a shared CPU; and 2. A back-end mySQL server, also shared, which holds the databases. The board data is fairly large by now, around 3 GB.
There's a correlation between the responsiveness of the board and CPU memory usage, which skyrockets to 2 or 3 times "normal" during slowdowns. Ordinarily, this might suggest a software memory leak. But re-booting our servers has no effect, so it's unlikely that's the root of the problem.
More likely, we think, the cause lies with our shared resources. That is, other users or sites sharing our CPUs and sucking up all available memory, cycles, and bandwidth. This would account for the seemingly random nature of the problem. If another site experiences excessive usage, or their software begins firing off rogue threads and processes that are left unchecked on the physical CPU, the Chiffboard grinds to a crawl. It's like a DOS (denial of service) attack as a side effect of our VPS configuration.
Yesterday we moved the databases to a different back-end SQL server with smaller network latency (faster response).
Dale and I have been discussing other steps we hope to take after the holidays to address the problem, and we'll keep you posted. Meanwhile, keep thinking happy, happy thoughts.

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Vivat diabolus in musica! MTGuru's (old)
GG Clips /
Blackbird ClipsJoel Barish: Is there any risk of brain damage?
Dr. Mierzwiak: Well, technically speaking, the procedure
is brain damage.