Excessive CPU use, half of hits own IP

My WordPress Multisite installation is behaving strangely, resulting in my ISP suspending my account for excessive CPU utilization.

Cpanel’s AwStats reports for Feb 1-17, that ip xxx.154.238.25 (my site’s public IP) has generated about 14,000 page requests – that is half of the TOTAL page requests for this period. This means half of the traffic on my site is self-generated.

The 24 hour graph of CPU utilization shows two modes: normal usage – an average of 250 CPU seconds per hour, and high usage – an average of ~6000 CPU seconds per hour (that rate don’t make sense, unless this is counting more than one CPU). But the real strangeness is this: normal usage level runs for about 12 hours, then ramps to high usage and stays at the high level for several more hours. Then it returns to normal. This pattern isn’t driven by any actual site visit activity. The graph looks like a Mesa in the desert.

The process admin-ajax.php shows up in the active process list and appears and is listed using many CPU seconds. If I kill that process, there is no change in response to either the public site response or WP admin panel.

I have tried using the Heartbeat control plugin (it didn’t make any difference.)

I have 2 MS instances – one in \public_html\ and the second in \public_html\foo. The seem to be working fine, but I wonder if this configuration might be related to the utilization problem.

Any suggestions to diagnose and resolve the cause of the CPU Mesas would be appreciated.