Hey David Horsey,
So you have already checked the ArcGIS Portal Status Page? View status reports—ArcGIS Online Help | Documentation
The content tab shows most popular and number of hits. If that doesn't suit your needs then you can write a python script to parse the IIS log files and count the number of requests each service gets.
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The client for whom I'm writing the report has just upgraded to Portal 10.7.1, so I wasn't aware of the breadth of information available in the Status tab. I think that'll be useful in the short term, longer term I'm looking to automate the process and others have suggested System Log Parser and the ArcGIS API for Python for ways to extract that information.
Hello David Horsey,
You may also want to look into ArcGIS Server statistics. This functionality is mainly accessed through the ArcGIS Server manager statistics tab. It helps provide information regarding, total requests, average response time and timeouts.
Hope this helps.
Thanks Thomas Jones. The Statistics are useful and in my experience are great for quickly seeing the general usage, but due to the number of map services I'm dealing with, it's not practical in this case to use it to identify the most popular map services.