How can I take the report of usage of maps like number of users and the peak time period of usage of maps in server 10.1???
This functionality to monitor server statistics using ArcGIS Server Manager is available out of the box from ArcGIS Server 10.3 onwards. ArcGIS Server records various service statistics, such as total requests, average response time, and timeouts, and reports this information in ArcGIS Server Manager as one or a series of graphs. Administrators and publishers can use this information to monitor service activity and better understand how clients are using services.
About server statistics—Documentation (10.3 and 10.3.1) | ArcGIS for Server
You can also check this link which was posted by a 10.1 user.
Any tool to monitor ArcGIS Server Usage - Geographic Information Systems Stack Exchange
Are we talking external, public facing maps or internal maps? For external, I think using some web stat keeping services like Google Analytics or Statcounter is the way to go. Server Statistics are certainly beneficial but I think not the full picture since one person's experience will end up being multiple requests against the Server as they pan & zoom the map.
Web stats from GA or Statcounter will help track number of unique visits, repeat visits, and technology like desktop vs mobile, browser type, OS, etc.
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