The Future Role of the SysAdmin
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Mail queues need to be monitored and cleared, kernels patched, customer issues need to be escalated. When you start getting into the size of a company with over 100 non-technical staff you will find you need at least one or two admins just to keep up with issues that come up.
I currently support over 150 production systems on Amazon, so your estimation of 10 servers per sysadmin is outdated and inaccurate. Even by Instagram's own admission they manage hundreds of systems with only 5 sysadmins. By your calculation, Instagram would have needed at least double that.