What version of Kentico are you running? That sounds like a bug.
Another possibility: Is your server perhaps going into sleep mode to save CPU when there is no traffic on the site? Some hosting services will put the site to sleep if there's nobody connected and will wake when the site gets a hit. This is typically if you have a hosted VM shared with other services, and it costs less. If this is the case you'd need to request to have the sleep mode removed from your VM.
In the event logs are you seeing any messages about the application stopping/starting when the user logs in?
If the server isn't stopping your app, it's almost certainly a bug because the scheduled task should run on time and you might need to fill out a bug report.