How Much Time spent on websites of user
Seems the Time Spent  module is what you're looking for

This module tracks, by Ajax, how long a registered user took to read a page and the amount of time he spent on the entire site.
You can configure which content type and role should be tracked and view the report, exhibiting the total hours (or minutes/seconds) a user spent reading a page and at the entire site.

Similar module -Time Tracker

Features

Track Time on any entity and or comments
Activities and Time Entries are now both fieldable entities.
Track time using time intervals (start and end times) or durations    (admin/settings/time_tracker)
Integrated Timer to track time as you go
Time estimating
User Time Sheets to display weekly logged time (user/%uid/time_sheet)
View for reporting on tracked time (time/report)