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Fair-share scheduling is a method of allotting CPU time to multiple processes in a defined way. When multiple processes need a CPU, operating systems frequently allot CPU time slices on a round robin, equal time basis for processes with the same priority. An exmaple example is that if there are 4 processes wanting the CPU then each process will get 1/4 of a second every second. This is not to say that each process will get 1/4 of a second all at one time. Most operating systems define a time slice, something like 50 msec (arbitrary number that doesn't necessarily represent any OS) and will give each process a time slice. So in this completely made up case, Each process will get 5 time slices every second.
Long ago someone said that we need a way of giving people with legitimate requirements more of the time than less needy people. "Need" often reflects what level of service the user wants to pay for. The fair-share scheduler was born. There have been a number of implementations over the years, but on of the most common was to simply assign weights to users such that one user may get twice as many time slices in a given time period as others. This has been expanded often to groups so that processes belonging to a group of users would be given the same number of time slices as another group.