Can you explain this exercise solution to me? A process running on CTSS (Compatible Time Sharing System) needs 30 quanta to complete. How
many times must it be swapped in, including the very first time (before it has run at all)?

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Answer : The first time it gets 1 quantum. On succeeding runs it gets 2, 4, 8, and 15, so it must be swapped
in 5 times.

why does it get 2,4,8,15 on the succeeding runs?
According to this article: F. J. Corbató, M. M. Daggett, R. C. Daley, An Experimental Time-Sharing System (IFIPS 1962), CTSS uses a "multi-level scheduling algorithm":


  The basis of the multi-level scheduling algorithm is to assign each user program as it enters the system to be run (or completes a response to a user) to an $\text{ell}$-th level priority queue. The process starts with the time-sharing supervisor operating the program at the head of the lowest level occupied queue, $\text{ell}$, for up to $2^\text{ell}$ quanta of time and then if the program is not completed (i.e. has not made a response to the user) placing it at the end of the $\text{ell}+1$ level queue. Programs are initially entered into a level $\text{ell} = 0$.


A program at the ell-th level is assigned $2^{\text{ell}}$ quanta.

Now you know how to solve your exercise.