I think the idea here is that you can have a process, e.g. a nightly update or batch update, that populates a stored field with it's unstirred equivalent. Performing this over the LAN/WAN is much slower when using PSOS.
Users would search on the stored/indexed value which is populated via a PSOS as necessary.
It's also the asynchronous nature of PSOS.
Go to a little used part of the system. Trigger an update script which will go and touch all records. You don want to wait for that to finish before you can move on. Enter PSOS with the Wait For It To Finish button untucked. You can carry on, it does its work, but on demand, rather than writing a schedule for it - and you don't necessarily want the client anywhere near that.
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Searching on stored values will always be faster than searching on unstored values, it has nothing to where the script executed . Is there something I don't get ?
The text makes an implicit jump to the conclusion; what they are trying to say is that a PSoS script can be fired from a client to update those stored values faster or even without making the client wait for the script to finish, and at the same time - when the script is done - make the system behave faster because everyone can work with the stored values.