Time Machine. I use Synology's "Time Backup" which does file versioning for as many versions as you specify, and for which folders / shares you choose. I'm not Apple-literate.
Time Machine is the built-in file backup/imaging program for OS X. My guess is from reading your descriptions of Time Backup, Synology has done a very good job of mimicking Time Machine's functionality, however that's a NAS-based tool, it's not Time Machine.
Using OS X Time Machine with a network share requires a few things. For it to work reliably, it needs to use AFP. Both Synology and QNAP support AFP.
However, QNAP's Time Machine module only supports one user and that user is named "Time Machine". It's quite limited in its implementation. Not only does it cause issues with multiple machines/users, it prevents a single user from using other AFP shares on the same NAS. I'm really displeased with QNAP's Time Machine support (or lack thereof) and if they don't fix it, I'll probably be going Synology the next time around.
Over here, the QNAP's are not going anywhere - but the Apple products are becoming more and more scarce (on their own).
I think you're planning to replace the wrong device(s).
I never had problems with Time Machine on Synology, also it did not prevent using other shares on the NAS. I don't know how they implemented it (don't have it enabled on the Qnap)...
But personally in the same case I would definitely not be replacing my computer just because a NAS would be implementing a feature badly if I had an option of replacing the actual NAS... I have yet to see a backup solution that would work as easily and well as time machine (I still prefer not to use it through a hack).
We use the BTRFS filesystem in ReadyNAS OS 6. BTRFS is a great filesystem, especially when it comes to data integrity. We've got features such as unlimited snapshots and bitrot protection against media degradation. These are great features and assist with protecting your data, but backups are still important. Don't store important data on just the one device. Note that if the primary copy of the data is on your PC then the copy on a NAS is a backup.Netgear has the better backup file system
Unfortunately, QNAP uses a special user account (TimeMachine) for Time Machine functionality
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