Where does the transport layer work?
I would like to know where the Transport layer of the OSI model is in the computer system.
This is not true. The OSI model applies to the OSI protocol suite, which is defunct and does not work anywhere in AFAICS. However, TCP / IP has its own model, which also includes the transport layer. I will assume what you mean hereafter.
Is it part of the operating system?
Yes.
Is it running in its own process or thread?
No, it works as part of the operating system.
How it communicates information to other applications
Through system calls, for example. Berkeley Sockets API, WinSock, etc.
or to other layers?
Through internal core APIs.
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The fact that the OSI model calls the transport layer quite closely matches the TCP layer in TCP / IP. That is, it provides guaranteed delivery / error recovery and transparent transmission between hosts - you don't need to pay attention to how data is routed from one host to another - you just specify the destination, and the network figures out how to get it there.
As far as implemented: well, mostly in the TCP / IP stack, which is usually part of the OS. Modern hardware can implement at least a few bits and pieces in hardware (eg, TCP checksum and flow control). The networking stack will offload those parts of the TCP operation to the hardware through the device driver.
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