In a single-processor architecture, A process might be New, Ready, Running, Blocked and Terminated.
Are there any additional states in a multi-processor architecture?
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More states aren't technically needed, depending on an operating system's implementation.
A kernel can have a task scheduler running on each processor. This allows for the use of the same process states, since each scheduler is running independent of each other. The kernel can have "load balancing" between the processors so that not too many processes are queued-up on one processor
xv6 (UNIX V6 port for x86 architecture) is a bare-bones kernel that does just this, and so, it has the same process states like in a single-processor environment.