What I get from you description is:
Time 0:
Process 1 starts using its time slice, e.g. 5 units.
Time 5:
Process 2 arrives.
Process 1 used up its time slice and is replenished.
A round robin scheduler checks the ready queue by selecting the next process with time left.
At time 0 your ready queue looks like this:
P1
At time 5:
P1 going over to P1 -> P2
Tanenbaum writes in Modern Operating Systems: When a process used up it's time slice, it is enqueued at the end. So what happens is, that P1 is enqueued again before P2 is enqueued for the first time. This saves an additional context switch from P1 to P2 and back to P1.
If you want P2 to be run first, then you can use priority base round robin.
