I cannot find out how can I specify a person in a queue, his turn.
I've produced just the sentences below:
the student whose turn came
the student who has the current turn
I wonder that which one is true and is there a better way?
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Your sentences do get the point across, but they both seem a bit clunky to me. Depending on how the phrase is used in context, something like this might work...
the student currently being seen (or served or helped)
the student at the front of the queue
the first in the queue
I'll note that I'm an American English speaker, and we use the word line where others use queue, so it's possible I am misusing queue somewhat.