Why is the threshold set to one half of the current congestion window size when the timer expires at the sender?
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1) It's not (always).
2) If it is, it's because TCP falls back and attempts another "slow start" of the connection, looking for a point where there is no packet loss.
Absent any other information may I suggest a copy of TCP/IP Illustrated and perhaps a read through the TCP Congestion Control RFCs(follow references too!) or the short version on Wikipedia which describe some (common) congestion control algorithms?