Although both the subject and the description of the problem may suggest that he's after something like the code you suggest, I don't think he really wants to split his input into several different files. I think it's more reasonable to assume he's after something along the lines of:

#!/usr/bin/perl

use strict;
use warnings;

$\ = $, = "\n";
chomp(my $head=<>);

my ($last,@buffer);
while (<>) {
    chomp;
    my $time = (split)[0];
    if ( $time eq ($last ||= $time) ) {
        push @buffer, $_;
    } else {
        print $head, @buffer;
        ($last,@buffer)=($time,$_);
    }
}
print $head, @buffer;

__END__
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Note that this assumes the input is ordered, which seems reasonable too, if you read the description carefully. Of course I do not claim it to be the cleanest or smartest way to do it, either.