• Work in minutes, not HH:MM
• read arrival times into an array

which leads to something like...

use constant START_MINS => 180;
use constant PERIOD     => 20;

my @arrivals = qw/03:05 03:17 03:23/;

@arrivals = map { as_minutes( $_ ) } @arrivals;

for my $time ( START_MINS+1 .. START_MINS+8 ) {
    my @arrivals_this_period = grep
      { $_ > $time && $_ <= $time + PERIOD } @arrivals;
    print as_hhmm( $time ) , ": ", scalar @arrivals_this_period, "\n";
}

sub as_minutes {
    my ( $hh, $mm ) = split /:/, shift;
    return $hh*60 + $mm;
}

sub as_hhmm {
    my $minutes = shift;
    my $hh = int $minutes / 60;
    my $mm = $minutes - $hh * 60;
    return sprintf "%02d:%02d", $hh, $mm;
}
[download]

Output:

03:01: 2
03:02: 2
03:03: 3
03:04: 3
03:05: 2
03:06: 2
03:07: 2
03:08: 2
[download]

There's still the problem of your 20 minute period crossing midnight though

update: Fixed the for my $time ... line to match the output I gave