I couldn't take the sub calculate_time as written. Below is a version w/ duplications reduced, does not solve the actual problem of printing data in the intervals (see OP).

#  Jun 20 2003 - create hash w/ "map" instead of explicit creation
my %MonthToNumber;
@MonthToNumber{qw (Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec)} =
  map { sprintf "%02d" , $_; } (1..12);

my %NumberToMonth =
  map { $MonthToNumber{$_} => $_ } keys %MonthToNumber;

sub calculate_time {
  my $get_sec =
    sub {
      my @time = reverse @{ parse_date($_[0]) };
      return
        #         second, minute, hour, day     month          year
        timelocal(@time[0..(scalar @time -3)] , $time[-2] -1 , $time[-
+1]);
    };

  my ($begin_sec , $end_sec) =
    ( $get_sec->($begin_time) , $get_sec->($dateproc) );

  my $elapsed = $end_sec - $begin_sec;

  #printf "BEGIN: %s(%s)  END: %s(%s)\nELAPSED: %s\n"
  #  , $begin_sec , $begin_time
  #  , $end_sec , $dateproc
  #  , $elapsed;

  if ( $elapsed < $interval ) {
    push (my @visual_page_values, {$processed_visual_pages{$dateproc}}
+);
    print "The End seconds are: $dateproc  @{$processed_visual_pages{$
+dateproc}}\n";
  } else {
      $begin_time = $dateproc;
      push (@final_visual_pages, $dateproc);
      print " Final Visual pages are: @final_visual_pages\n";
  }
}

sub parse_date {
  my $date = shift;

  return [ ] unless defined $date;

  my ($day, $month, $year, $hour, $minute, $second) =
      split '[/:]' , $date;

  return
    [ $year , $MonthToNumber{$month} , $day
    , $hour , $minute , $second
    ];
}
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Other Notes (Jun 20 2003):

• If parse_date() is not going to be used elsewhere, contents of the returned array reference should be reversed (to avoid reverse()-ing later for timelocal()).
• Similar like above, if %MonthToNumber is used for the sole purpose to convert a month name to number for timelocal(), one could just use the hash values 0-11 instead of 1-12. In which case there also would be no need to use sprintf. More importantly, @time can be passed as it is, w/o the need of adjustment to any individual value.
• %NumberToMonth seems unnecessary if/when it is employed few times, for some definitions of few.