The code below will do what you want.

Several things could be done to improve the code: 1) the pid could be output with the queue numer ($lock) so that the kill 0 recomened by RazorbladeBidet could be used; 2) better exit handling is needed so that improvement 1 is not needed; 3) the queue (countfile) could have a more robust structure.

use strict;

use Fcntl ':flock';
use Time::HiRes qw ( usleep );

sub check_queue {
    my $lock = shift;
    my $top;
    open(FILE, "<countfile") || die;
    flock(FILE, LOCK_SH);
    seek(FILE, 0, 0);
    $top = readline FILE;
    flock(FILE, LOCK_UN);
    close FILE;
    return $lock == $top;
}

sub add_to_queue {
    my $lock = 0;
    if (open FILE, "+<countfile") {
        flock(FILE, LOCK_SH);
        seek(FILE, 0, 0);
        for (readline FILE) {
            $lock = $_;
        }
        $lock++;
    } else {
        open(FILE, ">>countfile") || die;
    }
    flock(FILE, LOCK_EX);
    seek(FILE, 0, 2);
    printf FILE "%d\n", $lock;
    flock(FILE, LOCK_UN);
    close FILE;
    return $lock;
}

sub remove_from_queue {
    my $lock = shift;
    my @procs = ();
    if (open FILE, "+<countfile") {
        flock(FILE, LOCK_SH);
        seek(FILE, 0, 0);
        for (readline FILE) {
            if ($_ == $lock) {
                next;
            }
            push @procs, $_;
        }
        flock(FILE, LOCK_EX);
        seek(FILE, 0, 0);
        truncate(FILE, 0) || die;
        print(FILE @procs);
        flock(FILE, LOCK_UN);
        close FILE;
    } else {
        die;
    }
}

my $done = 0;
my $lock = add_to_queue();
eval {
    open LOCK, "./lockfile" || die;

    while (!$done) {
        flock(LOCK, LOCK_EX);
        last if check_queue($lock);
        flock(LOCK, LOCK_UN);
        usleep 50000;
    }
    print STDERR "Doing work on $lock ", scalar localtime, " \n";
# Do your work here
    sleep 3;
    print STDERR "Done working on $lock ", scalar localtime, " \n";
    remove_from_queue($lock);
    flock(LOCK, LOCK_UN);
};
if ($@) {
    remove_from_queue($lock);
}
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