You should use the warnings and strict pragmas so the second and third lines of your program should be:

use warnings;
use strict;
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chomp($pid  = $$);
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$$ does not contain a newline so there is no point in using chomp().

$LOCK_EXCLUSIVE = 2;
$UNLOCK         = 8;
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You should use the constants from the Fcntl module:

use Fcntl ':flock';
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$fifo             = "ENV{HOME}/a_file";
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You probably meant to use the %ENV hash there:

$fifo             = "$ENV{HOME}/a_file";
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close(FILE) || warn "update queue file exited $?\n";
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The $? variable will only contain useful information if you are running an external program. which you are not. However, you should include the $! variable in any system error messages.

close(LOCK) || warn "lock file exited $?\n";;
$str=`cat $fifo.lock`;

flock LOCK, $UNLOCK;
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close(LOCK) will also unlock the file so the subsequent attempt to unlock it is superfluous.

system("rm $fifo.lock");
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Is there a reason that you couldn't use the built-in unlink function?

Have you read the entries in perlfaq5 on file locking?

You are locking the PID file but not the data file. This may work better:

#!/usr/local/bin/perl
use warnings;
use strict;

use Fcntl ':flock';
use Tie::File;

my $fifo = "$ENV{HOME}/a_file";
my $lock = "$fifo.lock";

tie( my @pid, 'Tie::File', $lock )->flock( LOCK_EX )
    or die "Cannot open '$lock' $!";

@pid = $$;

tie( my @lines, 'Tie::File', $fifo )->flock( LOCK_EX )
    or die "Cannot open '$fifo' $!";

( my $pop, @lines ) = sort @lines;

print "$hostname is poping $pop\n";

untie @lines;

print "PID lock file is going to be removed\n";

untie @pid;
unlink $lock or die "Cannot unlink '$lock' $!";

__END__
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