This should do the trick.

my %idx;
open FH, "logfile" or die "can't open logfile";
while (<FH>) {
    my ($ser,$msg) = $_ =~ /^(.+?-.+?)\s+(\S+)/;
    $idx{$ser}->{$msg}++;
    if ($idx{$ser}->{begin} && 
        $idx{$ser}->{doing-work} &&
        $idx{$ser}->{complete} ) {
        # found all the stuff, remove the entry.
        delete $idx{$ser};
    }
}
print "didn't complete:\n";
foreach my $ser (keys %idx)
{
    print "\t$ser\n";
}
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