I'm pleased you liked my solution and wanted to take it further but I think you've introduced a slight bugette. Because you have used .+? R.+? in your pattern it has matched from the first "R" it encounters after the "QUEUE(...)" sequence so the output from your script is actually

Queue details.QUEUE(MQSI.3PL846) ----> REMOTE)RNAME(MQSI.3PL846)
Queue details.QUEUE(MQSI.3PL944) ----> REMOTE)RNAME(MQSI.3PL944)
[download]

I am not familiar with whatever application produced the text so I don't know if it is wise to rely on the "QUEUE" and "RNAME" being the same. By the same token, I don't know if "QUEUE" and "RNAME" can appear more than once in one line. If they could then a different approach with a global match might be appropriate. However, if they are unique in a line then you can avoid non-greedy matching.

 my $data = '
Timed out (reason: in while loop)
::expect_out(0,string) = >
::expect_out(1,string) = RF use CSWT#RF###
dis qremote(MQSI.3PL846) RNAME1 : dis qremote(MQSI.3PL846) RNAME
AMQ8409: Display Queue details.QUEUE(MQSI.3PL846)TYPE(QREMOTE)RNAME(MQ
+SI.3PL846)
No commands have a syntax error.
AMQ8409: Display Queue details.QUEUE(MQSI.3PL944)TYPE(QREMOTE)RNAME(MQ
+SI.3PL944)
end2 : end
';

my $match = qr
   {(?mx)
       ^
       AMQ8409
       .+
       (QUEUE\([^)]+\))
       .+
       (RNAME\([^)]+\))
   };

for ( split /\n/, $data )
{ 
    print "$1 ----> $2\n" if /$match/  
}
[download]

This produces

QUEUE(MQSI.3PL846) ----> RNAME(MQSI.3PL846)
QUEUE(MQSI.3PL944) ----> RNAME(MQSI.3PL944)
[download]

I hope this is of interest.

Cheers,

JohnGG