DOes the error indicate if the problem is that the alarm systme cannot call the unit (e.g. sometimes cannot reach that DID) or that the unit cannot always call out.
If the former you may not have enough inbound trunks.
You say this is in a "block of DID's". Do you mean that it comes in on a T1? If so how are the outbound calls routed?
Alternaitvely, if the error codes menas the dialer cannot call out, then possibly you have insufficient outbound trunking.
If this is T1 then you may be able to configure "two way DID" or even better ISDN PRI in which case all channels would be available for both inbound (using DID numbers for routing) and outbound.
One possibility is that during the busy hour the alarm system is getting an "all trunks busy" due to no available outbound trunks. So, look at the times and dates and look to see if this may only occur during your busy hour.
If this is the case freqently, it would seem that others within the company are probably losing productivity due to lack of trunks for outbound calls.
IN that case you need to look at more trunks, or creating a two-way trunk group (if this is not the case currently) if possible.
Another possibility is that there is either
1) A device wired in parallel to the alarm system on the alarm system side of the jack
or
2) The alarm system jack is incorrect and does not loop the line through the alamr dialer - if the RJ 31x is wired properly, other devices that share this line/PBX port loop through the dialer. It can then interupt the other equipment (forcing the line on-hook) and then go off hook and dial out.
3) The final possibility is simply an intermittent on the PBX port and or line and or inside wire.
good luck