I remember an install about 15 years ago we performed for a lady who was a little irrational… She was being attacked by aliens, and when she called 911 and the police showed they would go up in the attic and hide behind the insulation where no  one could find them. We contacted every exterior opening, the master bedroom door and master closet door, glass breaks in every room with a window, 4 or 5 motions in a 2500’ foot house, contacted the attic hatch, 3 sirens in the attic, keypads at every exit and 1 in the bedroom, several panic buttons, 2 wireless panic- in case 1 didn’t work. We spent almost 2 hours showing her how to use the system, under every scenario she could come up with (she even wanted us to test the panic button while standing in the middle of the pool in the back yard, I refused, but we showed it worked on all sides of it).     This lady was well known by the local police, she called them daily. To shorten this story, she set of the panic button before we even made it back to the shop that evening, and almost weekly thereafter. I guess the moral of the story is, if you have been in this business long enough you learn to pick your battles. This lady paid very well, although she was crazy in my book. If it walks like a duck and talks like a duck….probably not a cat!
    We had another one that was even worse. 2 competing brothers in the HVAC business, 1 swore the other was trying to kill him…….. this would take all day to explain….Have a great day, thanks for the knowledge you share with us.
     I've been following the John from NJ "problem customer" saga as I have 2 customers that fall into the same category.  After dealing with both these ladies for 10+ years, I would like to give him a little feedback on the actions I have taken.  For the record, they are both still my clients, but I now sleep better at night since the nuisance calls have decreased and the monies generated from their accounts have increased.
    John, you have clearly gotten involved above & beyond the standard business relationship.  While this is not so good for business financially, it is a generally good human practice, it builds good karma, and most of all it's what sets us independent companies apart from the national corporations that just don't care.  You need to put your foot down:
1.    For service requests, continue to be absolutely willing to send a tech out, but if it's not an emergency then you need to insist on setting an appointment time and date that works in your schedule.  Hold firm & remind her that there will be a charge for the call. 
2.    Don't be afraid to tell her NO to an "emergency" service request that is not really an emergency.  Stand your ground.  Remind her that while she is a priority, her systems are functioning and she is not your only customer.  Remind her that when you are working on her account she is your main focus, but that your other customers expect the same level of service.   
3.    Offer to add daily communication tests from the alarm panel at an additional monthly fee.  Remind her that "it will provide additional piece of mind" without her having to actually do anything (and it increases your RMR which will help you feel better).      
4.    Flip the personal.  If she's dragging you into her personal details, then you might remind her that you are a person as well.  "No, I cannot drop what I'm doing and run out to your home to discuss your concerns.  The systems are functioning properly, and right now I'm at my kid's basketball game.  A regular service appointment time will need to be set, and there will be a charge".  Abusive customers sometimes forget that we have our own lives.  
5.    Stop putting ideas in her head like "the only way that is possible is if ......blah blah blah". Don't do it, it just makes her mind go off on another tangent and creates new scenarios for her to freak out about.  Instead just confirm for her that your tests show her theory is incorrect and her systems are working correctly.
6.    Stop trying to explain things to her so thoroughly in the hopes that she will understand and let up.  She's clearly unable (or unwilling) to think logically, so stop trying to reason with her.
    Bottom line, you've got to handle her like she's a child:  Say no when it's appropriate, don't give her too many choices, keep explanations simple, give clear and direct instructions.... and do it all in a kind way but set reasonable boundaries. 
    If she just can't handle it and feels you are being unreasonable then suggest to her that while you would hate to loose her as a client, you may no longer be the right company for her.  Of course you will continue to monitor her account while she finds another company that she's more comfortable with.  Of course you will set the installer & dealer codes back to factory so that the incoming company can utilize what equipment she already owns.  And of course plan to let her out of her contract with no penalty because her needs (and your sanity) trump your need to get into a legal battle over what is most likely a couple hundred bucks.  Either she will settle down & be glad that you are willing to continue working with her...or she'll walk.  If she stays then your billable hours should increase, your RMR should increase, and your BS should decrease.  If she walks then you can at least sleep sound knowing you did everything you could to help her without continuing to sacrifice your sanity or your business.  Good Luck!         
Jessica in NM