Whelen has this in the installation and service manual.
http://whelen.com/pb/aviation/11051.pdf
"WARNING: Strobe light power supplies are meant to be used, not to
remain in an inactive state. Use them at all times, this will improve their
proper functioning. Any strobe light power supply that has been out of
service for a long period of time is subject to failure because the electrolytic
condenser loses the polarity formation. A strobe light power
supply not having been used for one year or longer is vulnerable to
failure, applies to 1993 & older units.
If this is the case, it is recommended to start operating the system on
a voltage that is reduced by 25 percent for 10 to 15 minutes before putting
the power supply into normal service. This will prevent overheating
of the condenser while they reform. If the power supply, after a long
period of non use, is operated at full voltage immediately, there is an
excellent possibility that the condenser will become overheated."
It also has this:
"NOTE: Installing one new flash tube in any multi-head strobe light system,
will sometimes cause the remaining old flash tube to misfire or
skip. This signifies that the old flash tube is nearing the end of it?s service
life. However, to check the questionable flash tube, install it in a system
and apply a reduced voltage, approximately 20 percent, to the
input to the power supply. If the flash tube will operate at this reduced
level, it still has a great deal of service life in it."
It is interesting that the method for both is to run the system at reduced voltage.
Can you eliminate parts of the lights in the circuit to see if each runs OK by itself? Or is that not feasible?
I am going to use the same system but have not bought my power supply for this very reason. I will be interested to find out what you discover. Good luck.