I'm glad someone brought up the OSP VRS.
I spend a good amount of my day at a hospital and see a number of OSP vehicles parked in the ED lot and I have been scanning the UHF (and old 700 MHz) frequencies and have never heard a VRS active. Is it possible the troopers in the area have the portables on MARCS directly and are not using the VRS? The hospital has very good coverage from the Hamilton County site, they have a tower right up the street.
They have both. OSP portables are dual-band which include both UHF (for the repeater) and 700-800 MHz (for MARCS). If MARCS coverage at the hospital is sufficient, there's no need for the troopers to use vehicle repeaters and they can put their portable radio directly on the post talkgroup. If the trooper is on a traffic stop on Route 37 in East BFE, they'll almost certainly use the repeater.
The beeps are most likely the "Singletone" automatic sensing tone that mobile repeaters send out to see if there's another repeater on the air. The more sophisticated VRS units will shut down if another repeater on their system is in the immediate area. This was the difference between the Motorola PAC-RT systems and the more basic PAC-PL.
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Just a guess but I believe this is what you're hearing.
Rick is correct; OSP uses Pyramid vehicle repeaters which send out the tones just as described.
The description of the difference between the old Motorola PAC-RT and PAC-PL systems is also an interesting historical tidbit. Many volunteer fire departments experienced the very problems described; since PAC-PL was less expensive than PAC-RT that's what most volunteer departments bought; many used 153.830 MHz as their portable frequency, and when everybody was using the same portable frequencies (even with different PL tones on the portable frequency), partial chaos usually resulted.
Another historical tidbit... back in the low-band days, OSP PAC-RT units were wired to an interesting circuit (which I'm told was devised in-house by OSP radio techs) that was connected to the dome light in the vehicle; when the trooper opened the door and got out of the car, it activated the PAC-RT which responded with a "beep" on the portable frequency to let the trooper know the repeater was active. When the trooper opened the door and got back in the car, the clever little circuit deactivated the repeater.