What happens to systems such as Spanner under these circumstances?
Is it a headache or a non-issue
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It’s a huge problem. The most common approach to address it is called smearing; the duration of each second for a 24 hour period ahead of the “leap” is adjusted. For strict ordering systems this works as each device maintains time sync with the global clock, the duration of a clock cycle is just slightly different. I think this was in the original Spanner paper, actually.
Some rare systems use monotonic oscillator seconds and ignore the earth rotation second, but if you ever have to translate those to real time, you get an accumulating disaster over time and it’s generally regarded as not a good idea.
I wonder if that's what electricity producers do? If you are selling 50 or 60 Hz service, an extra second here or there must really mess things up.
A few years ago, a dispute between Kosovo and Serbia caused the entire European grid to drift away from 50.000Hz down to 49.996Hz. Millions of microwave clocks across the continent ended up 6 minutes late: https://hackaday.com/2018/03/09/europe-loses-six-minutes-due....
Clocks used to be able to use the 60Hz cycle to track time, and grid providers would run slightly slow or fast ("time error correction") to get back into sync. A leap second would just be part of this.
I believe in the US this error correction has been discontinued in the East and in Texas, but is still done in the West for some kind of non-clock "inadvertent interchange" reasons I don't understand.
Wait is that why my oven and microwave clocks are constantly getting out of sync by multiple minutes every year?
Out of sync with each other, or are they drifting together in lockstep? In the latter case, yes, that's the most likely explanation.
Leap seconds are not added on a regular schedule like leap days, they depend on physical measurements of Earth. So high reliability systems with comprehensive timekeeping would not be perturbed by these choices, I would think.