I have come across the following probability problem in my research. To avoid going into unnecessary background, I will present it under a different guise, but the mathematics is the same.
I'm fairly sure that this must be a well-studied problem, but unfortunately I lack the necessary background to search for the relevant literature. Any pointers will be appreciated.
There is a watch tower, ideally to be manned $24$ hours a day. There are $6$ guards willing to volunteer, with each guard $k\in\{1,\dots,6\}$ willing to contribute $k$ contiguous hours every $24$.
If these guards make random choices*, independently of each other, as to which time slot they will take, then the worst case scenario is that the watch tower is manned for only $6$ hours a day, and the best case scenario is that it is manned for $6+5+\dots+1=21$ hours a day. But what is the expected period for which there will be at least one guard at the tower?
Of course, I am really interested in the general case where the circle is divided into $N$ segments, and there are $K$ contiguous slots of lengths $1,\dots,K$.
*In my actual scenario, the guards would show up (and therefore also leave) on the hour, but I don't mind if they are approximated to show up at any time chosen continuously between $0$ and $24$.
The chance that a given hour is not manned is $\frac {18}{24}\cdot \frac {19}{24}\cdot \frac {20}{24}\cdot \frac {21}{24}\cdot \frac {22}{24}\cdot \frac {23}{24}\approx 0.38,$ so the expected number of unmanned hours is about $9.2$