Brain Teaser #91: Brain Thumper #5: How Long is the Average Break?

[davilla]

Brain Thumper #5

Five transit centers are evenly spaced near the perimeter of a small city. Any two transit centers are connected by exactly one route. Additionally, from each transit center there is one route that extends into the suburbs in the shape of a large colorful lollipop. The five routes passing through the downtown area take 50 minutes to travel for all-stops buses and 40 minutes for express buses that stop only at transfer points. Buses on the remaining routes serve all stops. On perimeter routes they take half an hour to travel between transit centers, and on suburb routes 45 minutes to complete the loop and return to the same transit center.

Buses that enter a transit center are dynamically rerouted using a simple pattern. Each transit center employs the following mechanism independently. Regarless of the originating route, the first two buses are designated as express buses and the next two buses serve all stops in the heart of the city. The next three buses proceed in a clockwise manner among the three remaining routes, and then we start over. The pattern repeats in this rotation, sending out each bus as soon as it arrives.

If a bus driver comes in along a route of 40 minutes or less, he or she must continue operating the bus on whatever route it's reassigned to. Otherwise, the driver gets a break while a relief operator takes over the vehicle. The driver on break will serve as the relief for the next bus coming in on a route of 45 minutes or more. It's like a game of musical chairs, with buses always in use and hopefully only one person driving.

There are 62 operators on duty and as many buses in operation as necessary to keep them busy. How long is the average break?

 

[NateTG]

114/11 minutes is the average break length.

 

[davilla]

There are 60 operators on duty and as many buses in operation as necessary to keep them busy. How long is the average break?

114/11 minutes is the average break length.

I think I know how you arrived at this number and the logic doesn't seem correct. However, I made a mistake myself when first formulating the problem, so if you would like to explain how you arrived at the answer I might be proved wrong. Please forgive me for changing the number of operators from 60 to 62. This should make the numbers come out cleaner.

 

[NateTG]

With 62 drivers:

62 drivers means that there are 62 driver minutes per minute

Despite the complicated system for scheduling busses, the conditions imply that there will be one driver at each station at all times.

That means that there are 5 breaking driver minutes per minute
so there are 57 driving driver minutes per minute.

Since there are three breaks per 285 minutes of driving (Consider batches of seven busses sent out from each station) that means that there are 3 breaks / 285 driving minutes * 57 driving minutes / minute = 3/5 breaks per minute.

So there are three new breaks every five minutes, and there are 25 minutes of break time every five minutes, so the average break time is 25/3 minutes.

I expect that you wanted the average break time to come out to three minutes.

 

[davilla]

Yes! Using a different method I also got an average break of 8:20.

Using your method for the case of 60 operators I came up with 95/11 or a little closer to 9 minutes, so I think you may have made an arithmetic mistake earlier.

I expect that you wanted the average break time to come out to three minutes.

No, I originally wanted the average headway to be 10-15 minutes, but in summing I neglected to double the time for bidirectional routes.

 

[NateTG]

No, I originally wanted the average headway to be 10-15 minutes, but in summing I neglected to double the time for bidirectional routes.

By the sound of it, I prefer my approach.

P.S. Redid the work, and got 95/11 also.

BTW:
It turns out not to be important, but your description of the routes is a bit confusing. I assume that the graph should be isomorphic to K5 with tails, but you mention 'five routes passing through downtown' when there should be ten. You also have 'connected by exactly one route' but it's ambiguous whether route means path or edge in the sense of graph theory.

If you re-use this, you might consider replacing "route" with "direct route", and replacing five with ten, or omitting it entirely.

 

[davilla]

You mention 'five routes passing through downtown' when there should be ten.

I'm not sure how you're counting ten. The five perimeter routes are crosstown since the downtown area is the center of the city.

The express buses are a variation of the regular buses. It's really the same numbered route, just a different type of service.

It's ambiguous whether route means path or edge in the sense of graph theory.

By "route" I meant a bus route, which doesn't fall under either definition. In this case each route is an edge, but in general it is a named path.

I can see how this could be confusing. Thanks for putting up with the strange jargon.

 

[NateTG]

I'm not sure how you're counting ten. The five perimeter routes are crosstown since the downtown area is the center of the city.

I was counting both directions of the routes - must have been a brain fart (=. I suppose there could be five to ten routes really.