🚀 go-pugleaf

tk@dcs.ed.ac.uk (Tommy Kelly) writes:

>Every morning, I take a train to the city, then have to take a bus to
>Uni.  There are two bus stops within easy walking distance of the train
>station.

>Bus stop 1 is 4mins 20 seconds away, and is the earlier on the route (the bus
>visits it before bus stop 2).

>Bus stop 2 is 3mins 20 seconds away.

>The bus takes about 30 seconds to travel from stop 1 to stop 2.

>Bus interarrival time is 15 minutes, and can effectively be at any point
>in their run when I begin my walk from the train station.

>a. On average, which bus stop should I walk to minimize my time in travelling
>   to the University?

Surely it is stop 2. Even if they are at the same distance, you can
catch an earlier bus at stop 2. Since stop 2 is also closer than stop
1, there is no contest.

>b. In general, for times :
>	Tn - time to walk to stop n
>	dn - time for bus to travel from stop n to stop n+1
>	l  - bus interarrival time

>	which stop should I choose?

If Tn+dn > T(n+1) choose stop n+1 over stop n. Any bus you could catch
at stop n, you can also catch at stop n+1. You might even catch an
earlier bus.

If Tn+dn < T(n+1) choose stop n over stop n+1. Similar reasoning.

You can compare two non-adjacent stops in a similar way. If
Tn+dn+...+d(n+m-1) > T(n+m) choose stop n+m over stop n, and vice
versa. Since you can compare any two stops, you can sort the stops and
choose (one of) the best.

The interarrival time is irrelevant.

	Torben Mogensen (torbenm@diku.dk)