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Re: Waiting For the Bus
Posted:
Feb 14, 2002 9:32 PM
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This is a symmetrical situation and buses do not have accidents and stop
entirely.
Therefore the expectated travelling time is proportional to the number of stops
to travel.
Therefore never cross the road, no matter how long it feels like its taking.
In fact, the longer you wait, the more likely it is that you will be making a
mistake crossing the road.
Raymond Kennington
library.treasures@NOSPAMsaqnet.co.uk wrote:
>
> I thought of this problem as I myself often wait for the bus for too
> long times.
>
> Suppose there is a route that operates N buses between A and B.
> The route runs along a single, two-way road with M pairs besides the
> two at A and B.
>
> The bus stops that belong to any one pair serve a single location and
> are exactly opposite each other on each side of the road.
>
> The locations A and B have just one stop each where the buses turn
> around in a tight loop without even temporarily terminating there.
>
> A and B are not termini in the conventional sense therefore, and the
> buses run in an effectively non-stop circular route that is only
> linear in appearance.
>
> Now, the service times are diabolical, for there does not seem to be a
> planned or followed time table. Instead, the buses arrive at random,
> albeit with some definite mean time T between two consecutive buses at
> any one bus stop.
>
> Also, for reasons of other traffic on the road, any one bus may
> overtake any others and the buses may arrive at the next stop in an
> order that is different from that in which they arrived at the
> previous stop.
>
> A passenger is waiting for a bus at location C.
> His destination is at place D between C and A.
> It is generally reasonable for him therefore to wait at that stop
> which serves the buses that go directly towards A, and not at the bus
> stop across the road that serves the buses going directly towards B.
>
> However, he is observant of the number K of buses that go in the
> opposite direction, directly towards B, during the long time T/waiting
> that he is waiting.
>
> He is determined to get to his destination by bus only, and so as soon
> as possible.
>
> He reasons that after a certain T/waiting and above a certain K<>N,
> there must be a calculable probability that he will arrive at his
> destination D sooner if he crosses the road and goes to the opposite
> bus stop and boards the next bus that goes to B then to D from there,
> unless a bus arrives here first, where he is right now waiting.
>
> While he knows the total number of buses, N, he has no knowledge of
> exactly how many buses may be between C and A, or between C and B.
>
> He also knows M, that is the number of places between A and B, and
> that how many places are there between C and A or C and B.
> Furthermore, he knows the similar relative place of D among all the
> other places on the route.
>
> The buses, otherwise than their reliability of arriving, are ideal.
> They are never filled up to full capacity and the passenger boarding
> and alighting times are of zero duration. It is also safe to cross the
> road quickly and so it is always possible to catch the bus that
> arrives first.
>
> I am wondering if I am missing something here, e.g. a mean time for
> the buses for completing one round, say from A back to A?
>
> I did not intend to present a problem here, where any single bus may
> leave one stop and arrive at the next one after an infinite time. That
> would effectively mean that the total number of the buses, N, is
> effectively not N, but may become less!
>
> The plot seems complicated enough and at times of my fleeting insight
> I see now redundancy, now underdefinition, now some other flaw!
>
> So, if there is something of interest here, I would be curious whether
> it is possible to calculate it from the given parameters, when is it,
> if there is such a time at all, that the passenger is better off by
> hopping on a bus that first diverges from his destination?
>
> Thomas <library.treasures@NOSPAMsaqnet.co.uk>
>
> (Remove NOSPAM from email address if replying by email)
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