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From: jbriggs444 on 21 Jan 2010 08:45
On Jan 20, 9:27 pm, "Anthony Buckland"
<anthonybucklandnos...(a)telus.net> wrote:
> "mike" <m....(a)irl.cri.replacethiswithnz> wrote in message
>
> news:MPG.25c0f5389eaf649d989773(a)news.comnet.net.nz...
>
> > ...
> > So for 2n urinals (mounted in a circle so each has two neighbours), what
> > is the expected number of arrivals before occupancy = n, as a function
> > of n?
> > ...
>
> Unless the arrivals are by elevator inside the circle, the occupancy
> stays at zero, or the number of unfortunate men present when
> the circle was closed.

There is no topological distinction between "inside" and "outside" of
a circle drawn on sphere. And in any case nobody said that the
urinals had to face inward.

See:

http://komplexify.com/epsilon/category/lion-hunting/

"2. The method of inversive geometry. Place a locked, spherical cage
in the desert, empty of lions, and enter it. Invert with respect to
the cage. This maps the lion to the interior of the cage, and you
outside it."
From: Anthony Buckland on 21 Jan 2010 14:06

"jbriggs444" <jbriggs444(a)gmail.com> wrote in message
news:98721929-893a-4013-bdde-6bc813f610d5(a)k17g2000yqh.googlegroups.com...
On Jan 20, 9:27 pm, "Anthony Buckland"
<anthonybucklandnos...(a)telus.net> wrote:
>> ...
>> Unless the arrivals are by elevator inside the circle, the occupancy
>> stays at zero, or the number of unfortunate men present when
>> the circle was closed.
>
>There is no topological distinction between "inside" and "outside" of
>a circle drawn on sphere. And in any case nobody said that the
>urinals had to face inward.


Oops. I should have thought of the girls' washroom
in the Harry Potter movies.


From: Ilmari Karonen on 21 Jan 2010 15:14
On 2010-01-19, mike <m.fee(a)irl.cri.replacethiswithnz> wrote:
> In article <slrnhl8vp4.tvn.usenet2(a)melkki.cs.helsinki.fi>, usenet2
> @vyznev.invalid says...
>> On 2010-01-18, Jim Ferry <corklebath(a)hotmail.com> wrote:
>> > Suppose a bathroom has n urinals in a row which are initially
>> > unoccupied. Men approach one by one, balking at any urinal adjacent
>> > to one in use. Assuming they select urinals uniformly at random from
>> > the set of those neither in use nor adjacent to one in use (and, for
>> > simplicity, that they never leave -- think Austin Powers), what is the
>> > expected fraction of urinals in use when the row fills up, in the
>> > limit n -> infinity?
>>
>> The assumption that nobody ever leaves makes a big difference. If,
>> instead, we assume that men do occasionally leave, but that there's a
>> queue from which any vacancies are immediately filled, then the number
>> of occupied urinals will almost surely converge to n/2 for even n or
>> (n+1)/2 for odd n.
>>
> So for 2n urinals (mounted in a circle so each has two neighbours), what
> is the expected number of arrivals before occupancy = n, as a function
> of n?

Good question. I don't have a full answer, but even just the mean
number of arrivals needed to go from n-1 occupied urinals to n seems
to scale as O(n^4). It's that last gap that takes the longest to
fill, though. I have a vague hunch that the total number of arrivals
needed to go from 0 to n might be something like O(n^4 log n), but I
could be completely wrong there.

--
Ilmari Karonen
To reply by e-mail, please replace ".invalid" with ".net" in address.
From: Richard Heathfield on 21 Jan 2010 18:33
Anthony Buckland wrote:
> "mike" <m.fee(a)irl.cri.replacethiswithnz> wrote in message
> news:MPG.25c0f5389eaf649d989773(a)news.comnet.net.nz...
>> ...
>> So for 2n urinals (mounted in a circle so each has two neighbours), what
>> is the expected number of arrivals before occupancy = n, as a function
>> of n?
>> ...
>
> Unless the arrivals are by elevator inside the circle, the occupancy
> stays at zero, or the number of unfortunate men present when
> the circle was closed.


No, you can just have them arranged so that the chaps face inwards
rather than outwards.

--
Richard Heathfield <http://www.cpax.org.uk>
Email: -http://www. +rjh@
"Usenet is a strange place" - dmr 29 July 1999
Sig line vacant - apply within
From: Phil Carmody on 22 Jan 2010 03:53
Ilmari Karonen <usenet2(a)vyznev.invalid> writes:
> On 2010-01-19, mike <m.fee(a)irl.cri.replacethiswithnz> wrote:
>> In article <slrnhl8vp4.tvn.usenet2(a)melkki.cs.helsinki.fi>, usenet2
>> @vyznev.invalid says...
>>> On 2010-01-18, Jim Ferry <corklebath(a)hotmail.com> wrote:
>>> > Suppose a bathroom has n urinals in a row which are initially
>>> > unoccupied. Men approach one by one, balking at any urinal adjacent
>>> > to one in use. Assuming they select urinals uniformly at random from
>>> > the set of those neither in use nor adjacent to one in use (and, for
>>> > simplicity, that they never leave -- think Austin Powers), what is the
>>> > expected fraction of urinals in use when the row fills up, in the
>>> > limit n -> infinity?
>>>
>>> The assumption that nobody ever leaves makes a big difference. If,
>>> instead, we assume that men do occasionally leave, but that there's a
>>> queue from which any vacancies are immediately filled, then the number
>>> of occupied urinals will almost surely converge to n/2 for even n or
>>> (n+1)/2 for odd n.
>>>
>> So for 2n urinals (mounted in a circle so each has two neighbours), what
>> is the expected number of arrivals before occupancy = n, as a function
>> of n?
>
> Good question. I don't have a full answer, but even just the mean
> number of arrivals needed to go from n-1 occupied urinals to n seems
> to scale as O(n^4). It's that last gap that takes the longest to
> fill, though. I have a vague hunch that the total number of arrivals
> needed to go from 0 to n might be something like O(n^4 log n), but I
> could be completely wrong there.

There is no 'last gap', surely? There are two last gaps, and they
slowly perform a random walk towards and away from each other until
they finally meet, at which point there's a 2/3 chance that the new
arrival will leave space for another new arrival and finish the
circle.

Phil
--
Any true emperor never needs to wear clothes. -- Devany on r.a.s.f1
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