  &nbsp; aside: you know you've done well with your lunch when the stacked sandwiches are as tall as they are wide... as i sit planning - yes, planning - what to write i am torn as to what my actual aim is; if i am diverting then that is more than good enough, but surely i have something more to share? to this end i will attempt, from time to time,&nbsp;to expound on some of the theories i have formed about how things work. if nothing else, it will be a useful insight to the way my mind works, and how i perceive the world. the first that comes to mind is 'leapfrogging buses', or 'how to not get infuriated on your way home'.
more on that in&nbsp;a minute. &nbsp; hold your thumb and first finger out, very close together. now, without them touching, try to move them closer. closer still. yes? now, that is how close i was to buying a polaroid instamatic camera on thursday. i saw one (two, actually, but one appeared to date from the paeleolithic era and wouldn't have looked out of place with a pin-hole) in a charity shop, and spent a good five minutes toying with the idea.
i've always wanted one, that i can remember; something about the self-containment appeals to me. but i didn't buy it in the end out of respect for my ever-dire financial situation (camera = cheap; film = expensive) and also the thought that i may be mistaken for someone taking 'candid' (nudge nudge, wink wink, say no more squire) photos at home. a label i wouldn't mind so much if it was a remote possibility, but sadly not. today's list is of things that are not really fruit but can be classed as one of the day's five portions since they are healthy: a glass of milk vitamin supplement pill opening a window taking the stairs thinking about having some fruit its true; you know it is.
&nbsp; leapfrogging buses: a theory. more of an observation, really. some simplifying assumptions: i) all buses travel the same route, at the same driving speed. ii) all passengers are travelling to the end of the line. iii) all distributions are normal. iv) there are no price/ticket considerations in choosing buses. v) all stops are the same duration. (these&nbsp;will be dismissed in turn, but to illustrate the idea i've&nbsp;started it as simple as possible) imagine there are people waiting for the bus at every stop. one bus travelling the route will stop at every stop to pick people up, and take a certain amount of time to do the run. this is obvious; but more interesting is if there are two buses travelling next to each other. assuming i) to be valid (as is the case on my way to/from work), people will signal for the first bus, which will stop. assuming ii) to be valid, the second bus will not stop and will overtake the first. the buses have now swapped positions, and will 'leapfrog' each other down the route, stopping at every other stop and thus&nbsp;completing the run&nbsp;faster.
however, buses tend not to set off together; it makes no business sense. but, if iii) is valid then there will be a statistical variation on the number of people waiting for the bus at any stop at any time. in truth, the variation is not truly random but follows a certain pattern (rush hour, etc) with an amount of variation built-in. and if there is a variation, then we can say that v) is not valid.&nbsp;so; if we are to imagine there is, at a certain stop at a certain time, significantly more people than the mean number of people at any stop at any time(which is statistically very likely) then the solitary bus will be signalled and will have a long stop for everyone to board.
this will delay that particular bus, and the next bus along the route will gain some time on it. a feedback loop is created: the first bus is taking longer to do the run since its stops are longer, this leads to a greater build-up of people at the following stops and will cause those stops to be longer also. the second bus may not encounter the same heavy numbers of passengers, and so will gain on the first bus, moreso as the first bus is continuing to make longer and longer stops.
eventually, the second bus will catch up, and the two will begin to leapfrog. &nbsp; so we have leapfrogging buses; if these are running faster than a single bus would, then it is conceivable, is it not, that the two would eventually catch up with any single bus ahead on the same route? bingo! three buses in a row. assumption iv) need not be discussed heavily since people will either simply signal the first bus that comes along (which is statistically more likely to be a 'first' bus of a pre-leapfrogging pair), thus not affecting the leapfrogging behaviour,&nbsp;or will have a ticket for a certain bus&nbsp;company, which modifies the idea from being 'any bus' to 'buses of a certain company', treating each company separately.
whilst assumption ii) is not valid in the general case, it does not affect the behaviour in the long run .
also required is the possibility of there being no people waiting at the stop; all the possibilities of first bus/second bus stopping are thus: yes/yes: no distance/order change between buses yes/no : decrease in distance between buses, possible leapfrogging no/yes: increase in distance between buses, no leapfrogging no/no:&nbsp;&nbsp;no distance/order change between buses if all these occur randomly then their overall effect is that described earlier, to promote leapfrogging. &nbsp; and there we have it; my idea as to why (particularly at peak times where it is more likely that there are people at each stop) buses can bunch up in twos and threes. i have no idea if this has been postulated anywhere else; i certainly haven't looked for it. but do let me know what you think, or if i don't make sense somewhere, which is quite likely. anyhow, i'm making chop suey. excuse me.
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