Let intuition take over, intelectualisation to its limits. (Are you sure that intuition is responsible here? Anyway, there it goes....)
"Chaos theory looks at how very simple things can generate very complex outcomes". The mention here of the workings of chaos does not discriminate on the simple things. It robs chaos of its process essence. And the simple things are the simple rules, or there is not a simple thing as such and the simplicity is derived by abstraction, and abstraction is a mental process we employ to make sense of the world around us. It is the right abstractions we should always use. By saying simple things, we employ abstraction too. But that abstraction is not helpful as it muddles up and leaves it unclear about what chaos is.
Motorway traffic swarms.
The example of the swarms, gives the essence of chaos as it traces its foundations. The rules, the simple rules which are: maintain a distance between itself and its neighbours and fly or swim in the average direction of its neighbours. From this alone the wonderful, swirling, complex patterns the birds or fish make are seen.
Look at that, like a spark traveling at lightning speed in my brain circuitry, elucidated, gave a clear projection, a clearer picture on an idea that was hovering around my mind for some time now. This involves motorway traffic and how to manage it. I always felt sorry to tens some times hundreds of stranded drivers in their cars in long queue of traffic especially during the morning and evening journeys back and forth to work.
I maintained that in order to avoid motorway pile-ups drivers had to keep a distance with the cars in front and rear, a distance which should be governed by braking speed and distance traveled once the brakes are applied. Safe braking distance in all and in essence. The term used in that website namely average speeds is what the acquired continuous speed might be too. So in order to have an acquired continuous speed of 40 miles/hour, the distance between any two cars in a column or lane it should be 36 metres or nine car lengths, for a 50 miles/hour 53 metres and at 60 miles/hour 73 metres at 70 miles/hour 96 metres. Though I fathomed the significance of a safe braking distance and somehow it was evident that by having an adequate distance between each traveling car, it would have allowed the traffic to pass through motorway traffic jams hotspots without braking speed maintaining an uninterrupted traffic flow without halting at any point through the hotspot. Even on motorway entry points where the rush traffic enters the motorway in droves. The distance kept by the drivers already in the motorway would have allowed incoming drivers to join the flow without braking speed by entering through the ample, wide windows in the traffic flow maintaining the momentum of the flow.
The number of cars allowed to join in, in a traffic flow window would be controlled by traffic lights, operating under swimming pool fun shoot rules, permitting a new batch of cars only if the cars in the batch before have cleared the entry way. Allowing only enough cars, according to the traffic flow from the junction (in cars/min or cars per minute) divided by an average car length to determine how many cars would be allowed in each traffic light interval. The number of cars allowed would be worked as minimum and maximum values, calculated from the parameters of cars/min traffic flow and car average length. Car average length will be calculated by taking an average from articulated lorries to motorbikes usage of the motorway traffic. Since it will take into account their contribution in the traffic it will be dynamically determined based on their traffic flow contribution. Let us work an example:
I have to think from the scratch. I should let some traffic expert to do that. Let us put down what are the parameters. There are two or three lanes traversing a motorway cross-section. So there is a set maximum of two or three cars traversing the cross-section in a given instance. Can this be set as the instantaneous speed? Would any notions of instantaneous speed be significant in these circumstances? What we are interested is cars per a unit of time. Do cars per unit of time resemble flow; let’s say of water, through a pipe’s cross-section? Water flow is designated as volume of water passing in a unit of time. Let us say 50 cm3/sec, which is 50cm3 of water passes through a cross-section in one second. Likewise in a motor way tract, replacing volume by number of cars in a unit of time, let that be minutes. Let us say 50 cars/min in blocks of two or three cars at any given instance. How can we calculate the number of cars passing through a cross-section assuming that we know the allocated speed, the car-to-car distance and the number of lanes? We should assume as well that traffic operates under saturated conditions. Let us say, maintaining a 40 miles/hour speed, then distance between cars should be 36 metres. We aspire to attain a quantity that has cars per unit of time units. The 36 metres car-to-car distance can be translated into car lengths. There are 9 car lengths for the 36 metres car-to-car distance. How can this be used to derive car/hour or car/min or cars/sec units? Traffic flow web search? Yes.
So if we speak about an average speed of 40 miles/hour traveling at a distance of 9 car lengths, let us say an average car length is 4 metres, then each car will occupy 9 + 1= 10 (itself) car lengths of the road or 40 metres. A distance of 40 metres would have one car, of 400 metres 10 cars, of 800 metres 20 cars and 1,000 metres 25 cars. So there will be 25 cars in one kilometre tract for each lane, times two, 50 cars for two lanes and times three, 75 cars for three lanes.
And each of these lanes flows at 40 miles/hour speed. How can this be used to find out the car flow from a given cross-section?
If it was 40 km/hour that means that in an hour that same column would extend for 40 kilometres it should have 40 times 75 equals 3000 cars (40X75= 3000) in one hour, and 3000 cars in 60 mins or 50 cars per minute, or 5 cars every 6 seconds or less than one car per second.
If it was 50 km/hour, a column extending for 50 kilometres it would include 50X75=3750 in one hour, 3750 in 60 mins or 62.5 cars per minute, or a slightly more than one car every second.
There are two issues to be considered. First the reaction rate of a driver, who is required to join the traffic without changing its average flow of cars, and second the length of the tract of the flow which will determine its success or not. For two lanes the cars accommodated 6,000 cars/hour and 9,000 cars/hour for the three lanes.
(Find out number of cars involved in a traffic jams. And the reaction time of a driver.)
There is the added bonus of how long it takes for some one to find its place in the traffic column without hindering the flow of the traffic. And another point which it should come out after I find the average reaction time, is the distance might be in need to increase to more than safe braking distance, to make it easier for the average driver to integrate in the flow of the traffic.