Monday, February 14, 2011

Latency Versus Bandwidth – what Jeremy Clarkson and David Mackay need to understand

I was given a Jeremy Clarkson book, “Driven to Distraction” for Christmas.  In the opening article he casts doubt over the wisdom of the variable speed limits around the M25,  calling it “a new state control system to quash individualism on the motorway”.  Much as I hate to criticize a fellow Doncasterian I can only conclude that he doesn’t understand the maths.

Back in 1987 I worked for a consultancy.  One of my co-workers was a software engineer engaged in writing fluid modelling programs.  He told me that up to a certain point  the flow in a water pipe is linear and it travels with very little resistance.   Above a critical velocity the flow becomes turbulent and the water molecules bump into each other such that the flow is no longer a linear function of the pressure exerted.  My co-worker also told me that the flow of traffic on a motorway could be modelled in a very similar way to water in a pipe.  So much for individualism!

Let us apply the latency versus bandwidth analysis I used in the last blog to this new mathematical problem; how to get around the M25 as quickly as possible. The latency is what Mr Clarkson perceives, it’s the journey time.  To measure the bandwidth of a motorway you need to draw a virtual line across the road and count how many cars per second, per minute or per hour that cross that line.  For instance if we all obeyed the 2 second rule there would be 3/2 cars per second (3 lanes * 2 seconds), which is 90 cars per minute, 5400 cars per hour, or 130,000 cars a day.  Actually that’s not a lot considering how many cars use the M25 every day but then they don’t all use the same bit of motorway.  Of course many of us don’t obey the 2 second rule so the bandwidth is actually higher than that.  The point is that the bandwidth is most definitely finite. Whats more, in a traffic jam the bandwidth of the motorway decreases dramatically!

So how do variable speed limits help? Well, it appears that the M25 traffic planners understand the difference between latency and bandwidth and a bit about fluid dynamics.  From my analysis above the peak bandwidth of the motorway is largely independent of traffic speed and by reducing the speed of the traffic they are more likely to achieve a non-turbulant flow which keeps the traffic moving.  In other words the traffic planners are optimising bandwidth, not latency.   However the more vehicles they can get past a given point, the more they reduce the average latency too.  Left to their own devices individual motorists might indeed achieve a lower latency, but because they reduce the bandwidth of the motorway the average latency actually increases.

From a fuel consumption point of view traffic jams are a worst case scenario.  Its simple mathematics that for a journey of X miles and consuming Y gallons the average MPG is X/Y.  Of course the fashion is to talk about litres per 100km which is just the other way around.  However standing still with the engine running is time at 0MPG or infinite l/100km.  In order words, time spent stationary or even in a gear below the top is fuel wasted.  Coming back to the linear versus turbulent flow analogy, if every vehicle travels 56mph a linear flow would be achieved and the overall fuel consumption would be minimised.

However, I for one would not like to drive everywhere at 56mph.  There is also a carbon cost to me being on the planet so time wasted is also carbon spent.   The good news is that providing that a linear flow can be maintained the actual carbon cost of the higher speed is secondary.  That is, if everyone went exactly 70mph then that would still be lower entropy than some people doing 85mph and others doing 50mph.

This brings me neatly on to the second part of my article – road trains.  One way to increase the bandwidth of a motorway is to simply make more lanes.  This is both very expensive and very environmentally unfriendly.  The second and probably more fruitful way is to reduce the time distance between cars.  Imagine if there was a way to make a road train such that each car travels safely just 30cm behind each other, or, say, 5m apart.  For the simplicity of the mathematics lets say they are travelling at 120km/h (75mph) or 2km/minute, 33.3m/second.  The bandwidth of this motorway would be 20 cars/second or 13x the current safe bandwidth. Actually you don’t have to imagine road trains as they are a practical reality right now:

In David Mackay’s otherwise excellent book “Sustainability without the hot Air” he makes mention of several promising low carbon technologies but road trains are not mentioned.  That Dr Mackay is anti-car is quite evident.  However public transport, as alluded to in my earlier blogs is not a panacea and I doubt if people will be willing to give up on personal transport anytime soon.  Road trains represent a pragmatic solution that can reduce car journey times and keep traffic flowing, thus reducing the carbon footprint.

Further, by getting the cars so close together one of the major contributions to fuel consumption, wind resistance, is dramatically reduced.  Anyone who watches motor racing understands slipstreaming.  Dr Mackay uses this precise argument to postulate why trains are more efficient than cars.

There will always be some people who want to go faster than others.  So let me put out a straw man for people – keep the motorways 3 lanes, make the inside line 60mph, the middle lane 70mph and the outside lane 80mph, and make road trains of 10-20 cars followed by a gap.   To change lane the driver indicates his or her intention and the on-board computers negotiate to find a gap and fit in the car.  As the relative speeds between lanes are slow this can be done quite easily.  Indeed with all that bandwidth there will be gaps in the trains. After all each lane has more bandwidth than the current motorway has.  Now Mr Clarkson and Dr Mackay, at opposite ends of the carbon-political spectrum are both happy :o).