Abercrombie wrote:
My simulation software (which remains vapour ware, until I get some clarity) suggests that, once you have pulled in, you make matters worse if you let more in. In fact, the (mental) calculation suggests that, once you have pulled over, the least selfish thing to do is the most selfish thing (!), i.e. let no more in. If everyone ALREADY in L1 let exactly one in from lane 2, and everyone who was in lane 2 let no-one else in, assuming both lanes carry the same volume originally, a sort of balance might be achieved. That is my expectation, anyway, although I'll have to write some queues to see it in action with my own eyes, so to speak.
Fortunately, I seem to be gifted with a sort of "in head debugger" which means I can prototype and debug moderately complex software without actually writing any code!
As far as I can see you are absolutely correct. If every car in L1 lets 1 car from L2 in, but NO-ONE joining from L2 does so, then you will achieve equal flow in each lane, which gives the highest net flow AND the shortest queue, so clearly that is ideal behaviour.
There are two other extremes. One is that no-one in L1 lets anyone in, in which case L2 comes to a stand. Apart from leaving a batch of cars "hung out to dry" this actually clears itself, because as soon as L2 becomes slower than L1 it ceases to become attractive to join it, so you just end up with a long queue in L1. The other is that someone from L2 joins, then "repays the favour" by allowing someone else from L2 to overtake him and pull in. This is the worst possible case, as it reaches a ridiculous position where L1 will stop completely and only the cars that pull out and overtake will actually get anywhere.
The one common thread running in all 3 scenarios is that the selfish person benefits, as well as tending to exhibit the best "queue minimising" behaviour.
If I wrote the Highway Code, advice to people in a 2 lane queue would be:
1. Upon encountering the queue get in the fastest moving lane
2. When the head of the queue approaches, if in L1 allow ONE car from L2 to join ahead of you, but no more.
3. If in L2 look for a gap as late as you comfortably can, and once you've merged back to L1 don't allow anyone else in at any cost!
Blimey - all 3 are recommendations to be selfish! Didn't that mentally ill mathematician say something about this?
Oh, and my advice to highways agencies would be to lay out the merge so that L1 has to join L2, not the other way round. Solves the whole problem at a stroke... (run that one past your mental software simulator, BW!)