I have been re-reading Philip Ball’s Critical Mass, his book about “social physics”, how the study of physics can lead us to understand aggregate human behaviour better. It’s very wide-ranging and interesting, with pretty obvious implications for my day job. One of Ball’s early chapters is about path-formation and “flocking” (eg. how a mass of people can most efficiently leave a room – vital to understand this when planning fire exits etc.). He doesn’t actually mention queue formation but it’s the same sort of thing, and it’s a problem that strikes me when I go into Pret A Manger for the occasional breakfast bacon and egg baguette of a morning (om nom nom).

The London Bridge Pret A Manger is managed by a gentleman of boundless enthusiasm, both for his job and for the rules it requires him to enforce. I often see him upbraiding staff – in a friendly way, naturally – for incorrect cap position, insufficient service speed, etc. His main role in the customers’ lives, though – a thankless one – is to inform us that there are six open tills and can we please form six queues, instead of, as invariably happens, one big long queue which fills individual tills from the front. Is he right?

sammiches.jpg I have drawn an scientific DIAGRAM of the Pret a Manger London Bridge layout which I reproduce here. The black dots represent queueing individuals in the formation that they commonly (without prior planning) adopt. The red dots represent the queuing layout that the manager wants them to adopt.

The manager’s version is clearly more efficient when it comes to minimising queue length and from a management perspective it allows him a fairer assessment of the performances of individual workers. But the longer queue is in the collective interests, not only of current customers in the shop but also of prospective shop entrants (I’m assuming, and my own experience tallies with this, that neither method has a positive impact on the time it takes you to get served, though see below for more on this). The six short queues favoured by the manager would clearly make access to the tables, and to one end of the sandwich cabinet, more difficult. (Very importantly, this end of the cabinet has the Pret Chocolate Bars in). Whereas the one long queue allows better access to and from table areas and cabinets, with a minimum of queue crossing even at busy times. The crowd here have organised themselves into the most mutually beneficial system possible in terms of using the shop space.

But is this system unstable? Every individual beyond a certain distance in the queue will find their own utility maximised (i.e. will get their nosh faster) if they leave the queue and go immediately to one of the more empty tills. We now have a kind of prisoner’s dilemma in action – Ball in his book talks about the p.d. and game theory but it’s in the bit I haven’t read yet. Some people who are left behind in the main queue or do not redistribute fast enough will then find their utility lowered (i.e. it takes longer to get the sammich).

Left to its own devices the queue doesn’t collapse into sub-queues – whether it’s a spirit of mutual co-operation that drives this, or a kind of selfishness (the benefit created by quicker sandwiches is not equal to the disbenefit created by breaking the social convention) doesn’t really matter. What matters is that the manager’s encouragement to move from long queue to sub-queues creates this collapse, and therefore invariably causes an increase in queueing time for some customers. He really should leave well alone.