In the little town of Analogia there is now a supermarket. It is just off the main square and it rapidly became popular with the Analogians—but it was less so, of course, with the five local shopkeepers, and especially their wives, who began to see it as serious threat to their livelihoods. The manager of the supermarket was a local entrepreneur called Giovanni. It wasn't his idea to open anything quite as grand—but as his wife insisted, its main function was to provide respectable employment for his 15 young, unmarried nieces. They were all there at their checkouts from 8 o'clock in the morning until closing time, irrespective of how many customers there were. There is never much for them to do for the first couple of hours but they have never minded because the store was warm and they could drink coffee and talk to each other.
Early in the morning is the best time to shop. Because there are no queues at the checkouts at this time, the process is very rapid. The number of shoppers being served each hour depends only upon the rate at which they arrive at the checkouts with their purchases. As the morning progresses, the store becomes busier, and toward mid morning the first queues begin to develop. When this happens shopping becomes more time-consuming. The word “queue” is hardly accurate. It implies the formation of an orderly line of customers, each patiently waiting their turn. Nothing could be more alien to the Analogians. They mill about in a disorderly crowd, blocking up the aisles and the approaches to the checkouts. When a cashier is free, they more or less fall into the vacant places and are served. They frequently arrive at a till, wait for a while and then drift randomly away.
The girls can only operate the ancient checkout machinery at a fixed rate, and the “queues” rapidly become worse. Giovanni has a mathematical sort of mind and he is interested in the relationship between the number of customers in the store and the rate at which they are served. Pinned to his office wall he has a graph showing the way in which the pattern of shopping changes throughout the morning as the number of customers steadily increases. His graph shows how busy the shop is, represented by the concentration of shoppers [S] (i.e. the number of shoppers per square meter of floor space), against the velocity of processing, ν (i.e. the total number of customers passing through the checkouts per hour). This graph clearly shows the change from the early morning situation, where ν is determined by the rate at which customers arrive at the checkouts (which is simply related to the customer density), and the way in which the limiting rate, νmax, is reached once queues have developed at every checkout.
Eventually, the loss of trade irritated the wives of the other five shopkeepers so much that they insisted that their husbands do something. Like all Analogians, they were very law-abiding citizens and they wondered what they could do that would be both effective and legal. They hit on two different strategies. In the first of these, all five of them appeared in the store simultaneously and they each monopolized a checkout for the complete morning by engaging the checkout girls in conversation. This was surprisingly easy, especially because they thoughtfully went armed with photographs showing their summer holiday visits to Euro Disney. Giovanni was, of course, furious when he realized what had happened—but he was interested to analyze the effect of this ploy and he plotted a second graph. This showed that at first it made little difference because the customers had all found a free checkout, but the queues developed earlier and with five of the 15 checkouts inoperative the value of νmax was down to two-thirds of the normal level.
News of dire warnings of dismissal, and deprivation of dowries, reached the ears of the five shopkeepers. It was a small town. They all had sons and the nieces were attractive girls. A different, more subtle ploy would have to be used next time. Accordingly, a week later, after much prompting from their wives, the five shopkeepers arrived at the supermarket wearing a variety of false beards and moustaches. They pretended to be bona fide shoppers and they mingled with the other customers. They wheeled trolleys of food to the checkouts, let the girls check all of their purchases and produce the bill and then they all discovered that they had left their wallets at home. They then changed their disguises, re-entered the store, and repeated the procedure. By lunchtime, Giovanni had realized what was happening. His irritation was somewhat reduced by his interest in the statistics. It gave him a third graph to study. He noticed that there was a noticeably different effect. When the queues were short, the five shopkeepers had appreciably slowed down the velocity of processing, but as the queues lengthened, there were so many genuine customers that the five represented an insignificant fraction of the total and their effect became negligible. This time the νmax value was unchanged.
Giovanni retired to his office, reached for a bottle of Chianti, and thought hard. He decided that life was too short and Analogia too small for a trade war. He offered the five shopkeepers favorable concessions within his supermarket and they all lived happily thereafter. The three graphs are still there pinned to his notice board, gently yellowing in the Analogian sunshine. One day, he thought, someone will be interested in my mathematical modeling—but they never were.