Game 1 results and behaviour summary

The initial configuration of the system is 10 stations with 4 people between each station, giving a starting point of 36 people as work in progress.

The number of people processed through each station each round is determined by the roll of a six-sided die. The average roll value of such a die is 3.5, so one might expect each station to process on average 3.5 people per round. This means that over 20 rounds, 70 people should have been processed and left the system.

In practice, what happens is that random variations in the rolls mean that some players roll a higher value than the number of people they have in their queue, while others roll a lower value than the number of people they have in their queue. Those that roll a higher value do not process to their full potential as they do not always have the people available to work with. Those that roll lower values than the number of people waiting can produce temporary bottlenecks. Such temporary bottlenecks not only mean that a queue builds behind that particular station, but they also choke off supply to the rest of the system, meaning that subsequent stations are even more likely not to process to their potential capacity.

The above effects can be clearly seen in the game snapshot below, where the player at station 6 has rolled a number of low values leading to a temporary bottleneck. All but one of the players after station 6 has rolled a higher value than the number of people waiting, leading to a potential capacity of these four stations of 5 + 1 + 5 + 4 = 15 being realised as an actual movement of only 7 people. Had more people been processed by station 6, more people would have been available in stations 7 to 10 to move.

The temporary bottleneck may subside after only a few rounds, but to fully recover from it and clear the backlog, the bottleneck player and all those down-stream need to roll several consecutive high values. This is statistically less likely than any single player rolling a few low values, which is what caused the problem in the first place.

The temporary bottlenecks mean that although the average roll on each round is 3.5, the average number of people moved is significantly less than this. In fact, the most likely outcome of a game is that you get 53 through the system in 20 rounds; an average movement of 2.65 people per round, as shown in the diagram below.

An analysis of the results shows that about 99% of all games played will get between 44 and 62 people through the system. The above diagram also shows it is very unlikely that you will get 70 people through, although it is possible, but with odds that are about 4 times longer than winning the UK National Lottery.

It should be noted that there is no restriction on people entering the system. This means that the roll value and movement value for the first station are equal, and will average 3.5. Because this is larger than the average number of people moved through all the subsequent stations, the number of people as work in progress in the system will gradually increase as the game progresses. Put simply – people enter the system faster than they leave it so the number in the system must grow.

Similarly, because queues start to grow as the work in progress grows and combined with the creation of temporary bottlenecks, the average passage time through the system also increases as the game progresses. These effects are shown for an example game in the diagrams below.

We can summarise by saying that the capacity of each stage of the system (local optima) does not add up to the capacity of the whole system (global optimum).

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