2.3. Structure and Behaviour Through Time - Feedback Loops and the Dynamics of a Slow-to-respond Shower
Causal loop diagrams are a stepping-stone to interpreting and communicating dynamics or performance through time. The best way to appreciate this point is to see a worked example. Here I present a hot water shower like the one at home or in a hotel room. In this example, we start from dynamics of interest and then construct a causal loop diagram that is capable of explaining the dynamics. Our analysis begins with a time chart as shown in Figure 2.7. On the vertical axis is the water temperature at the shower head and on the horizontal axis is time in seconds. Imagine it is a hot summer's day and you want to take a nice cool shower at 25°C. When you step into the cubicle the shower is already running but the water temperature is much too cold. The time chart shows three alternative time paths or trajectories for the water temperature labelled 'ideal', 'common sense', and 'most likely'. The ideal outcome is that you quickly adjust the tap setting by just the right amount and the water temperature immediately rises to the desired 25°C after which it remains rock steady. You are comfortably cool. Common sense says this ideal can't happen because, like most showers, this one is slow to respond. There is a time delay of a few seconds between adjusting the tap and a change in the water temperature. To begin with, the common-sense trajectory is flat and the water temperature remains too cold. Then, after a while the temperature begins to rise and quite soon settles at the desired 25°C. Unfortunately, bitter experience contradicts common sense. The most likely trajectory is quite different. Again the temperature starts too cold. You adjust the tap and gradually the temperature rises. After a few seconds the temperature is just right. But annoyingly it continues to rise. Before long you are much too hot, so you reverse the tap. It makes no immediate difference. So you reverse the tap even more. At last the temperature begins to fall and after a few more seconds you are again comfortably cool at 25°C. However, your comfort is short-lived as the water temperature continues to fall and you are right back where you started – too cold. The cycle continues from cold to hot and back again.
Figure 2.7. Puzzling dynamics of a slow-to-respond shower
The most likely trajectory is a classic example of puzzling dynamics, performance over time that is both unintended and surprising. Who would deliberately set-out to repeatedly freeze and scald themselves? The feedback systems thinker looks for the structure, the web of relationships and constraints involved in operating a shower, that cause normal people to self-inflict such discomfort. It is clear from Figure 2.7 that the dynamic behaviour is essentially goal seeking. The shower taker wants the water temperature to be 25°C, but the actual water temperature varies around this target. The feedback structure that belongs with such fluctuating behaviour is a balancing loop with delay, and that's exactly what we are looking for in modelling or representing the shower 'system'. This notion of having in mind a structure that fits (or might fit) observed dynamics is common in system dynamics modelling. It is known formally as a 'dynamic hypothesis', a kind of preliminary guess at the sort of relationships likely to explain a given pattern of behaviour through time.
Figure 2.8 shows a causal loop diagram for a slow-to-respond shower. First consider just the words. Five phrases are enough to capture the essence of the troublesome shower: desired temperature, actual water temperature, temperature gap, the flow of hot water and the flow of cold water. Next consider the causal links. The temperature gap depends on the difference between desired and actual water temperature. The existence of a temperature gap influences the flow of hot water. This link represents the decision making and subsequent action of the shower taker. You can imagine a person turning a tap in order to change the flow of hot water and to get comfortable. The flow of hot water then influences the actual water temperature, but with a time delay because the shower is slow-to-respond. Also shown is a separate inflow of cold water, represented as a link on the left. The water temperature obviously depends on both water flows, hot and cold.
Figure 2.8. Causal loop diagram of a slow-to-respond shower
The end result is a balancing feedback loop, labelled 'comfort seeking', which is just what we are looking for to explain cyclical behaviour. The loop type can be confirmed by adding signs (positive or negative) to each link and telling a 'story' about the process of temperature adjustment around the loop. For convenience, imagine the desired water temperature is greater than actual at time zero – in other words the shower taker feels too cold and the temperature gap is greater than zero. Now consider the polarity of the first link. If the temperature gap increases then the flow of hot water becomes greater than it would otherwise have been. This is a positive link according to the polarity conventions. In the second link, if the flow of hot water increases, then the actual water temperature increases, albeit with a delay. This too is a positive link. (Note that in making the polarity assignment the flow of cold water, which also affects water temperature, is assumed to be held constant). In the third and final link, if the water temperature increases then the temperature gap becomes smaller than it would otherwise have been. This is a negative link according to the polarity conventions. The overall effect around the loop is for an increase in the temperature gap to result in a counteracting decrease in the temperature gap, which is the sign of a balancing loop.
Incidentally, there is another way to work out loop polarity besides telling a story around the loop. It is also possible to simply count the number of negative links around the loop. An odd number of negative links (1, 3, 5 ...) signifies a balancing loop while an even number of links (0, 2, 4 ...) signifies a reinforcing loop. The reason this rule-of-thumb works is that any story about propagation of change around a loop will result in a counteracting effect for an odd number of negative links and a reinforcing effect for an even number. In this case, there is one negative link around the loop (between actual water temperature and the temperature gap) and so it is a balancing loop. The other negative link in the diagram (between flow of cold water and actual water temperature) does not count since it is not part of the closed loop.
2.3.1. Processes in a Shower 'System'
A typical causal loop diagram shows a lot about connectivity in a small space. It is a purely qualitative model, a sketch of cause and effect, particularly good for highlighting feedback loops that contribute to dynamics and to dynamic complexity. Usually there are many practical operating details about causality that lie behind the scenes. Although not shown in the diagram, it is important to be aware of this detail, particularly when building an algebraic simulator of the same feedback structure. Then it is vital to be clear and precise about how such links actually work in terms of underlying behavioural responses, economic and social conventions and physical laws. It is also important to know the numerical strength of the effects. This skill of seeing the big picture while not losing sight of operating detail is a hallmark of good system dynamics practice, known as 'seeing the forest and the trees' (Senge, 1990; Sherwood, 2002). It is a skill well-worth cultivating.
One way to forge the connection from feedback loops to operations is to ask yourself about the real-world processes that lie behind the links. In the case of the shower there is an interesting mixture of physical, behavioural and psychological processes. Take, for example, the link from the flow of hot water to actual water temperature. What is really going on here? The diagram says the obvious minimum: if the flow of hot water increases then sooner or later, and all else remaining the same, the actual water temperature at the shower head increases too. The sooner-or-later depends on the time delay in the hot water pipe that supplies the shower, which is a factor that can be estimated or measured. But how much does the temperature rise for a given increase in water flow? The answer to that question depends on physics and thermodynamics – the process of blending hot and cold water. In a simulation model you have to specify the relationship with reasonable accuracy. You do not necessarily need to be an expert yourself, but if not then you should talk with someone who knows (from practice or theory) how to estimate the water temperature that results from given flows of hot and cold water – a plumber, an engineer or maybe even a physicist. Consider next the link from actual water temperature to the temperature gap. Algebraically the gap is defined as the difference between the desired and actual water temperature (temperature gap desired water temperature - actual water temperature). But a meaningful temperature gap in a shower also requires a process for sensing the gap. The existence of a temperature gap alone does not guarantee goal-seeking behaviour. For example, if someone entered a shower in a winter wetsuit, complete with rubber hood and boots, they would not notice a temperature gap, and the entire feedback loop would be rendered inactive. Although this case is extreme and fanciful, it illustrates the importance of credibly grounding causal links.
The final link in the balancing loop is from temperature gap to the flow of hot water. Arguably this is the single most important link in the loop because it embodies the decision-making process for adjusting the flow of hot water. There is a huge leap of causality in this part of the diagram. The common-sense interpretation of the link is that when any normal person feels too hot or too cold in a shower, he or she will take corrective action by adjusting the flow of hot water. But how do they judge the right amount of corrective action? How quickly do they react to a temperature gap and how fast do they turn the tap? All these factors require consideration. Moreover, the key to over-reaction in showers arguably lies in this single step of causality. Why do people get trapped into a repetitive hot-cold cycle when all they normally want to achieve is a steady comfortable temperature? The answer must lie in how they choose to adjust the tap setting, in other words in their own decision-making process. Later on, in Chapter 4, we will investigate a full-blown algebraic model of a shower and review the main formulations in the balancing loop. For now it's enough to know that behind the three links of this alluringly simple loop lie practical processes for blending hot and cold water, for sensing a temperature gap and for adjusting the flow of hot water.
2.3.2. Simulation of a Shower and the Dynamics of Balancing Loops
Figure 2.9 shows the simulated dynamics of a slow-to-respond shower over a period of 120 seconds generated by a simulation model containing all the processes mentioned above. As before, the desired water temperature is a cool 25°C. However, in this scenario the water temperature starts too high at 40°C. Corrective action lowers the temperature at the shower-head to the desired 25°C in about 10 seconds, but the temperature continues to fall, reaching a minimum just below 24°C after 12 seconds. Further corrective action then increases the temperature, leading to an overshoot that peaks at 27°C after 21 seconds. The cycle repeats itself twice in the interval up to 60 seconds, but each time the size of the temperature overshoot and undershoot is reduced as the shower taker gradually finds exactly the right tap setting for comfort. In the remainder of the simulation, from 60 to 120 seconds, the temperature at the shower-head remains steady at 25°C. The overall trajectory is a typical example of goal-seeking dynamics arising from a balancing loop with delay.
It is worthwhile to remember this particular combination of feedback structure and dynamic behaviour because balancing loops crop up all over the place in business, social, environmental and biological systems. Wherever people, organisations or even organisms direct their efforts and energy to achieving and maintaining specific goals in the face of an uncertain and changing environment there are balancing loops at work. Companies set themselves sales objectives, quality standards, financial targets and goals for on-time delivery. Governments set targets for economic growth, inflation, hospital waiting times, literacy, exam pass rates, road congestion, and public transport usage. The human body maintains weight, balance, temperature, and blood sugar. The ecosystem sustains an atmosphere suitable for the animals and plants within it. The vast global oil industry maintains a supply of oil sufficient to reliably fill our petrol tanks. The electricity industry supplies just enough electricity to keep the lights on. Economies generate enough jobs to keep most people employed. The list goes on and on. In some cases, like people's body temperature or domestic electricity supply, the balancing process works so well that it is rare to find deviations from the 'goal' – a degree or two from normal body temperature is a sign of illness and, in the electricity industry, it is unusual (at least in the developed world) for the lights to dim. In many cases, like sales objectives or hospital waiting times, the goals are known, but performance falls chronically short or else gently overshoots and undershoots. In other cases, however, like employment in the economy or inventory levels in supply chains, the balancing process is far from perfect. Performance deviates a long way from the goal, too much or too little. Corrective action leads to over- and undercompensation and the goal is never really achieved, at least not for long.
Figure 2.9. Simulated dynamics of a slow-to-respond shower
