What Accumulates

Online, with a lab for each exercise: thinking-in-diagrams.aryehnielsen.com

Stock & flow diagramsPart III · Accumulation · Chapter 6 of 12 · the first of two chapters on stock & flow diagrams

• Something builds up or runs down: savings, a backlog, a population.
• A cut in what comes in hasn’t stopped the total rising.
• Something leaves one place and has to turn up in another.
• Amounts have units, and the units have to agree.

With the plug in and the tap on, the water in a bathtub rises by the same amount every minute.

With the plug out, it rises more slowly as it deepens, since deeper water drains faster, until it drains as fast as it fills and holds one depth.

Turn off the tap and put the plug in: filling and draining stop, and the water is still there.

A stock is a quantity that accumulates: what would still be there if every flow stopped. The water in the tub can be measured at an instant; filling and draining exist only over time. A stock changes only through its flows.

It is drawn as a box around its name, its units beneath in smaller type. The world’s population is drawn the same way, filled by births and drained by deaths.

A flow moves material into or out of a stock. It is drawn as a double line, its material path, with an arrowhead where the material goes and, at its middle, a valve: a small circle crossed by an X, the flow’s name beside it.

A flow into a stock is one of its inflows, a flow out of it one of its outflows: filling is the tub’s inflow, draining its outflow. A flow’s value at an instant is its rate, such as 12 liters a minute.

A flow is named for what moves, filling or births, not for what sets it moving, such as tap.

A cloud, a small unnamed scalloped outline, is where a flow’s material comes from or goes to outside the model: the flow’s source where it starts, its sink where it ends. The bathtub’s model leaves out the water main and the sewer, so filling starts at a cloud and draining ends at one.

Each cloud ends one flow; whatever you name or count is a stock instead.

Every stock has units: liters for the water in the tub, people for the world’s population.

A diagram has one time unit, the unit every rate is per: the minute for the bathtub, the year for the population. A flow’s units are its stock’s units per time unit, so filling and draining run in liters per minute.

A transfer is a flow from one stock straight into another: an outflow of the first and an inflow of the second, such as pouring from water in bucket into water in tub.

Material that leaves a stock arrives somewhere, in another stock or at a cloud: conservation. So both stocks of a transfer count it in the same units, liters for the bucket and the tub.

An auxiliary is a variable that neither accumulates nor flows, such as a setting, a capacity or a ratio, drawn as its name alone: tap setting, drain opening, or the population’s births per person, the year’s births divided by the people alive. In Stella, a modeling program, an auxiliary is a converter.

A signal is a single thin line with an arrowhead, carrying a reading of one quantity to an auxiliary or to a flow’s valve; in the systems literature, an information link.

Tap setting → filling carries the tap’s setting, not water, and has a polarity as chapter 4’s effects thinking-in-diagrams.aryehnielsen.com/feedback/loops-that-amplify-loops-that-correct/#polarity do: +, since the further the tap is open, the faster the tub fills.

+ signals run from water in tub to draining and from world population to births and deaths. A signal from a stock to one of its own flows closes a loop through that stock (chapter 7 thinking-in-diagrams.aryehnielsen.com/accumulation/stocks-inside-feedback/#loops-through-stocks).

A stock equation, one line beneath a stock’s box, adds its inflows and subtracts its outflows:

change in water in tub = filling − draining

Its right-hand side is the stock’s net flow. With filling at 12 liters a minute and draining at 8, the net flow is 4 liters a minute. In 2023 about 132 million people were born worldwide and 62 million died: a net flow of about 70 million people a year.

Auxiliaries and signals never appear in it; they change a stock only through a flow’s rate. A stock with no flows reads change in … = 0.

A delay on a flow, two short strokes across its double line, means its material arrives well after it leaves, as a change arrives late along a delayed effect thinking-in-diagrams.aryehnielsen.com/feedback/loops-that-amplify-loops-that-correct/#delays. Grain shipped from a port reaches a mill weeks later: the flow from grain at the port to grain at the mill carries a delay, and the grain at sea is in neither stock.

A behavior-over-time graph shows one stock, flow or auxiliary over time: the quantity’s name (and a stock’s units) on the vertical axis, time on the horizontal, and one line of a named shape, rising or falling. The shapes are linear, exponential, approach (flattening toward a value), oscillation (swinging around a value), S-shaped and overshoot and collapse. A graph is a prediction from the structure, made before anything is measured.

A stock’s slope at any instant is its net flow: it rises while inflows exceed outflows, falls while outflows exceed inflows, and stays flat when they are equal.

With the plug in, the net flow is filling’s steady rate, so the water rises in a straight line.

With the plug out, draining speeds up as the water deepens, the net flow shrinks, and the water approaches the depth at which draining equals filling.

Asked to draw the water in a bathtub from graphs of its inflow and outflow, many graduate students at MIT’s business school drew a path its flows couldn’t produce (Booth Sweeney and Sterman, 2000); chapter 7’s Common misreading thinking-in-diagrams.aryehnielsen.com/accumulation/stocks-inside-feedback/#common-misreading gives a later study’s numbers.

When nothing accumulates, or the question is only which way one quantity moves another, a sentence does: opening the drain wider empties the tub faster. With several such effects running in a circle, a causal loop diagram does. When the question is how much there is now, not how it changes, a list of amounts does.

The diagram holds no numbers: rates and amounts go in a sentence or a table beside it.

What’s wrong with this diagram of a bathtub filling from the tap and emptying down the drain?

The tap, the water in the tub and the drain are three stocks in a row, joined by filling and draining. The tap is counted in liters per minute and the tub in liters, so filling’s units come out as liters per minute per minute. The tap holds no water: its setting fixes how fast filling runs, as the drain’s opening fixes draining. The lab’s check reports:

2 transfers join stocks in different units: filling, from tap (liters per minute) to water in tub (liters); draining, from water in tub (liters) to drain (liters per minute). What leaves one stock arrives in the other, so both count it in the same units.

The tap and the drain become auxiliaries, tap setting and drain opening, each with a + signal to its flow’s valve. Filling starts at a cloud and draining ends at one, leaving water in tub, in liters, the one stock, and both flows in liters per minute.

Passing the checks means meeting the brief, not being right.

• Donella H. Meadows, Thinking in Systems: A Primer (Chelsea Green, 2008), chapter 1, The Basics, on stocks, flows, clouds as sources and sinks, and the bathtub.
• John D. Sterman, Business Dynamics: Systems Thinking and Modeling for a Complex World (Irwin/McGraw-Hill, 2000), chapter 4, Structure and Behavior of Dynamic Systems, on the shapes behavior over time takes; chapter 6, Stocks and Flows, on the notation and on telling a stock from a flow by freezing a system in time; and chapter 7, Dynamics of Stocks and Flows, on reading a stock’s path from its net flow.
• Linda Booth Sweeney and John D. Sterman, “Bathtub Dynamics: Initial Results of a Systems Thinking Inventory” doi.org/10.1002/sdr.198, System Dynamics Review 16, no. 4 (2000), on how well students with no training in system dynamics read stocks and flows.
• isee systems, Working with Building Blocks iseesystems.com/resources/help/v3/Content/03-BuildingModels/Working_with_building_blocks.htm (Stella help), on the parts of a model in Stella, a modeling program.
• United Nations, World Population Prospects 2024, through Our World in Data, Number of births ourworldindata.org/grapher/number-of-births-per-year and Number of deaths ourworldindata.org/grapher/number-of-deaths-per-year, for the world’s births and deaths in 2023.