Bathtub Model
A simple stock-and-flow model that simulates the water level in a bathtub, demonstrating how the difference between inflow and outflow determines accumulation over time.
THE IDEA
Bathtub Model
The bathtub model is a classic example of a stock-and-flow system. It represents the accumulation of a resource (water in a tub) as the net result of inflow and outflow.
Equation
Stock: water in tub(t) = water in tub(t – dt) + (inflow – outflow) × dt
Parameters
- Initial stock: 50 gallons
- Time unit: minutes
- dt: 1 minute
- Run time: 10 minutes
Flow rates
- Inflow: 0 gal/min for t=0 to 5 minutes; 5 gal/min for t=6 to 10 minutes
- Outflow: constant 5 gal/min
Behavior
Initially, outflow exceeds inflow (5 > 0), so water level decreases. After 6 minutes, inflow equals outflow (5=5), so water level stabilizes.
This model illustrates how stocks respond to changes in flow rates, a fundamental concept in systems thinking.
When to use it
Managing inventory levels in a warehouse with incoming shipments and outgoing orders.
Tracking the number of customers in a queue with arrivals and departures.
Understanding population changes with birth and death rates.
Modeling the level of a pollutant in a lake with inflow from a factory and outflow to a river.
FROM THEORY TO EVERYDAY LIFE
In practice
Water Level Over Time
Start with 50 gallons of water. For the first 5 minutes, the faucet is off (inflow=0) and the drain is open (outflow=5), so water level decreases by 5 gallons per minute. At t=6, the faucet is turned on to 5 gal/min, equal to the drain rate, so water level remains constant.
Inventory Management
A warehouse has 50 units in stock. For the first 5 hours, no shipments arrive (inflow=0) but orders are shipped at 5 units per hour (outflow=5). After 5 hours, shipments arrive at 5 units per hour, matching the outflow, so inventory stabilizes.
A MOMENT TO REFLECT
Put it to work.
What happens to the water level if the inflow rate is set to 3 gal/min instead of 5 after 6 minutes?
How would the model change if the outflow depended on the water level (e.g., pressure)?
In what real-world systems does stock-and-flow behavior similar to this bathtub model appear?
CONTINUE THE CONVERSATION
Source texts
- Thinking in Systems A Primer