Mathematical models of dynamic systems

Discrete Event Simulation in Python

Diogo Costa (PhD, MSc)

Adjunct Professor, University of Saskatchewan, Canada & CEO of ImpactBLUE-Scientific

What is a mathematical model?

Description of a natural or human-driven system using mathematical concepts and language.

Models can be classified as:

Dynamic vs. non-dynamic
Discrete vs. continuous
Deterministic vs. probabilistic (stochastic)
Linear vs. nonlinear
Others

Mathematical models can be simple or complex.

Digital approximations of reality
Impossible to account for all processes

George Box: "All models are wrong, but some are useful."

Photo of a river flowing through a rocky area.

Example: model to predict natural processes

Model of flow in a river

A plot of observed and simulation streamflow over time at the Goaya hydrologic station in China. The model can capture the observed streamflow records. Streamflow varies substantially over time in both model and observations.

This model predicts streamflow for the Goaya hydrologic station, China.
The prediction is based on rainfall information that is used as model input.

Other examples

Weather forecast
Ocean swell forecast
Water quality in lakes
Urban flooding
and many other applications

¹ Chen, C., He, W., Zhou, H. et al. A comparative study among machine learning and numerical models for simulating groundwater dynamics in the Heihe River Basin, northwestern China. Sci Rep 10, 3904 (2020). https://doi.org/10.1038/s41598-020-60698-9

Example: model to predict human-driven activities

Predicting Inflation

A plot of UK inflation records (2002-2021) and forecast (2022) produced by BNP Paribas. The forecast shows a 7% inflation in 2022.

Predicting inflation is hugely important to help avoid economic crises

Other examples

Supply-Chain
Manufacturing
Logistics
Economic forecasts
Transport
Others

¹ https://www.ft.com/content/218d35ab-b044-467a-bb62-0d17547a4350

General code components and structure

3 main components
Input data and model parameters

# Define model parameters
processes = {"process_1": 5,
             "process_2": 2,
             "process_3": 3}

Run configuration

# Simulation period
simulation_time = 365

# Run model
discrete_model(processes, simulation_time)

Model engine

def discrete_model(processes, simulation_time):

  # 1) Run end-condition
  while (time < simulation_time):
      process_names = list(processes.keys())


      # 2) Loop over all processes
      for p in range(len(process_names)):
         process_name_p = process_names[p]


         # 3) Account for effect of each process
         time += processes[process_name_p]

Model outputs

Example of a discrete-event model output for a manufacturing activity

=> START OF SIMULATION  (Time = 0 days) 
Time =    6.00 days  |  Process Complete: Transport of raw material
Time =    9.00 days  |  Process Complete: Building components
Time =   11.00 days  |  Process Complete: Assembling parts
Time =   14.00 days  |  Process Complete: Selling product
=> COMPLETED: Supply-Chain cycle #1 | Time = 15.5 days
Time =   21.50 days  |  Process Complete: Transport raw material
Time =   24.50 days  | Process Complete: Building components
Time =   26.50 days  |  Process Complete: Assembling parts
Time =   29.50 days  |  Process Complete: Selling product
=> COMPLETED: Supply-Chain cycle #2 | Time = 31.0 days

Visualization of model results

Graphical visualizing of model results: helpful to identify patterns and tipping points in the system.
Visualization should be tailored to simulation objectives.
Many useful visualization packages available: matplotlib, seaborn, plotly.

Matplotlib library logo. Seaborn library logo. Plotly library logo.

Examples

2D line or scatter plots of the modeled data (y) vs. the corresponding time values (x)

plt.plot(x, y, color='green', marker='o', 
         markersize=12, linestyle='dashed', 
         linewidth=2)

Histogram to bin the data in x and count the number of values in each bin

plt.hist(x, 50, density=True, 
         facecolor='g', alpha=0.75)

Let's practice!

Discrete Event Simulation in Python