Modeling and Simulation in Python

Chapter 4

Copyright 2017 Allen Downey

License: Creative Commons Attribution 4.0 International

In [1]:
# Configure Jupyter so figures appear in the notebook
%matplotlib inline

# Configure Jupyter to display the assigned value after an assignment
%config InteractiveShell.ast_node_interactivity='last_expr_or_assign'

# import functions from the modsim library
from modsim import *

Returning values

Here's a simple function that returns a value:

In [2]:
def add_five(x):
    return x + 5

And here's how we call it.

In [3]:
y = add_five(3)

If you run a function on the last line of a cell, Jupyter displays the result:

In [4]:

But that can be a bad habit, because usually if you call a function and don't assign the result in a variable, the result gets discarded.

In the following example, Jupyter shows the second result, but the first result just disappears.

In [5]:

When you call a function that returns a variable, it is generally a good idea to assign the result to a variable.

In [6]:
y1 = add_five(3)
y2 = add_five(5)

print(y1, y2)

Exercise: Write a function called make_state that creates a State object with the state variables olin=10 and wellesley=2, and then returns the new State object.

Write a line of code that calls make_state and assigns the result to a variable named init.

In [7]:
# Solution goes here

In [8]:
# Solution goes here

Running simulations

Here's the code from the previous notebook.

In [9]:
def step(state, p1, p2):
    """Simulate one minute of time.
    state: bikeshare State object
    p1: probability of an Olin->Wellesley customer arrival
    p2: probability of a Wellesley->Olin customer arrival
    if flip(p1):
    if flip(p2):
def bike_to_wellesley(state):
    """Move one bike from Olin to Wellesley.
    state: bikeshare State object
    if state.olin == 0:
        state.olin_empty += 1
    state.olin -= 1
    state.wellesley += 1
def bike_to_olin(state):
    """Move one bike from Wellesley to Olin.
    state: bikeshare State object
    if state.wellesley == 0:
        state.wellesley_empty += 1
    state.wellesley -= 1
    state.olin += 1
def decorate_bikeshare():
    """Add a title and label the axes."""
    decorate(title='Olin-Wellesley Bikeshare',
             xlabel='Time step (min)', 
             ylabel='Number of bikes')

Here's a modified version of run_simulation that creates a State object, runs the simulation, and returns the State object.

In [10]:
def run_simulation(p1, p2, num_steps):
    """Simulate the given number of time steps.
    p1: probability of an Olin->Wellesley customer arrival
    p2: probability of a Wellesley->Olin customer arrival
    num_steps: number of time steps
    state = State(olin=10, wellesley=2, 
                  olin_empty=0, wellesley_empty=0)
    for i in range(num_steps):
        step(state, p1, p2)
    return state

Now run_simulation doesn't plot anything:

In [11]:
state = run_simulation(0.4, 0.2, 60)

But after the simulation, we can read the metrics from the State object.

In [12]:

Now we can run simulations with different values for the parameters. When p1 is small, we probably don't run out of bikes at Olin.

In [13]:
state = run_simulation(0.2, 0.2, 60)

When p1 is large, we probably do.

In [14]:
state = run_simulation(0.6, 0.2, 60)

More for loops

linspace creates a NumPy array of equally spaced numbers.

In [15]:
p1_array = linspace(0, 1, 5)

We can use an array in a for loop, like this:

In [16]:
for p1 in p1_array:

This will come in handy in the next section.

linspace is defined in You can get the documentation using help.

In [17]:

linspace is based on a NumPy function with the same name. Click here to read more about how to use it.

Exercise: Use linspace to make an array of 10 equally spaced numbers from 1 to 10 (including both).

In [18]:
# Solution goes here

Exercise: The modsim library provides a related function called linrange. You can view the documentation by running the following cell:

In [19]:

Use linrange to make an array of numbers from 1 to 11 with a step size of 2.

In [20]:
# Solution goes here

Sweeping parameters

p1_array contains a range of values for p1.

In [21]:
p2 = 0.2
num_steps = 60
p1_array = linspace(0, 1, 11)

The following loop runs a simulation for each value of p1 in p1_array; after each simulation, it prints the number of unhappy customers at the Olin station:

In [22]:
for p1 in p1_array:
    state = run_simulation(p1, p2, num_steps)
    print(p1, state.olin_empty)

Now we can do the same thing, but storing the results in a SweepSeries instead of printing them.

In [23]:
sweep = SweepSeries()

for p1 in p1_array:
    state = run_simulation(p1, p2, num_steps)
    sweep[p1] = state.olin_empty

And then we can plot the results.

In [24]:
plot(sweep, label='Olin')

decorate(title='Olin-Wellesley Bikeshare',
         xlabel='Arrival rate at Olin (p1 in customers/min)', 
         ylabel='Number of unhappy customers')


Exercise: Wrap this code in a function named sweep_p1 that takes an array called p1_array as a parameter. It should create a new SweepSeries, run a simulation for each value of p1 in p1_array, store the results in the SweepSeries, and return the SweepSeries.

Use your function to plot the number of unhappy customers at Olin as a function of p1. Label the axes.

In [25]:
# Solution goes here

In [26]:
# Solution goes here

Exercise: Write a function called sweep_p2 that runs simulations with p1=0.5 and a range of values for p2. It should store the results in a SweepSeries and return the SweepSeries.

In [27]:
# Solution goes here

In [28]:
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Optional Exercises

The following two exercises are a little more challenging. If you are comfortable with what you have learned so far, you should give them a try. If you feel like you have your hands full, you might want to skip them for now.

Exercise: Because our simulations are random, the results vary from one run to another, and the results of a parameter sweep tend to be noisy. We can get a clearer picture of the relationship between a parameter and a metric by running multiple simulations with the same parameter and taking the average of the results.

Write a function called run_multiple_simulations that takes as parameters p1, p2, num_steps, and num_runs.

num_runs specifies how many times it should call run_simulation.

After each run, it should store the total number of unhappy customers (at Olin or Wellesley) in a TimeSeries. At the end, it should return the TimeSeries.

Test your function with parameters

p1 = 0.3
p2 = 0.3
num_steps = 60
num_runs = 10

Display the resulting TimeSeries and use the mean function provided by the TimeSeries object to compute the average number of unhappy customers (see Section 2.7).

In [29]:
# Solution goes here

In [30]:
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Exercise: Continuting the previous exercise, use run_multiple_simulations to run simulations with a range of values for p1 and

p2 = 0.3
num_steps = 60
num_runs = 20

Store the results in a SweepSeries, then plot the average number of unhappy customers as a function of p1. Label the axes.

What value of p1 minimizes the average number of unhappy customers?

In [31]:
# Solution goes here

In [32]:
# Solution goes here