Figure 3.22: Queuing dynamics

Chapter	System Modeling
Figure number	3.22
Figure title	Queuing dynamics
GitHub URL	https://github.com/murrayrm/fbs2e-python/blob/main/example-3.15-queuing systems.py
Requires	python-control

Figure 3.22: Queuing dynamics. (a) The steady-state queue length as a function of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda/\mu_\text{max}} . (b) The behavior of the queue length when there is a temporary overload in the system. The solid line shows a realization of an event-based simulation, and the dashed line shows the behavior of the flow model (3.33). The maximum service rate is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mu_{max} = 1} , and the arrival rate starts at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda = 0.5} . The arrival rate is increased to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda = 4} at time 20, and it returns to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda =0.5} at # time 25.

# example-3.15-queuing_systems.py - Queuing system modeling
# RMM, 29 Aug 2021
#
# Figure 3.22: Queuing dynamics. (a) The steady-state queue length as a
# function of $\lambda/\mu_{max}$. (b) The behavior of the queue length when
# there is a temporary overload in the system. The solid line shows a
# realization of an event-based simulation, and the dashed line shows the
# behavior of the flow model (3.33). The maximum service rate is $\mu_{max}
# = 1$, and the arrival rate starts at $\lambda = 0.5$. The arrival rate is
# increased to $\lambda = 4$ at time 20, and it returns to $\lambda =0.5$ at
# time 25.
#

import control as ct
import numpy as np
import matplotlib.pyplot as plt

# Queing parameters 

# Queuing system model (KJA, 2006)
def queuing_model(t, x, u, params={}):
    # Define default parameters
    mu = params.get('mu', 1)

    # Get the current load
    lambda_ = u

    # Return the change in queue size
    return np.array(lambda_ - mu * x[0] / (1 + x[0]))

# Create I/O system representation
queuing_sys = ct.NonlinearIOSystem(
    updfcn=queuing_model, inputs=1, outputs=1, states=1)

# Set up the plotting grid to match the layout in the book
fig = plt.figure(constrained_layout=True)
gs = fig.add_gridspec(3, 2)

#
# (a) The steady-state queue length as a function of $\lambda/\mu_{max}$.
#

fig.add_subplot(gs[0, 0])       # first row, first column

# Steady state queue length
x = np.linspace(0.01, 0.99, 100)
plt.plot(x, x / (1 - x), 'b-')

# Label the plot
plt.xlabel(r"Service rate excess $\lambda/\mu_{max}$")
plt.ylabel(r"Queue length $x_{e}$")
plt.title("Steady-state queue length")

#
# (b) The behavior of the queue length when there is a temporary overload
# in the system. The solid line shows a realization of an event-based
# simulation, and the dashed line shows the behavior of the flow model
# (3.33). The maximum service rate is $\mu_{max} = 1$, and the arrival
# rate starts at $\lambda = 0.5$. The arrival rate is increased to $\lambda
# = 4$ at time 20, and it returns to $\lambda =0.5$ at time 25.
#

fig.add_subplot(gs[0, 1])       # first row, first column

# Construct the loading condition
t = np.linspace(0, 80, 100)

u =np.ones_like(t) * 0.5
u[t <= 25] = 4
u[t < 20] = 0.5

# Simulate the system dynamics
response = ct.input_output_response(queuing_sys, t, u)

# Plot the results
plt.plot(response.time, response.outputs, 'b-')

# Label the plot
plt.xlabel("Time $t$ [s]")
plt.ylabel(r"Queue length $x_{e}$")
plt.title("Overload condition")

# Save the figure
plt.savefig("figure-3.22-queuing_dynamics.png", bbox_inches='tight')