Murray at 18:46, 16 November 2024

2024-11-16T18:46:56Z

Murray at 16:36, 28 May 2023

2023-05-28T16:36:10Z

Murray: Created page with "{{Figure |Chapter=System Modeling |Figure number=22 |Figure title=Queuing dynamics |GitHub URL=https://github.com/murrayrm/fbs2e-python/blob/main/example-3.15-queuing_systems...."

2021-08-30T05:32:05Z

Created page with "{{Figure |Chapter=System Modeling |Figure number=22 |Figure title=Queuing dynamics |GitHub URL=https://github.com/murrayrm/fbs2e-python/blob/main/example-3.15-queuing_systems...."

New page

{{Figure
|Chapter=System Modeling
|Figure number=22
|Figure title=Queuing dynamics
|GitHub URL=https://github.com/murrayrm/fbs2e-python/blob/main/example-3.15-queuing_systems.py
}}
[[Image:figure-3.22-queuing_dynamics.png|640px]]

'''Figure 3.22:''' Queuing dynamics. (a) The steady-state queue length as a function of <math>\lambda/\mu_\text{max}</math>. (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 <math>\mu_{max} = 1</math>, and the arrival rate starts at <math>\lambda = 0.5</math>. The arrival rate is increased to <math>\lambda = 4</math> at time 20, and it returns to <math>\lambda =0.5</math> at # time 25.

<nowiki>
# 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')
</nowiki>

@@ Line 42: / Line 42: @@
 # Create I/O system representation
-queuing_sys = ct.NonlinearIOSystem(
+queuing_sys = ct.nlsys(
      updfcn=queuing_model, inputs=1, outputs=1, states=1)

@@ Line 1: / Line 1: @@
 {{Figure
 |Chapter=System Modeling
-|Figure number=22
+|Figure number=3.22
 |Figure title=Queuing dynamics
 |GitHub URL=https://github.com/murrayrm/fbs2e-python/blob/main/example-3.15-queuing_systems.py

Figure 3.22: Queuing dynamics - Revision history

Murray at 18:46, 16 November 2024

Murray at 16:36, 28 May 2023

Murray: Created page with "{{Figure |Chapter=System Modeling |Figure number=22 |Figure title=Queuing dynamics |GitHub URL=https://github.com/murrayrm/fbs2e-python/blob/main/example-3.15-queuing_systems...."