# Figure 3.22: Queuing dynamics

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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 |

Requires | python-control |

**Figure 3.22:** Queuing dynamics. (a) The steady-state queue length as a function of . (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 , and the arrival rate starts at . The arrival rate is increased to at time 20, and it returns to 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')