import numpy as np
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def budget(v: float, lam: list[float], v0: list[float], d: list[float], lengths: list[float], L: float) -> float:
"""
Calculates the remaining budget for road space allocation.
Args:
v: float or numpy array, representing the position of the vehicle(s).
lam: list of floats, representing the slopes for different vehicles.
v0: list of floats, representing the desired speeds for different vehicles.
d: list of floats, representing the jam spacings for different vehicles.
lengths: list of floats, representing the lengths of different vehicles.
L: float, representing the total road space.
Returns:
budget: float, representing the remaining budget for road space allocation.
"""
budget = L
for i, lam_i in enumerate(lam):
v0_i = v0[i]
d_i = d[i]
l_i = lengths[i]
s_i = vo(v, lam_i, v0_i, d_i)
budget -= s_i
budget -= l_i
return budget
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def f(x: float, lam: float, v0: float, d: float) -> float:
"""
Calculates the flow of vehicles.
Args:
x: float or numpy array, representing the position of the vehicle(s).
lam: float, representing the slope.
v0: float, representing the desired speed.
d: float, representing the jam spacing.
Returns:
y: float or numpy array, representing the flow of vehicles.
"""
y = np.heaviside((x - d), 1) * (lam * np.exp((-lam / v0) * (x - d)))
return y
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def ov(x: float, lam: float, v0: float, d: float) -> float:
"""
Calculates the optimal velocity of a vehicle.
Args:
x: float or numpy array, representing the position of the vehicle(s).
lam: float, representing the slope.
v0: float, representing the desired speed.
d: float, representing the jam spacing.
Returns:
y: float or numpy array, representing the optimal velocity of the vehicle(s).
"""
y = np.maximum(0, v0 - v0 * np.exp((-lam / v0) * (x - d)))
return y
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def vo(x: float, lam: float, v0: float, d: float) -> float:
"""
Calculates the velocity of a vehicle in free flow.
Parameters:
x: float or numpy array, representing the position of the vehicle(s).
lam: float, representing the slope.
v0: float, representing the desired speed.
d: float, representing the jam spacing.
Returns:
y: float or numpy array, representing the velocity of the vehicle(s) in free flow.
"""
y = (-v0 / lam) * np.log(1 - x / v0) + d
return y