TRAFFIC NETWORKS

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TRAFFIC NETWORKS
Basic Components, Linkages through Boundary
Conditions and Network Effects

• C. F. Daganzo
• (http//www.ce.berkeley.edu/daganzo/)

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GOAL
Review basic models and describe the macroscopic
phenomena that arises when we link them together
to form networks.
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A LANE-DROP BOTTLENECK
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OUTLINE
1. Unify eight single-lane models VT1, VT2,
Newell (queuing, lower order), ACT, CF(L), CA(L)
and CA(M). 2. Composition through lane
changes3. Network effects.
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A SINGLE STREAM THE DATA
n -1
n 0
n 1
xn(t)
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3-D VIEW MOSKOWITZS SURFACE
n
n

• Properties
• n is a function N(t, x)
• decreasing in x

• Properties
• x is a function X(t, n)
• decreasing in n

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THE QUESTION
s
n

• Given S (or some properties of it) in data
  domain

• Find S (i.e., the function X or N) in a
  solution domain

• Requirements well-posedness and realism

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CLASSIFICATION OF SOLUTION METHODS
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THE POINT OF REFERENCE PRIMAL VERSION
Nt or q

• N(t, x) is of H-J form
• Nt Q(- Nx)

r
qmax
Q
-w
u
ko
?
-Nx or k

• The arrow of time

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THE POINT OF REFERENCE DUAL VERSION

• X and N related by X(t, N(t, x)) x

• Nt Q(-Nx) becomes
• Xt V(-Xn) where
• V(s) sQ(1/s)

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CLASSIFICATION OF SOLUTION METHODS
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VARIATIONAL SOLUTION PRIMAL FORM
Theorem 1 all valid paths from B to P have
the same cost, tBP r(vBP), and the cost is
linear in the coordinates of B and P.

Corollary 1 if the boundary and the data are
PWL the minimization is an LP in the unknown B.
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VARIATIONAL SOLUTION DUAL FORM
u
r
-w
Transformed Fundamental Diagram

• Cost rate (speed) is also linear (inverse of the
  primal)
• Theorem All valid paths have the same cost.
• Corollary PWL problems reduce to LPs.

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CLASSIFICATION OF SOLUTION METHODS

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EXAMPLE OF DUAL PROBLEM Looking for X(t, n) ?
xn(t)

• xn(t) minxn(0) ut, x0(t-tn) wnt
  Newell-car-following. 2
• Primal analysis Newells queuing formula for
  N(t, x).3

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CLASSIFICATION OF SOLUTION METHODS

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SUFFICIENT NETWORKS Discrete space-time
PRIMAL
DUAL
x
n
u
r
-w
t
t

• Theorem 1 Geometric networks with wave speed
  slopes are sufficient.

• Theorem 1 If every corner of a PWL boundary is
  a node of a sufficient network, the network
  solution is exact.

• Translational symmetry allows efficient solution
  with DP.

• DP solution of primal leads to ACT method 4.

• DP solution of dual leads to the CF(L) rule 5

x(tt, n1) x(t, n1) mintu, x(t, n) x(t,
n1) sj
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RECAP SINGLE STREAM MODELS

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CELLULAR AUTOMATA FORMS

• DP recursion CF(L) model

x(tt, n1) x(t, n1) mintu, x(t, n) x(t,
n1) sj

• Dimensionless version

xn1(t1) xn1(t) minu, xn(t) xn1(t) 1

• CA(L) model

xn1(t1) xn1(t) minu, xn(t)
xn1(t) 1
Theorem 5 CA(L) solves LVP for all n, t with
error 1.
Result 5 CA(L) model has a dual--the CA(M)
model.
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RECAP SINGLE STREAM MODELS

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BOTTLENECKS A first step towards composition
r1()
xB(t)
r2()
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COMPOSITION IN PARALLEL LANE-CHANGING
(ii)

• Boundary conditions 6
• (i) Transfer of mass
• (ii) Transfer of momentum

(i)

• CF(L) CA(L,M) ACT if lane changes are given

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COMPOSITION IN PARALLEL LANE-CHANGING

• Endogenous treatment requires6
• Generation of mass transfers
• Optional moves (increase speed)
• Mandatory moves (destination-based)
• Constrained motion rules
• By acceleration/deceleration
• By inability to overtake

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COMPOSITION AT INTERSECTIONS
LC Ban
LC Ban
Mandatory LC
Merges
Diverges
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ON-RAMP MERGE DATA from 7 MODEL from 8
Data
Lane changes
Cumulative flows
Model
Cumulative flows
Lane changes
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NETWORK EFFECTS AND CHAOS 9
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OTHER ISSUES

• The conundrum of route choice (physics) 10
• Scalability conjectures 9

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• QUESTIONS

http//www.ce.berkeley.edu/daganzo/
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REFERENCES

• 1. Daganzo, C.F. A variational formulation of
  kinematic waves Basic theory and complex
  boundary conditions Transportation Research B 39
  (2), 187-196 (2005) and A variational
  formulation of kinematic waves Solution methods
  Transportation Research Part B (2005).
• 2. Newell, G.F. A lower order model
  Transportation Research Part B 36(3) (2002).
• 3. Newell, G. F. A simplified theory of
  kinematic waves, Parts I, II and III
  Transportation Research Part B, 27 281-313
  (1993).
• 4. Daganzo, C.F. A theory of supply chains
  Springer, Heidelberg, Germany (2003).
• 5. Daganzo, C.F. In traffic flow, cellular
  automata kinematic waves Transportation Researc
  h Part B, 40 (5) 396-403 (2006).
• 6. Laval, J. and Daganzo, C.F. Lane changing in
  traffic streams Transportation Research Part B,
  40, 251-264 (2006).
• 7. Cassidy, M. and J. Rudjanakanoknad. (2005)
  Increasing Capacity of an Isolated Merge by
  Metering its On-ramp Transportation Research B,
  39B(10) 896-913.
• 8. Menendez, M. (2006) An analysis of HOV lanes
  their impacts on traffic PhD Thesis, U. of
  California, Berkeley, CA
• 9. Daganzo, C.F. The nature of urban gridlock
  Transportation Research Part B, (in press).
• 10. Heydecker, B. and Addison, J. An exact
  expression of dynamic traffic equilibrium J.B.
  Lesort (editor) Proceedings 13th ISTTT, Elsevier
  (1996).

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QUEUES FROM NOWHERE?
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AGGREGATION HYPOTHESIS