Minplus Homogeneous Dynamical Systems N. Farhi, M. Goursat & J.-P. Quadrat
Minplus Homogeneous Dynamical Systems Growth rate, Eigenvalues and Traffic Applications
Introduction Growth Rate Eigenvalue The Growth Rate is not an Eigenvalue Traffic Application Bibliography
N. Farhi, M. Goursat & J.-P. Quadrat INRIA-Rocquencourt (France)
13/11/2008
Good Retirement Geert Jan !! Minplus Homogeneous Dynamical Systems N. Farhi, M. Goursat & J.-P. Quadrat Introduction Growth Rate Eigenvalue The Growth Rate is not an Eigenvalue Traffic Application Bibliography
Outline Minplus Homogeneous Dynamical Systems N. Farhi, M. Goursat & J.-P. Quadrat Introduction Growth Rate
1 Introduction 2 Growth Rate 3 Eigenvalue
Eigenvalue The Growth Rate is not an Eigenvalue
4 The Growth Rate is not an Eigenvalue 5 Traffic Application
Traffic Application Bibliography
6 Bibliography
Definitions Minplus Homogeneous Dynamical Systems N. Farhi, M. Goursat & J.-P. Quadrat
Minplus homogeneous dynamical systems:
x k+1 = f (x k ), with f : Rnmin 7→ Rnmin : f (λ ⊗ x) = λ ⊗ f (x) .
Introduction Growth Rate
Growth rate χ ∈ Rmin :
Eigenvalue The Growth Rate is not an Eigenvalue Traffic Application Bibliography
χ = lim xik /k, k
∀i = 1, · · · , n .
Eigenvalues λ ∈ Rmin : ∃x 6= ε : f (x) = λ ⊗ x.
Problems and Applications Minplus Homogeneous Dynamical Systems N. Farhi, M. Goursat & J.-P. Quadrat
Questions: ?∃χ,
, ?∃λ,
?χ = λ .
TRUE when f is monotone and G(f ) strongly connected. Introduction Growth Rate
Traffic Applications (f homogeneous not monotone):
Eigenvalue 1
The Growth Rate is not an Eigenvalue Traffic Application Bibliography
1
0 1
1
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0
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Canonical form of Homogeneous Systems Minplus Homogeneous Dynamical Systems N. Farhi, M. Goursat & J.-P. Quadrat Introduction
The dynamics x k+1 = f (x k ) is equivalent to ( x1k+1 /x1k = f1 (x k )/x1k , xik+1 /x1k+1 = fi (x k )/f1 (x k ), i = 2, · · · , n, using the homogeneity it can be written :
Growth Rate Eigenvalue The Growth Rate is not an Eigenvalue Traffic Application
Dynamics Canonical Form ( ∆k = h(y k ), y k+1 = g (y k ),
Bibliography
k = x k /x k and g with ∆k , x1k+1 /x1k , yi−1 i−1 = fi /f1 for 1 i i = 2, · · · , n.
Growth rate Minplus Homogeneous Dynamical Systems N. Farhi, M. Goursat & J.-P. Quadrat Introduction Growth Rate Eigenvalue The Growth Rate is not an Eigenvalue
As soon as the y k belong to a bounded closed (compact) set for all k, the set of measures: 1 PyN0 = δy 0 + δg (y 0 ) + · · · + δg N−1 (y 0 ) , N ∈ N , N is tight. Therefore we can extract convergent subsequences which converge towards invariant measures Qy 0 . Applying the ergodic theorem to the sequence (y k )k∈N : Growth Rate Existence
Traffic Application Bibliography
1 1 χ = (x1N − x10 ) = lim N N N
N−1 X k=0
! k
h(y )
Z =
h(y )dQy 0 (y ), Qy 0 a.e.
Remarks on Growth rate Existence Minplus Homogeneous Dynamical Systems N. Farhi, M. Goursat & J.-P. Quadrat
It would be very useful to prove that the limit exists for sequence starting from y 0 . 1
A priori homogeneous systems have not the uniform continuity property necessary to prove the convergence of the Cesaro means for y 0 .
2
In the case where the compact set is finite, we can apply the ergodicity results on Markov chains with a finite state number to show the convergence of PyN0 towards Qy 0 which proves the convergence of the Birkhoff average for the sequence starting from y 0 .
Introduction Growth Rate Eigenvalue The Growth Rate is not an Eigenvalue Traffic Application Bibliography
Non Everywhere Convergence of Birkhoff Averages Minplus Homogeneous Dynamical Systems N. Farhi, M. Goursat & J.-P. Quadrat
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f : x ∈ T1 → 2x ∈ T1 with: x 0 = 0.100111100000000 · · · 0.70
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Figure: Plot of S(n) with: S(n) ,
1 n
Pn−1 k=0
xk.
Eigenvalue of Homogeneous Systems Minplus Homogeneous Dynamical Systems N. Farhi, M. Goursat & J.-P. Quadrat Introduction Growth Rate Eigenvalue The Growth Rate is not an Eigenvalue Traffic Application
The eigenvalue problem a function f : Rnmin 7→ Rnmin can be formulated as finding x ∈ Rnmin non zero, and λ ∈ Rmin such that: λ ⊗ x = f (x) . Since f is homogeneous, we can suppose without loss of generality that if x exists then x1 6= ε and we have the: Eigenvalue Canonical Form: ( λ = h(y ) , y = g (y ) ,
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with yi−1 = xi /x1 , h(y ) = f1 (x)/x1 and gi−1 = fi /f1 for i = 2, · · · , n.
Eigenvalue Existence Minplus Homogeneous Dynamical Systems N. Farhi, M. Goursat & J.-P. Quadrat Introduction Growth Rate
Eigenvector Existence The existence of eigenvalue is reduced to the existence of the fixed point of g which gives an eigenvector. Standard Examples 1
f is a finite Markov chain transition operator.
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f is affine in standard algebra with dim(ker (f 0 − Id )) = 1.
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f is minplus linear.
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f is a dynamic programming function associated to a stochastic control problem.
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f is a dynamic programming function associated to a stochastic game problem.
Eigenvalue The Growth Rate is not an Eigenvalue Traffic Application Bibliography
Affine Example with dim(ker (f 0 − Id )) = 1. Minplus Homogeneous Dynamical Systems N. Farhi, M. Goursat & J.-P. Quadrat Introduction Growth Rate Eigenvalue The Growth Rate is not an Eigenvalue Traffic Application Bibliography
With standard notations we have to solve: λ + x = Mx + b, M1 = 1, Eigenvalue 1 simple . Using the variable change z = Px with: 1 0 · x −1 1 0 z = 1 = · y · · −1 0 ·
· 0 · 0 x . · · · 1
The system λP1+ z =PMP −1 z + Pb has a block triangular 1 c form PMP −1 = (thanks to the homogeneity M1 = 1), 0 N N has not the eigenvalue 1 (since 1 is a simple eigenvalue of PMP −1 ) and therefore g has a unique fixed point.
Tent Example Minplus Homogeneous Dynamical Systems N. Farhi, M. Goursat & J.-P. Quadrat Introduction
Let us consider the homogeneous system: ( x1k+1 = x2k , x2k+1 = (x2k )3 /(x1k )2 ⊕ 2(x1k )2 /x2k . We have h(y ) = y and g (y ) = y 2 ⊕ 2/y 2 (g is the tent transformation which is chaotic).
Growth Rate
g(g(x))
Eigenvalue
g(g(g(x))) 4 b
The Growth Rate is not an Eigenvalue
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M
M
Traffic Application
g(x)
g(x)
a
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g(x) 1
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N
5
c
N
N
Figure: Tent transformation and its iterates.
3 6
χ 6= λ Minplus Homogeneous Dynamical Systems N. Farhi, M. Goursat & J.-P. Quadrat
The eigenvalues are λ = y solution of y = y 2 ⊕ 2/y 2 that is: 2 λ ∈ 0, . 3 1
Introduction
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Growth Rate
4 χ= , 5
Eigenvalue The Growth Rate is not an Eigenvalue Traffic Application Bibliography
Starting from y 0 = 25 , the trajectory is periodic of period 2. The invariant measure is Qy 0 = 12 (δ 2 + δ 6 ) , therefore:
2
5
Qy 0 a.e.
The tent transformation admits the uniform law as invariant measure, therefore: Z 1 1 ydy = , a.e. for the Lebesgue measure. χ= 2 0
2 Circular Roads with 1 junction Minplus Homogeneous Dynamical Systems
a n+m
N. Farhi, M. Goursat & J.-P. Quadrat Introduction Growth Rate
q n+1 -1
an
-1
1
qn
1/2 1/2 1/2
an
q1
q
1/2
n+m
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a n-1 qn an
Eigenvalue The Growth Rate is not an Eigenvalue
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Traffic Application Bibliography
Figure: A junction with two circular roads cut in sections (top-right), its Petri net simplified modeling (middle) and the precise modeling of the junction (top left).
Dynamics Minplus Homogeneous Dynamical Systems N. Farhi, M. Goursat & J.-P. Quadrat Introduction Growth Rate Eigenvalue The Growth Rate is not an Eigenvalue Traffic Application Bibliography
The general Petri net equation: X X mpq q k−1 − q k = 0, ∀q ∈ Q, ∀k, min ap + p∈q in out in q∈p
q∈p
does not define completely the dynamics. We precise the dynamics by giving the turning probability (1/2) and the right priority to enter in the junction. k ⊕¯ k , i 6= 1, n, n + 1, n + m, qik+1 = ai−1 qi−1 ai qi+1 k k q k+1 = ¯an q1 qn+1 ⊕ an−1 q k , k n n−1 qn+m k qk q k+1 n+1 k ⊕ an+m−1 qn+m−1 , qn+m = ¯an+m 1 k+1 qn q k ⊕ ¯a1 q2k , q1k+1 = an qnk qn+m q q k+1 = an+m q k q k ⊕ ¯an+1 q k . n n+m n+2 n+1
Increasing trajectory property Minplus Homogeneous Dynamical Systems N. Farhi, M. Goursat & J.-P. Quadrat Introduction Growth Rate Eigenvalue The Growth Rate is not an Eigenvalue Traffic Application Bibliography
Theorem The trajectories of the states (qik )k∈N , starting from 0, are nondecreasing for all i. Proof by induction. For qn : k If qnk+1 = an−1 qn−1 k−1 ⇒ qnk+1 ≥ an−1 qn−1 ≥ fn (q k−1 ) = qnk .
k k If qnk+1 = ¯an q1k qn+1 /qn+m , k−1 k ⇒ qnk+1 ≥ ¯an qnk q1k qn+1 /¯an+m q1k−1 qn+1 k−1 k since qn+m ≤ ¯an+m q1k−1 qn+1 /qnk
⇒ qnk+1 ≥ qnk
Distances between states stay bounded Minplus Homogeneous Dynamical Systems N. Farhi, M. Goursat & J.-P. Quadrat
Theorem The distances between any pair of states stay bounded: ∃c1 : sup |qik − qjk | ≤ c1 , ∀i, j.
Introduction
k
Growth Rate Eigenvalue The Growth Rate is not an Eigenvalue Traffic Application Bibliography
Moreover: ∀T , ∃c2 : sup |qik+T − qik | ≤ c2 T , ∀i. k
Existence of the Growth Rate Minplus Homogeneous Dynamical Systems N. Farhi, M. Goursat & J.-P. Quadrat Introduction Growth Rate Eigenvalue The Growth Rate is not an Eigenvalue Traffic Application Bibliography
Theorem There exists an initial distribution on (qj0 /q10 )j=2,n+m , the Kryloff Bogoljuboff invariant measure, such that the average flow χ = lim qik /k, ∀i , k
exists almost everywhere.
Eigenvalue Formula Minplus Homogeneous Dynamical Systems N. Farhi, M. Goursat & J.-P. Quadrat Introduction
The eigenvalue problem can be solved explicitly. Theorem The nonnegative eigenvalues λ are solutions of: ffff 1 > −λ, ⊥ (1 − ρ) d − λ, − λ, r − (1 − ρ) d − (2r − 1 + 2ρ) λ =0 4
Growth Rate Eigenvalue The Growth Rate is not an Eigenvalue Traffic Application Bibliography
with N = n + m, ρ = 1/N, r = m/N, d the car density. N >> 1, r > 1/2 1 r −d λ ' max 0, min d , , . 4 2r − 1
Difference Between Eigenvalue and Growth Rate Minplus Homogeneous Dynamical Systems
1/4
N. Farhi, M. Goursat & J.-P. Quadrat Introduction Growth Rate
χ
Eigenvalue The Growth Rate is not an Eigenvalue Traffic Application Bibliography
λ 0
d 1
Figure: The traffic fundamental diagram χ(d ) when r = 5/6 (continuous line) obtained by simulation and its comparison with the eigenvalue λ(d ).
Phases Minplus Homogeneous Dynamical Systems
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N. Farhi, M. Goursat & J.-P. Quadrat
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Introduction Growth Rate Eigenvalue
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The Growth Rate is not an Eigenvalue Traffic Application Bibliography
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Free moving: When the density is small, 0 ≤ d ≤ α with 1 , after a finite time, all the cars move freely. α = 4(1−ρ) Saturation: When α ≤ d ≤ β with β = 12 r +1/2−ρ the 1−ρ junction is used at its maximal capacity without being bothered by downstream cars. r the crossing is Recession: When β < d < γ with γ = 1−ρ fully occupied but cars sometimes cannot leave it because the roads where they want to go are crowded. When γ < β, on the interval [γ, β] three eigenvalues exist. In this case the system is in fact blocked.
Blocking: When γ ≤ d ≤ 1, the road without priority is full of cars, no car can leave it and one car wants to enter.
Extension to Regular Towns Minplus Homogeneous Dynamical Systems
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N. Farhi, M. Goursat & J.-P. Quadrat Introduction Growth Rate
The Growth Rate is not an Eigenvalue Traffic Application Bibliography
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Figure: Roads on a torus of 4 × 2 streets with its authorized turn at junctions (left) and the asymptotic car repartition in the streets on a torus of 4 × 4 streets obtained by simulation.
Bibliography Minplus Homogeneous Dynamical Systems
F. Baccelli, G. Cohen, G.J. Olsder, and J.P. Quadrat: Synchronization and Linearity, Wiley (1992).
N. Farhi, M. Goursat & J.-P. Quadrat
N. Farhi, M. Goursat, J.-P. Quadrat: Fundamental Traffic Diagram of Elementary Road Networks algebra and Petri net modeling, in Proccedings ECC-2007, Kos, Dec. 2007.
Introduction Growth Rate Eigenvalue The Growth Rate is not an Eigenvalue Traffic Application Bibliography
N. Farhi: Modélisation minplus et commande du trafic de villes régulière, thesis dissertation, University Paris 1 Panthéon - Sorbonne, 2008. M. Fukui, Y. Ishibashi: Phase Diagram for the traffic on Two One-dimensional Roads with a Crossing, Journal of the Physical Society of Japan, Vol. 65, N. 9, pp. 2793-2795, 1996. S. Gaubert and J. Gunawerdena: The Perron-Frobenius theorem for homogeneous monotone functions, Transacton of AMS, Vol. 356, N. 12, pp. 4931-4950, 2004. B. Hassenblatt and A. Katok: A first course in Dynamics, Cambridge University Press, 2003.
