Fluid flows with jumps at the boundary Eleonora Deiana Guy Latouche - - PowerPoint PPT Presentation

Mathematical model Regenerative approach Stationary Distribution Concluding comments

Fluid flows with jumps at the boundary

Eleonora Deiana Guy Latouche Marie-Ange Remiche

Universit´ e de Namur

The Ninth International Conference on Matrix-Analytic Methods in Stochastic Models Budapest, June 28 - 30, 2016

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Mathematical model Regenerative approach Stationary Distribution Concluding comments

Classic fluid flow VS Fluid flow with jumps

t X(t) B t X(t) B

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Mathematical model Regenerative approach Stationary Distribution Concluding comments

Outline

Mathematical model Regenerative approach Stationary Distribution Concluding comments

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Mathematical model Regenerative approach Stationary Distribution Concluding comments

Definition and notations: FLUID FLOW

Two-dimensional process: {X(t), φ(t)}t≥0

◮ X(t) ∈ R+: level; ◮ φ(t) ∈ S = S+ ∪ S−: phase process.

Evolution of the level X(t):

◮ X(t) > 0, φ(t) = i ∈ S:

d dt X(t) = ci

◮ X(t) = 0: instantaneous jump to a fixed level B.

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Mathematical model Regenerative approach Stationary Distribution Concluding comments

Matrices

Transition and rate matrices: T =   T++ T+− T−+ T−−   , and C =   C+ C−   . Matrix of the change of phases in the jump: W =

• W−+

W−−

• .

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Mathematical model Regenerative approach Stationary Distribution Concluding comments

OBJECTIVE

◮ Calculation of the stationary distribution:

Πj(x) = lim

t→∞ P [X(t) < x, φ(t) = j] .

How? REGENERATIVE APPROACH

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Mathematical model Regenerative approach Stationary Distribution Concluding comments

Regenerative approach

t X(t) B x Sequence of regeneration points {hn}n≥0 defined as: h0 = 0, hn+1 = inf {t > hn|X(t) = 0} .

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Mathematical model Regenerative approach Stationary Distribution Concluding comments

Stationary distribution: Π(x) = (ν♠)−1νM(x).

◮ ν: stationary distribution of phases in the regeneration points:

νH = ν, where Hij = P[φ(hn+1) = j|φ(hn) = i], i, j ∈ S−.

◮ M(x): mean sojourn time in [0, x] between two regeneration points; ◮ ♠ = M(B)1: mean sojourn time between two regenerative points given

the phase of departure.

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Mathematical model Regenerative approach Stationary Distribution Concluding comments

Stationary distribution: Π(x) = (ν♠)−1νM(x).

◮ ν: stationary distribution of phases in the regeneration points:

νH = ν, where Hij = P[φ(hn+1) = j|φ(hn) = i], i, j ∈ S−.

◮ M(x): mean sojourn time in [0, x] between two regeneration points; ◮ ♠ = M(B)1: mean sojourn time between two regenerative points given

the phase of departure.

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Mathematical model Regenerative approach Stationary Distribution Concluding comments

◮ ν: stationary distribution of phases in the regeneration points:

Transition matrix of phases between two regeneration points: H =

• W−+

W−−

• 

 Ψ I   eUb. Where:

◮ Ψ: probability of the first return to the initial level, ◮ eUx: probability, starting from a fixed level x, to reach level 0 in a finite

time.

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Mathematical model Regenerative approach Stationary Distribution Concluding comments

H =

• W−+

W−−

• 

 Ψ I   eUb. t X(t) B x W++ W+−

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Mathematical model Regenerative approach Stationary Distribution Concluding comments

Mean sojourn time M(x)

M(x) =

• W−+

W−−

• 



• M+(x)
• M−(x)

  . t X(t) B x

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Mathematical model Regenerative approach Stationary Distribution Concluding comments

Mean sojourn time M(x)

M(x) =

• W−+

W−−

• 



• M+(x)
• M−(x)

  . t X(t) B x W++ W+−

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Mathematical model Regenerative approach Stationary Distribution Concluding comments

Γ(x): mean sojourn time in [0, x] before the first return to the initial level, starting in level 0; x B defined as: Γ(x) = x eKudu

• C −1

+

Ψ|C −1

− |

• ,

with eKx: expected number of crossing of level x, starting from level 0 before the first return to the initial level.

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Mathematical model Regenerative approach Stationary Distribution Concluding comments

Hb

+(x): mean sojourn time in [0, x] before reaching either level 0 or level B,

starting in level 0; x x B B

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Mathematical model Regenerative approach Stationary Distribution Concluding comments

Similarly...

◮

Γ(x): mean sojourn time in [0, x] before the first return to the initial level, starting in level B;

◮ Hb −(x): mean sojourn time in [0, x] before reaching either level 0 or level

B, starting in level B. These quantities can be putted together in the system:   Γ(x)

• Γ(x)

  =   I eKbΨ e

• Kb

Ψ I     Hb

+(x)

Hb

−(x)

  ,

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Mathematical model Regenerative approach Stationary Distribution Concluding comments

0 < x < b

M+(x) = Ψ M−(x)

• M−(x) = Hb

−(x) +

Ψb M+(x) t X(t) B x

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Mathematical model Regenerative approach Stationary Distribution Concluding comments

0 < x < b

M+(x) = Ψ M−(x)

• M−(x) = Hb

−(x) +

Ψb M+(x) t X(t) B x

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Mathematical model Regenerative approach Stationary Distribution Concluding comments

x ≥ b

M+(x) = Γ(x − b) + Ψ M−(x)

• M−(x) = Hb

−(b) +

Ψb M+(x). t X(t) B x

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Mathematical model Regenerative approach Stationary Distribution Concluding comments

x ≥ b

M+(x) = Γ(x − b) + Ψ M−(x)

• M−(x) = Hb

−(b) +

Ψb M+(x). t X(t) B x

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Mathematical model Regenerative approach Stationary Distribution Concluding comments

If 0 < x < b: M+(x) = Ψ(I − ΨbΨ)−1Hb

−(x)

• M−(x) = (I −

ΨbΨ)−1Hb

−(x).

If x ≥ b M+(x) = (I − Ψ Ψb)−1 Γ(x − b) + ΨHb

−(b)

• M−(x) = (I −

ΨbΨ)−1

• ΨbΓ(x − b) + Hb

−(b)

• .

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Mathematical model Regenerative approach Stationary Distribution Concluding comments

Further work:

◮ random size of the jumps; ◮ jumps after a random interval of time; ◮ brownian motion.

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Mathematical model Regenerative approach Stationary Distribution Concluding comments

Thank you for your attention!

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