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 1 / 19

Mathematical model Regenerative approach Stationary Distribution Concluding comments Classic fluid flow VS Fluid flow with jumps X ( t ) B t X ( t ) B t 2 / 19

Mathematical model Regenerative approach Stationary Distribution Concluding comments Outline Mathematical model Regenerative approach Stationary Distribution Concluding comments 3 / 19

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 d t X ( t ) = c i ◮ X ( t ) = 0: instantaneous jump to a fixed level B . 4 / 19

Mathematical model Regenerative approach Stationary Distribution Concluding comments Matrices Transition and rate matrices:      T ++ T + −  C + 0  , and C =  . T = 0 T − + T −− C − Matrix of the change of phases in the jump: � � W = . W − + W −− 5 / 19

Mathematical model Regenerative approach Stationary Distribution Concluding comments OBJECTIVE ◮ Calculation of the stationary distribution: Π j ( x ) = lim t →∞ P [ X ( t ) < x , φ ( t ) = j ] . How? REGENERATIVE APPROACH 6 / 19

Mathematical model Regenerative approach Stationary Distribution Concluding comments Regenerative approach X ( t ) B x t Sequence of regeneration points { h n } n ≥ 0 defined as: h 0 = 0 , h n +1 = inf { t > h n | X ( t ) = 0 } . 7 / 19

Mathematical model Regenerative approach Stationary Distribution Concluding comments Stationary distribution: Π ( x ) = ( ν ♠ ) − 1 ν M ( x ) . ◮ ν : stationary distribution of phases in the regeneration points: ν H = ν , where i , j ∈ S − . H ij = P [ φ ( h n +1 ) = j | φ ( h n ) = i ] , ◮ 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. 8 / 19

Mathematical model Regenerative approach Stationary Distribution Concluding comments Stationary distribution: Π ( x ) = ( ν ♠ ) − 1 ν M ( x ) . ◮ ν : stationary distribution of phases in the regeneration points: ν H = ν , where i , j ∈ S − . H ij = P [ φ ( h n +1 ) = j | φ ( h n ) = i ] , ◮ 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. 8 / 19

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:   � �  Ψ  e Ub . H = W − + W −− I Where: ◮ Ψ: probability of the first return to the initial level, ◮ e Ux : probability, starting from a fixed level x , to reach level 0 in a finite time. 9 / 19

Mathematical model Regenerative approach Stationary Distribution Concluding comments   � �  Ψ  e Ub . H = W − + W −− I X ( t ) W ++ W + − B x t 10 / 19

Mathematical model Regenerative approach Stationary Distribution Concluding comments Mean sojourn time M ( x )   � � � M + ( x )   . M ( x ) = W − + W −− � M − ( x ) X ( t ) B x t 11 / 19

Mathematical model Regenerative approach Stationary Distribution Concluding comments Mean sojourn time M ( x )   � � � M + ( x )   . M ( x ) = W − + W −− � M − ( x ) X ( t ) W ++ W + − B x t 11 / 19

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; B x defined as: � x � � e Ku d u C − 1 Ψ | C − 1 Γ( x ) = , − | + 0 with e Kx : expected number of crossing of level x , starting from level 0 before the first return to the initial level. 12 / 19

Mathematical model Regenerative approach Stationary Distribution Concluding comments H b + ( x ): mean sojourn time in [0 , x ] before reaching either level 0 or level B , starting in level 0; B B x x 13 / 19

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 ; ◮ H b − ( 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:       e Kb Ψ  H b  Γ( x ) I + ( x )  =    , Kb � � � H b Γ( x ) e Ψ − ( x ) I 14 / 19

Mathematical model Regenerative approach Stationary Distribution Concluding comments 0 < x < b � � M + ( x ) = Ψ � M − ( x ) Ψ b � � − ( x ) + � M − ( x ) = H b M + ( x ) X ( t ) B x t 15 / 19

Mathematical model Regenerative approach Stationary Distribution Concluding comments 0 < x < b � � M + ( x ) = Ψ � M − ( x ) Ψ b � � − ( x ) + � M − ( x ) = H b M + ( x ) X ( t ) B x t 15 / 19

Mathematical model Regenerative approach Stationary Distribution Concluding comments x ≥ b � � M + ( x ) = Γ( x − b ) + Ψ � M − ( x ) Ψ b � � − ( b ) + � M − ( x ) = H b M + ( x ) . X ( t ) x B t 16 / 19

Mathematical model Regenerative approach Stationary Distribution Concluding comments x ≥ b � � M + ( x ) = Γ( x − b ) + Ψ � M − ( x ) Ψ b � � − ( b ) + � M − ( x ) = H b M + ( x ) . X ( t ) x B t 16 / 19

Mathematical model Regenerative approach Stationary Distribution Concluding comments If 0 < x < b : � � M + ( x ) = Ψ( I − � Ψ b Ψ) − 1 H b − ( x ) M − ( x ) = ( I − � � Ψ b Ψ) − 1 H b − ( x ) . If x ≥ b � � Ψ b ) − 1 � � M + ( x ) = ( I − Ψ � Γ( x − b ) + Ψ H b − ( b ) Ψ b Ψ) − 1 � � M − ( x ) = ( I − � � � Ψ b Γ( x − b ) + H b − ( b ) . 17 / 19

Mathematical model Regenerative approach Stationary Distribution Concluding comments Further work: ◮ random size of the jumps; ◮ jumps after a random interval of time; ◮ brownian motion. 18 / 19

Mathematical model Regenerative approach Stationary Distribution Concluding comments Thank you for your attention! 19 / 19