Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM) Markov modulated Brownian motion and the flip-flop fluid queue Guy Latouche Universit´ e libre de Bruxelles Joint work with Giang T. Nguyen The 9th International Conference on Matrix-Analytic Methods in Stochastic Modeling Budapest, 28th–30th of June, 2016 MMBM and the flip-flop MAM9 — June 2016 1
Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM) Outline Regulated process 1 Regenerative Approach 2 The flip-flop 3 Two boundaries 4 Sticky boundary (BM) 5 Sticky boundary (MMBM) 6 MMBM and the flip-flop MAM9 — June 2016 2
Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM) Brownian motion X ( · ) 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2.5 t 0 1 2 3 4 5 6 Approximate simulation of a BM µ = 0, σ = 1. MMBM and the flip-flop MAM9 — June 2016 3
Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM) Regulator X ( · ) 1.5 1 M ( · ) 0.5 0 −0.5 −1 −1.5 −2 −2.5 0 1 2 3 4 5 6 t M ( t ) = min { X ( s ) : 0 ≤ s ≤ t } Regulator: R ( t ) = | M ( t ) | regulated process: Z ( t ) = X ( t ) + R ( t ) MMBM and the flip-flop MAM9 — June 2016 4
Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM) Regulated BM Z ( · ) 4 3.5 R ( · ) 3 2.5 2 1.5 1 0.5 0 t 0 1 2 3 4 5 6 R increases only when Z ( t ) = 0 MMBM and the flip-flop MAM9 — June 2016 5
Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM) MMBM and FQ Z ( · ) Z ( · ) 0 0 t t ϕ ( · ): Markov process of phases, ϕ ( s ) = i → BM( µ i , σ i σ 1 = · · · = σ m = 0 → Fluid Queue Intervals of sojourn at zero for fluid No sojourn at zero for BM Focus on stationary distribution: drift is negative BM: assume σ i > 0 for all i MMBM and the flip-flop MAM9 — June 2016 6
Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM) Regenerative Approach MMBM and the flip-flop MAM9 — June 2016 7
Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM) Reversed process X ∗ ( · ) X ( · ) 0 0 X ( · ): Markov modulated process. Z ( · ): regulated process, boundary at zero ϕ ( · ): phase (control) process with stationary distribution α Reversed process: X ∗ ( t ) = − X ( − t ), ϕ ∗ ( t ) = ϕ ( − t ) Rogers ’94, Asmussen ’95 X ∗ ( u ) ≤ x | ϕ ∗ (0) = i ] t →∞ P[ ϕ ( t ) = i , Z ( t ) ≤ x ] = α i P[sup lim u ≥ 0 MMBM and the flip-flop MAM9 — June 2016 8
Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM) Matrix-analytics for fluid queues Starting with Ram’s paper at ITC 1999. Z ( t ) x 0 t two subsets of phases: S + and S − such that fluid ↑ or ↓ mass at 0 : γ ( T −− + T − + Ψ) = 0 density at x = γ T − + e Kx � C − 1 Ψ | C − | − 1 � x > 0 + · Ψ first return probability to level 0 · ( e Kx � � I Ψ ) ij = E[number of crossings] of ( x , j ), taboo of 0 physically meaningful clean separation between boundary x = 0 and interior x > 0 MMBM and the flip-flop MAM9 — June 2016 9
Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM) Opportunities for extensions Finite buffer Reactive boundaries change of phase upon hitting boundary, change of generator while at boundary Piecewise level-dependent fluid rates Two-dimensional fluid model Algorithms to compute the key matrix Ψ What about MMBMs? Ψ does not make sense as such. By design, MMBMs are about intervals MMBM and the flip-flop MAM9 — June 2016 10
Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM) Opportunities for extensions Finite buffer Reactive boundaries change of phase upon hitting boundary, change of generator while at boundary Piecewise level-dependent fluid rates Two-dimensional fluid model Algorithms to compute the key matrix Ψ What about MMBMs? Ψ does not make sense as such. By design, MMBMs are about intervals MMBM and the flip-flop MAM9 — June 2016 10
Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM) Markov-regenerative approach Y ( · ) 1 0.5 × × × × × 0 t 0 1 2 3 4 5 6 7 8 9 10 θ n +1 θ n Need Regenerative epochs { θ n } ρ = stationary distribution of ϕ ( θ n ) M ij ( x ) = E [time spent in [0 , x ] × j between θ n and θ n +1 | ϕ ( θ n ) = i ] Then t →∞ P [ Y ( t ) ≤ x , ϕ ( t ) = j ] = ( γ ρ M ( x )) j lim MMBM and the flip-flop MAM9 — June 2016 11
Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM) The flip-flop MMBM and the flip-flop MAM9 — June 2016 12
Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM) Fluid queues and the BM Ram at MAM-in-NY (2011): fluid queue with 2 phases. � − λ λ � Transition matrix: T = λ − λ √ √ Fluid rates: c + = µ + σ λ , c − = µ − σ λ . 1.5 1 Oscillates faster as λ → ∞ , 0.5 0 Amplitude increases −0.5 −1 Converges to BM( µ , σ ) −1.5 −2 Example: λ = 100, µ = 0, σ = 1 −2.5 0 1 2 3 4 5 6 MMBM and the flip-flop MAM9 — June 2016 13
Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM) Markov Modulated flip-flop Flip-flop parameters: � T − λ I λ I � generator T λ = λ I T − λ I √ � ∆ µ + � λ ∆ σ fluid rates C ∗ = √ ∆ µ − λ ∆ σ with ∆ v = diag ( v 1 , . . . , v m ) Two copies of the phase Markov process, κ λ tells us whether copy + or copy − is active three-dimensional process { L λ ( t ) , ϕ λ ( t ) , κ λ ( t ) } to be projected on { L λ ( t ) , ϕ λ ( t ) } MMBM and the flip-flop MAM9 — June 2016 14
Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM) Convergence (G.L. & G.N., 2015) Projection { L λ ( t ) , ϕ λ ( t ) : t ≥ 0 } converges weakly to { X ( t ) , ϕ ( t ) : t ≥ 0 } . Weak convergence for process regulated at 0, as well as for finite buffer. Convergence of stationary distributions Establish connection to earlier results (duality / time and level reversal, spectral decomposition) MMBM and the flip-flop MAM9 — June 2016 15
Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM) Evolution of road map Take λ big (and finite) and apply algorithms from fluid queue theory to compute approximations for MMBMs. Determine characteristic for the flip-flop and formally take lim λ →∞ Use flip-flop to construct building blocks example: first passage probability matrix from regulated level 0 to level x Work on new processes (two examples later) MMBM and the flip-flop MAM9 — June 2016 16
Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM) Matrix U For a while: sidetracked by the importance of Ψ for the fluid queues. For MMBM, fundamental matrix is U : b t 0 1 2 3 4 5 6 ( e Ub ) ij = P[reach 0 in phase j | start from ( b , i )]. U ( λ ) for flip-flop is analytic function around 1 /λ = 0, converges to U of MMBM MMBM and the flip-flop MAM9 — June 2016 17
Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM) Two boundaries MMBM and the flip-flop MAM9 — June 2016 18
Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM) Regeneration for two boundaries Y ( · ) × × × 2 1.5 1 0.5 × × × 0 t 0 0.2 0.4 0.6 0.8 1 θ n − 1 θ n +1 θ n Alternate first visits to 0 and first visits to upper bound b Need transition probabilities P 0 ❀ b and P b ❀ 0 and expected time in [0 , x ] × j during an excursion Obtained from flip-flop Reactive boundary for free. Example: one set of parameters between θ n − 1 and θ n and another one between θ n and θ n +1 MMBM and the flip-flop MAM9 — June 2016 19
Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM) Sticky boundary (BM) MMBM and the flip-flop MAM9 — June 2016 20
Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM) Regulator Z ( · ) 4 3.5 R ( · ) 3 2.5 2 1.5 1 0.5 0 t 0 1 2 3 4 5 6 R increases only when Z ( t ) = 0 MMBM and the flip-flop MAM9 — June 2016 21
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