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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

1

Regulated process

2

Regenerative Approach

3

The flip-flop

4

Two boundaries

5

Sticky boundary (BM)

6

Sticky boundary (MMBM)

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Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM)

Brownian motion

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t X(·) Approximate simulation of a BM µ = 0, σ = 1.

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Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM)

Regulator

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t X(·) M(·) M(t) = min{X(s) : 0 ≤ s ≤ t} Regulator: R(t) = |M(t)| regulated process: Z(t) = X(t) + R(t)

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Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM)

Regulated BM

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t Z(·) R(·) R increases only when Z(t) = 0

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Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM)

MMBM and FQ

t t Z(·) Z(·) ϕ(·): Markov process of phases, ϕ(s) = i → BM(µi, σi σ1 = · · · = σm = 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 σi > 0 for all i

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Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM)

Regenerative Approach

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Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM)

Reversed process

X ∗(·) X(·) 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 lim

t→∞ P[ϕ(t) = i, Z(t) ≤ x] = αiP[sup u≥0

X ∗(u) ≤ x|ϕ∗(0) = i]

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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. t x Z(t) two subsets of phases: S+ and S− such that fluid ↑ or ↓ mass at 0 : γ(T−− + T−+Ψ) = 0 density at x = γT−+ eKx C −1

+

Ψ|C−|−1 x > 0 · Ψ first return probability to level 0 · (eKx 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

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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

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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

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Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM)

Markov-regenerative approach

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t Y (·) × × × × × θn θn+1 Need Regenerative epochs {θn} ρ = stationary distribution of ϕ(θn) Mij(x) = E[time spent in [0, x] × j between θn and θn+1 | ϕ(θn) = i] Then lim

t→∞ P[Y (t) ≤ x, ϕ(t) = j] = (γ ρ M(x))j

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Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM)

The flip-flop

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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. Transition matrix: T = −λ λ λ −λ

• Fluid rates: c+ = µ + σ

√ λ, c− = µ − σ √ λ. Oscillates faster as λ → ∞, Amplitude increases Converges to BM(µ, σ) Example: λ = 100, µ = 0, σ = 1

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Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM)

Markov Modulated flip-flop

Flip-flop parameters: generator Tλ = T − λI λI λI T − λI

• fluid rates C ∗ =

∆µ + √ λ∆σ ∆µ − √ λ∆σ

• with ∆v = diag(v1, . . . , vm)

Two copies of the phase Markov process, κλ tells us whether copy + or copy − is active three-dimensional process {Lλ(t), ϕλ(t), κλ(t)} to be projected on {Lλ(t), ϕλ(t)}

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Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM)

Convergence (G.L. & G.N., 2015)

Projection {Lλ(t), ϕλ(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)

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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 limλ→∞ 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)

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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:

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t b (eUb)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

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Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM)

Two boundaries

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Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM)

Regeneration for two boundaries

0.2 0.4 0.6 0.8 1 0.5 1 1.5 2

t Y (·) × × × × × × θn−1 θn θn+1 Alternate first visits to 0 and first visits to upper bound b Need transition probabilities P0❀b and Pb❀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 θn−1 and θn and another one between θn and θn+1

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Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM)

Sticky boundary (BM)

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Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM)

Regulator

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t Z(·) R(·) R increases only when Z(t) = 0

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Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM)

Change of clock

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t R(·) V (·) Γ(·) V (t) = t + R(t)/ω for some ω > 0 — grows faster than t when Z(t) = 0 Γ such that V (Γ(t)) = t New clock: slowed down when Z(t) = 0 Y (t) = Z(Γ(t)) BM with sticky boundary

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Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM)

BM with sticky boundary

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t Z(·) Y (·) V (t) = t + R(t)/ω for some ω > 0 here, ω = 1.5 Γ such that V (Γ(t)) = t New clock: slowed down when Z(t) = 0 Y (t) = Z(Γ(t)) BM with sticky boundary

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Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM)

Sticky boundary (MMBM)

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Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM)

MMBM with sticky boundary

Markov modulation: {ϕ(t)} on {1, 2, . . . , m}; mean µi, variance σ2

i

R(t) = | min0≤s≤t X(s)|, Z(t) = X(t) + R(t) ri(t) = t 1[ϕ(s) = i] dR(t) V (t) = t +

• i

ri(t)/ωi ωi: stretching of time may depend on the phase. Γ(t) such that V (Γ(t)) = t Y (t) = Z(Γ(t))

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Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM)

Markov-regenerative process

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t Y (·) θn θn + δn θn+1 create artificial intervals between “successive” visits to 0 Use a timer δn i.i.d. exponential (q) θn+1 = inf {t > θn + δn+1 : Y (t) = 0} θ0 = 0.

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Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM)

MM-flip-flop with sticky boundary

MM flip-flop parameters: generator Tλ = T − λI λI λI T − λI

• fluid rates C ∗ =

∆µ + √ λ∆σ ∆µ − √ λ∆σ

• with ∆v = diag(v1, . . . , vm)

At level 0: T0 =

• λI

T − λI

• Stretching effected through

T ∗

0 = (1/

√ λ)∆ωT0

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Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM)

Stationary distribution

lim

t→∞ P[ϕ(t) = i, Z(t) ≤ x] = γν[∆−1 ω + 2(−K)−1(I − eKx)∆−1 σ ]

where ν is solution of ν∆σU = 0, ν1 = 1, U is “minimal” solution of ∆2

σX 2 + 2∆µX + 2Q = 0

and K = ∆−1

σ U∆−1 σ

+ 2∆−2

σ ∆µ

Identical to stationary distribution for MMBM except for probability mass at 0

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Regulated process Regenerative Approach The flip-flop Two boundaries Sticky boundary (BM) Sticky boundary (MMBM)

Conclusion

Easy to think about physical behaviour of flip-flop fluid queue and to take limits. We could revisit results obtained from “traditional” approach and improve

• n them.

Opens path to analysis of new processes and raises new questions for investigation.

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