Multi-stage Stochastic Fluid Models for Congestion Control Magorzata - - PowerPoint PPT Presentation

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Multi-stage Stochastic Fluid Models for Congestion Control Magorzata OReilly * * University of Tasmania, Australia Australia New Zealand Applied Probability

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control

Multi-stage Stochastic Fluid Models for Congestion Control

Małgorzata O’Reilly*

*University of Tasmania, Australia

Australia New Zealand Applied Probability Workshop Brisbane 2013

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control

ANZAPW Auckland 2012

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control

ANZAPW Auckland 2012

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control

ANZAPW Auckland 2012

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control

ANZAPW Auckland 2012

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control

Outline

1

Introduction: Stochastic Fluid Model

2

Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control

Definition of a SFM

Let {(ϕ(t), X(t)), t ≥ 0} be a process such that: {ϕ(t), t ≥ 0} is an irreducible CTMC with a (finite) set of phases S and generator T {ϕ(t), t ≥ 0} is the driving process Level X(t) records some performance measure When ϕ(t) = i, the rate at which X(t) is changing is ci

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control

SFM with boundaries 0 and B

ϕ(t) - phase, X(t) - level

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control

Definition of a bounded SFM

Let {(ϕ(t), X(t)), t ≥ 0} be a process such that: {ϕ(t), t ≥ 0} is an irreducible CTMC with a (finite) set of phases S and generator T When ϕ(t) = i then X(t) = 0, ci < 0 = ⇒ dX(t)/dt = 0 X(t) = B, ci > 0 = ⇒ dX(t)/dt = 0 Otherwise, dX(t)/dt = ci

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control

Some Notation

S1 = {i ∈ S : ci > 0} S2 = {i ∈ S : ci < 0} S0 = {i ∈ S : ci = 0} C1 = diag(ci) for all i ∈ S1 C2 = diag(|ci|) for all i ∈ S2 T11 = [Tij] for all i ∈ S1, j ∈ S1 T12 = [Tij] for all i ∈ S1, j ∈ S2 T10 = [Tij] for all i ∈ S1, j ∈ S0 etc

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control

Fluid generator Q(s) (Bean, O’Reilly, and Taylor 2005)

Assume Re(s) ≥ 0 Q11(s) = C−1

1 [(T11 − sI) − T10(T00 − sI)−1T01]

Q22(s) = C−1

2 [(T22 − sI) − T20(T00 − sI)−1T02]

Q12(s) = C−1

1 [T12 − T10(T00 − sI)−1T02]

Q21(s) = C−1

2 [T21 − T20(T00 − sI)−1T01]

Definition Q(s) = Q11(s) Q12(s) Q21(s) Q22(s)

  • Q = Q(0)

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control

In-Out Fluid

Figure: Start in (i, 0), end in (j, y) at time ˆ θ(y)

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control

Corresponding Laplace-Stieltjes Transform (LST)

|Y(t)| = t

u=0 |cϕ(u)|du

ˆ θ(y) = inf{t ≥ 0 : |Y(t)| = y} Definition Let ˆ ∆y(s) = [ ˆ ∆y(s)ij] be such that for all i, j ∈ S1 ∪ S2 ˆ ∆y(s)ij = E(e−sˆ

θ(y) : ϕ(ˆ

θ(y)) = j|ϕ(0) = i, Y(t) = 0) Fact ˆ ∆y(s) = eQ(s)y

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control

Return to Level Zero

Figure: Start in (i, 0), end in (j, 0) at time θ(0)

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control

Matrix Ψ(s) (Bean, O’Reilly, and Taylor 2005)

Let θ(0) = inf{t ≥ 0 : X(t) = 0} Definition For s with Re(s) ≥ 0, i with ci > 0, j with cj < 0, let Ψ(s)ij = E(θ(0) < ∞, θ(0) = i|ϕ(0) = i, X(0) = 0) Fact For s ≥ 0, Ψ(s) is the minimum nonnegative solution of Q12(s) + Q11(s)Ψ(s) + Ψ(s)Q22(s) + Ψ(s)Q21(s)Ψ(s) = 0

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control

ˆ Gx,y(s) - Draining with a Taboo

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control

ˆ Hx,y(s) - Filling in with a Taboo

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control

Draining and Filling - with taboo

For i, j ∈ S1 ∪ S2, 0 < x < y [ˆ Gx,y(s)]ij = E[e−sθ(0) : θ(0) < θ(y), ϕ(θ(0)) = j | Y(0) = x, ϕ(0) = i] [ˆ Hx,y(s)]ij = E[e−sθ(y) : θ(y) < θ(0), ϕ(θ(y)) = j | Y(0) = x, ϕ(0) = i]

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control

ˆ Gx,y(s) and ˆ Hx,y(s) (Bean, O’Reilly, and Taylor 2005)

Fact

  • ˆ

Gx,y(s) ˆ Hx,y(s) I ˆ Hy(s) ˆ Gy(s) I

  • =
  • ˆ

Gx(s) ˆ Hy−x(s)

  • where

ˆ Gx(s) =

  • ˆ

Gx

12(s)

ˆ Gx

22(s)

  • =
  • Ψ(s)e(Q22(s)+Q22(s)Ψ(s))x

e(Q22(s)+Q22(s)Ψ(s))x

  • ˆ

Hx(s) = ˆ Hx

11(s)

ˆ Hx

21(s)

  • =
  • e(Q11(s)+Q12(s)Ξ(s))x

Ξ(s)e(Q11(s)+Q12(s)Ξ(s))x

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control

Remark

Using the above building blocks (Q(s), Ψ(s), ˆ Gx,y(s) and ˆ Hx,y(s)), and arguments based on appropriate partitioning of sample paths, the (transient and stationary) analysis of (different classes of) SFMs follows. We use these building blocks in the analysis of the multi-stage SFMs with congestion control, which is discussed below.

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

Outline

1

Introduction: Stochastic Fluid Model

2

Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

Two-stage buffer

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

Two-stage SFM (with lower boundary 0)

Thresholds b1, b2, 0 < b1 < b2, for controlling congestion. The process starts from Stage 1 in level 0 Stage 1 → Stage 2 when reaching b2 from below Stage 2 → Stage 1 when reaching b1 from above Matrices P(b2), P(b1) record the probabilities of these transitions While in Stage ℓ ∈ {1, 2}, the process evolves according to a traditional SFM with a set of phases Sℓ, generator T ℓ and fluid rates cℓ

i

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

Mutli-stage SFMs

This class of models contains a model introduced by Malhotra, Mandjes, Scheinhardt and van den Berg (2009). Generalizations here: Any real fluid change rates c(ℓ)

i

(including zero), where i ∈ S(ℓ), and ℓ = 1, 2 is the current stage. The transition between the stages may involve not only the change in T(ℓ), but also in S(ℓ). The change in c(ℓ)

i

at the moment of the transition between the stages allows all possible types of changes of sign (from + or − to +, − or 0). We treat the model with an upper boundary B > b2. We consider a generalization to multi-stage SFMs.

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

Mutli-stage SFMs

This class of models contains a model introduced by Malhotra, Mandjes, Scheinhardt and van den Berg (2009). Generalizations here: Any real fluid change rates c(ℓ)

i

(including zero), where i ∈ S(ℓ), and ℓ = 1, 2 is the current stage. The transition between the stages may involve not only the change in T(ℓ), but also in S(ℓ). The change in c(ℓ)

i

at the moment of the transition between the stages allows all possible types of changes of sign (from + or − to +, − or 0). We treat the model with an upper boundary B > b2. We consider a generalization to multi-stage SFMs.

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

Mutli-stage SFMs

This class of models contains a model introduced by Malhotra, Mandjes, Scheinhardt and van den Berg (2009). Generalizations here: Any real fluid change rates c(ℓ)

i

(including zero), where i ∈ S(ℓ), and ℓ = 1, 2 is the current stage. The transition between the stages may involve not only the change in T(ℓ), but also in S(ℓ). The change in c(ℓ)

i

at the moment of the transition between the stages allows all possible types of changes of sign (from + or − to +, − or 0). We treat the model with an upper boundary B > b2. We consider a generalization to multi-stage SFMs.

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

Mutli-stage SFMs

This class of models contains a model introduced by Malhotra, Mandjes, Scheinhardt and van den Berg (2009). Generalizations here: Any real fluid change rates c(ℓ)

i

(including zero), where i ∈ S(ℓ), and ℓ = 1, 2 is the current stage. The transition between the stages may involve not only the change in T(ℓ), but also in S(ℓ). The change in c(ℓ)

i

at the moment of the transition between the stages allows all possible types of changes of sign (from + or − to +, − or 0). We treat the model with an upper boundary B > b2. We consider a generalization to multi-stage SFMs.

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

Mutli-stage SFMs

This class of models contains a model introduced by Malhotra, Mandjes, Scheinhardt and van den Berg (2009). Generalizations here: Any real fluid change rates c(ℓ)

i

(including zero), where i ∈ S(ℓ), and ℓ = 1, 2 is the current stage. The transition between the stages may involve not only the change in T(ℓ), but also in S(ℓ). The change in c(ℓ)

i

at the moment of the transition between the stages allows all possible types of changes of sign (from + or − to +, − or 0). We treat the model with an upper boundary B > b2. We consider a generalization to multi-stage SFMs.

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

Mutli-stage SFMs

Methodology: The analysis in Malhotra, Mandjes, Scheinhardt and van den Berg (2003) was based on solving appropriate balance equations using a spectral expansion. Here, we use the building blocks discussed earlier, and matrix-analytic methods. Model with no upper boundary is discussed below. The analysis for the bounded model is similar.

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

Mutli-stage SFMs

Methodology: The analysis in Malhotra, Mandjes, Scheinhardt and van den Berg (2003) was based on solving appropriate balance equations using a spectral expansion. Here, we use the building blocks discussed earlier, and matrix-analytic methods. Model with no upper boundary is discussed below. The analysis for the bounded model is similar.

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

Mutli-stage SFMs

Methodology: The analysis in Malhotra, Mandjes, Scheinhardt and van den Berg (2003) was based on solving appropriate balance equations using a spectral expansion. Here, we use the building blocks discussed earlier, and matrix-analytic methods. Model with no upper boundary is discussed below. The analysis for the bounded model is similar.

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

Outline

1

Introduction: Stochastic Fluid Model

2

Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

LSTs of the times spent at the boundaries

¯ P(b2)

11 (s)

= P(b2)

11

+ P(b2)

10 (sI − T(2) 00 )−1T(2) 01

¯ P(b2)

12 (s)

= P(b2)

12

+ P(b2)

10 (sI − T(2) 00 )−1T(2) 02

¯ P(b1)

21 (s)

= P(b1)

21

+ P(b1)

20 (sI − T(1) 00 )−1T(1) 01

¯ P(b1)

22 (s)

= P(b1)

22

+ P(b1)

10 (sI − T(1) 00 )−1T(1) 02

and ¯ P(0)

21 (s)

=

  • I
  • T(1)

22

T(1)

20

T(1)

02

T(1)

00

  • − sI

−1 T(1)

21

T(1)

01

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

LST of the times spent between the boundaries

Lb2b1(s) =

  • ¯

P(b2)

12 (s) + ¯

P(b2)

11 (s)Ψ(2)(s)

  • G(2);(b2−b1)

22

(s) Eb2b1 = −d/ds Lb2b1(s)|s=0

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

LST of the times spent between the boundaries

Lb1b2(s) = ¯ P(b1)

22 (s)H(1);(b1,b2) 21

(s) + ¯ P(b1)

21 (s)H(1);(b1,b2) 11

(s) Eb1b2 = −d/ds Lb1b2(s)|s=0

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

LST of the times spent between the boundaries

˜ Lb1b2(s) = Lb1b2(s) +

  • ¯

P(b1)

22 (s)G(1);(b1,b2) 22

(s) + ¯ P(b1)

21 (s)G(1);(b1,b2) 12

(s)

  • ×
  • I − ¯

P(0)

21 (s)G(1);(0,b2) 12

(s) −1 H(1);(0,b2)

11

(s) ˜ Eb1b2 = −d/ds ˜ Lb1b2(s)|s=0

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

Busy Period

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

LST of the Busy Period

Theorem We have Ψ(s) = G(1);(0,b2)

12

(s) +H(1);(0,b2)

11

(s)Lb2b1(s)

  • I − Lb1b2(s)Lb2b1(s)

−1 ×

  • ¯

P(b1)

22 (s)G(1);(b1,b2) 22

(s) + ¯ P(b1)

21 (s)G(1);(b1,b2) 12

(s)

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

Outline

1

Introduction: Stochastic Fluid Model

2

Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

Stationary Analysis: Existence

The stationary distribution of the two-stage SFM exists when the drift µ(2) =

  • i∈S(2)

πic(2)

i

, corresponding to the SFM in Stage 2, is strictly negative.

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

Stationary Analysis: Steps of the Method

  • 1. Derive the stationary distribution vector

ξ =

  • ξ(0)

ξ(b1) ξ(b2)

  • f a DTMC observed at the

moments when hitting level 0 (from above), or b1 from above while in Stage 2, or b2 from below

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

Stationary Analysis: Steps of the Method

The one-step transition probability matrix A of this chain, partitioned in an analogous manner, is given by A =   A00 A0b2 Ab10 Ab1b2 Ab2b1   where A00 = ¯ P(0)

21 G(1);(0,b2) 12

A0b2 = ¯ P(0)

21 H(1);(0,b2) 11

Ab2b1 = Lb2b1 Ab1b2 = Lb1b2 Ab10 = ¯ P(b1)

22 G(1);(b1,b2) 22

+ ¯ P(b1)

21 G(1);(b1,b2) 12

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

Stationary Analysis: Steps of the Method

  • 2. Write expressions for the probability mass vectors
  • p(0)2

p(0)0

  • , p(b2)0, and p(b1)0, in terms of ξ and

some normalizing constant α

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

Stationary Analysis: Steps of the Method

We have

  • p(0)2

p(0)0

  • =

α

  • ξ(0)
  • −
  • T(1)

22

T(1)

20

T(1)

02

T(1)

00

−1 p(b2)0 = αξ(b2)P(b2)

10 (−T(2) 00 )−1

p(b1)0 = αξ(b1)P(b1)

20 (−T(1) 00 )−1

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

Stationary Analysis: Steps of the Method

  • 3. Write the set of equations for the vectors π(2)(b−

2 )2,

π(2)(b+

2 )1, π(1)(b+ 1 )1 and π(1)(b− 1 )2 in terms of the above

probability mass vectors where π(2)(b−

2 )2

= lim

x→b−

2

π(2)(x)2 π(2)(b+

2 )1

= lim

x→b+

2

π(2)(x)1 π(1)(b−

1 )2

= lim

x→b−

1

π(1)(x)2 π(1)(b+

1 )1

= lim

x→b+

1

π(1)(x)1

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

Stationary Analysis: Steps of the Method

π(2)(b−

2 )2

=

  • p(b2)0T(2)

02 + π(2)(b+ 2 )1C(2)Ψ(2)

+π(1)(b+

1 )1C(1) 1 H(1);(0,b2−b1) 11

P(b2)

12

  • (C(2)

2 )−1

π(2)(b+

2 )1

=

  • p(b2)0T(2)

01 + π(1)(b+ 1 )1C(1) 1 H(1);(0,b2−b1) 11

P(b2)

11

+π(2)(b−

2 )2C(2) 2 H(2);(b2−b1,b2−b1) 21

P(b2)

11

  • (C(2)

2 )−1

π(1)(b+

1 )1

=

  • p(b1)0T(1)

01 + p(0)0T(1) 01 C(1) 1 H(1);(b1,b1) 11

+π(1)(b−

1 )2C(1) 2 H(1);(b1,b1) 21

  • (C(1)

1 )−1

π(1)(b−

1 )2

=

  • p(b1)0T(1)

02 + π(1)(b+ 1 )1C(1) 1 G(1);(0,b2−b1) 12

  • (C(1)

2 )−1

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

Stationary Analysis: Steps of the Method

  • 4. Write expressions for the remaining probability density

vectors π(1)(x) and π(2)(x) in terms of the above probability mass and density vectors

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

Stationary Analysis: Steps of the Method

For 0 < x < b1,

  • π(1)(x)1

π(1)(x)2

  • =

p(0)2 p(0)0

  • T(1)

21

T(1)

01

  • N(1)

1 (0; x)(C(1))−1

+π(1)(b−

1 )2C(1) 2 N(1) 2 (b1; x)(C(1))−1

for b1 < x < b2,

  • π(1)(x)1

π(1)(x)2

  • =

π(1)(b+

1 )1C(1) 1 N(1) 1 (b1; x)(C(1))−1

and

  • π(2)(x)1

π(2)(x)2

  • =

π(2)(b−

2 )2C(2) 2 N(2) 2 (b2; x)(C(2))−1

for x > b2,

  • π(2)(x)1

π(2)(x)2

  • =

π(2)(b+

2 )1C(2) 1 N(2) 1 (b2; x)(C(2))−1

and for ℓ ∈ {1, 2}, 0 < x < b1, b1 < x < b2 and x > b2, π(ℓ)(x)0 =

  • π(ℓ)(x)1

π(ℓ)(x)2

  • T(ℓ)

10

T(ℓ)

20

  • (−T(ℓ)

00 )−1. 48 / 57

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

Stationary Analysis: Steps of the Method

  • 5. Evaluate normalizing constant α using the fact that total

probability mass must be equal to 1 b2

x=0

π(1)(x)dx1 + ∞

x=b1

π(2)(x)dx1 +

2

  • i=0

p(bi)1 = 1.

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

Outline

1

Introduction: Stochastic Fluid Model

2

Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

Long-run proportion of time spent in a stage

Using the above results, we can evaluate p(1) = b2

x=0

π(1)(x)dx1 + p(0)1 + p(b1)1, p(2) = ∞

x=b1

π(2)(x)dx1 + p(b2)1, interpreted as the long-run proportion of time spent in Stage 1 and Stage 2, respectively.

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

Transient tendency of the switches

between the two stages can be assessed using δ2→1 = 1 (1/|S(1)

1 |)Eb2b11

, δ1→2 = 1 (1/|S(2)

2 |)˜

Eb1b21 , with δ2→1 and δ1→2 interpreted as the transient rate of the switch from Stage 2 to 1 and from Stage 1 to 2, respectively, and where a higher rate means a faster switch to the other stage.

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

Stationary tendency of the switches

between the two stages can be assessed using r2→1 = 1 π(1)(b−

2 )1Eb2b11,

r1→2 = 1 π(1)(b+

1 )2˜

Eb1b21 , with r2→1 and r1→2 interpreted as the long-run rate of the switch from Stage 2 to 1 and from Stage 1 to 2, respectively, and where a higher rate means a faster switch to the other stage.

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

Outline

1

Introduction: Stochastic Fluid Model

2

Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

Multi-stage buffer

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

Multi-stage SFMs

Thresholds uk, dk, k = 2, . . . , n, with 0 < dk < uk < dk+1 Hitting uk from below while in Stage (k − 1) results in Stage (k − 1) → Stage k Hitting dk from above while in Stage k results in Stage k → Stage (k − 1) The analysis is built upon arguments similar to before, and is more complex. Related models (with dk = uk): Bean and O’Reilly (2008), Da Silva Soares and Latouche (2009)

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Introduction: Stochastic Fluid Model Multi-stage SFMs with congestion control Two-stage SFMs Transient Analysis Stationary Analysis Additional measures Multi-stage SFMs

References

1

O’Reilly, M. M. Multi-Stage Stochastic Fluid Models for Congestion Control (2013). Submitted (EJOR).

2

Bean, N. G., O’Reilly, M. M. and Taylor, P . G. (2005). Hitting probabilities and hitting times for stochastic fluid flows. Stoch. Proc. Appl. 115, 1530–1556.

3

Bean, N. G., O’Reilly, M. M. and Taylor, P . G. (2009). Hitting probabilities and hitting times for stochastic fluid flows: the bounded model. Probab. Engrg.

  • Inform. Sci., 23(1):121–147.

4

  • R. Malhotra, M. R. H. Mandjes, W. R. W. Scheinhardt and J. L. van den Berg

(2009). A feedback fluid queue with two congestion control thresholds. Math Meth Oper Res, 70:149–169.

5

  • N. G. Bean and M. M. O’Reilly. Performance measures of a multi-layer

Markovian fluid model (2008). Ann. Oper. Res., 160:99–120.

6

  • A. Da Silva Soares and G. Latouche. Fluid queues with level dependent

evolution (2009). European Journal of Operational Research, 196:1041–1048. Thanks for listening!

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