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Page 1: Reflected Brownian Motions, Dirichlet Processes and ...Reflected Brownian Motions, Dirichlet Processes and Queueing Networks K. Ramanan (Carnegie Mellon University) includes joint

Reflected Brownian Motions, Dirichlet Processesand Queueing Networks

K. Ramanan(Carnegie Mellon University)

includes joint work withWeining Kang and Martin Reiman

K. Ramanan (Carnegie Mellon University) includes joint work with Weining Kang and Martin Reiman (Carnegie Mellon UnivReflected Brownian Motions, Dirichlet Processes and Queueing Networks 1 / 23

Page 2: Reflected Brownian Motions, Dirichlet Processes and ...Reflected Brownian Motions, Dirichlet Processes and Queueing Networks K. Ramanan (Carnegie Mellon University) includes joint

Queueing Networks and Diffusion Approximations

Some Early Influential Papers

1 M. Harrison. The diffusion approximation for tandem queues inheavy traffic. Adv. Appl. Probab., 10 (1978), 886–905.

K. Ramanan (Carnegie Mellon University) includes joint work with Weining Kang and Martin Reiman (Carnegie Mellon UnivReflected Brownian Motions, Dirichlet Processes and Queueing Networks 2 / 23

Page 3: Reflected Brownian Motions, Dirichlet Processes and ...Reflected Brownian Motions, Dirichlet Processes and Queueing Networks K. Ramanan (Carnegie Mellon University) includes joint

Queueing Networks and Diffusion Approximations

Some Early Influential Papers

1 M. Harrison. The diffusion approximation for tandem queues inheavy traffic. Adv. Appl. Probab., 10 (1978), 886–905.

2 M. Harrison and M. Reiman. Reflected Brownian motion on anorthant. Ann. Probab., 8 (1981), 302–308.

K. Ramanan (Carnegie Mellon University) includes joint work with Weining Kang and Martin Reiman (Carnegie Mellon UnivReflected Brownian Motions, Dirichlet Processes and Queueing Networks 2 / 23

Page 4: Reflected Brownian Motions, Dirichlet Processes and ...Reflected Brownian Motions, Dirichlet Processes and Queueing Networks K. Ramanan (Carnegie Mellon University) includes joint

Queueing Networks and Diffusion Approximations

Some Early Influential Papers

1 M. Harrison. The diffusion approximation for tandem queues inheavy traffic. Adv. Appl. Probab., 10 (1978), 886–905.

2 M. Harrison and M. Reiman. Reflected Brownian motion on anorthant. Ann. Probab., 8 (1981), 302–308.

3 M. Reiman. Open queueing networks in heavy traffic. Math. Oper.Res., 9 (1984), 441–458.

K. Ramanan (Carnegie Mellon University) includes joint work with Weining Kang and Martin Reiman (Carnegie Mellon UnivReflected Brownian Motions, Dirichlet Processes and Queueing Networks 2 / 23

Page 5: Reflected Brownian Motions, Dirichlet Processes and ...Reflected Brownian Motions, Dirichlet Processes and Queueing Networks K. Ramanan (Carnegie Mellon University) includes joint

Queueing Networks and Diffusion Approximations

Some Early Influential Papers

1 M. Harrison. The diffusion approximation for tandem queues inheavy traffic. Adv. Appl. Probab., 10 (1978), 886–905.

2 M. Harrison and M. Reiman. Reflected Brownian motion on anorthant. Ann. Probab., 8 (1981), 302–308.

3 M. Reiman. Open queueing networks in heavy traffic. Math. Oper.Res., 9 (1984), 441–458.

4 M. Harrison and M. Reiman. On the distribution ofmultidimensional reflected Brownian motion. SIAM J. Appl. Math.,41 (1981) 345–361.

K. Ramanan (Carnegie Mellon University) includes joint work with Weining Kang and Martin Reiman (Carnegie Mellon UnivReflected Brownian Motions, Dirichlet Processes and Queueing Networks 2 / 23

Page 6: Reflected Brownian Motions, Dirichlet Processes and ...Reflected Brownian Motions, Dirichlet Processes and Queueing Networks K. Ramanan (Carnegie Mellon University) includes joint

Queueing Networks and Diffusion Approximations

Some Early Influential Papers

1 M. Harrison. The diffusion approximation for tandem queues inheavy traffic. Adv. Appl. Probab., 10 (1978), 886–905.

2 M. Harrison and M. Reiman. Reflected Brownian motion on anorthant. Ann. Probab., 8 (1981), 302–308.

3 M. Reiman. Open queueing networks in heavy traffic. Math. Oper.Res., 9 (1984), 441–458.

4 M. Harrison and M. Reiman. On the distribution ofmultidimensional reflected Brownian motion. SIAM J. Appl. Math.,41 (1981) 345–361.

5 M. Harrison and R. Williams Multidimensional reflected Brownianmotions having exponential stationary distributions. Ann. Probab.,15 (1987) 115–137.

K. Ramanan (Carnegie Mellon University) includes joint work with Weining Kang and Martin Reiman (Carnegie Mellon UnivReflected Brownian Motions, Dirichlet Processes and Queueing Networks 2 / 23

Page 7: Reflected Brownian Motions, Dirichlet Processes and ...Reflected Brownian Motions, Dirichlet Processes and Queueing Networks K. Ramanan (Carnegie Mellon University) includes joint

Queueing Networks and Diffusion Approximations

Some Early Influential Papers

1 M. Harrison. The diffusion approximation for tandem queues inheavy traffic. Adv. Appl. Probab., 10 (1978), 886–905.

2 M. Harrison and M. Reiman. Reflected Brownian motion on anorthant. Ann. Probab., 8 (1981), 302–308.

3 M. Reiman. Open queueing networks in heavy traffic. Math. Oper.Res., 9 (1984), 441–458.

4 M. Harrison and M. Reiman. On the distribution ofmultidimensional reflected Brownian motion. SIAM J. Appl. Math.,41 (1981) 345–361.

5 M. Harrison and R. Williams Multidimensional reflected Brownianmotions having exponential stationary distributions. Ann. Probab.,15 (1987) 115–137.

6 M. Harrison and R. Williams Brownian models of open queueingnetworks with homogeneous customer populations. Stochastics,22 (1987) 77-115.

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Reflected Processes – what are they?

Gd(x) φ

G is the closure of some connected domain in Rn

d(·) is a vector field specified on the boundary ∂Gd(x) is a cone for every x ∈ ∂G, graph of d(·) is closed

φ satisfies some specified interior dynamicsWant φ(t) ∈ G for all t ∈ [0,∞)

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The Skorokhod Problem – Multidimensional Version

Given (G,d(·)), for any continuous ψ, find a continuous φ such that

G

1 φ(t) = ψ(t) + η(t) ∈ G

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The Skorokhod Problem – Multidimensional Version

Given (G,d(·)), for any continuous ψ, find a continuous φ such that

G

1 φ(t) = ψ(t) + η(t) ∈ G2 |η|(t) <∞ for every t ∈ [0,∞)

K. Ramanan (Carnegie Mellon University) includes joint work with Weining Kang and Martin Reiman (Carnegie Mellon UnivReflected Brownian Motions, Dirichlet Processes and Queueing Networks 4 / 23

Page 11: Reflected Brownian Motions, Dirichlet Processes and ...Reflected Brownian Motions, Dirichlet Processes and Queueing Networks K. Ramanan (Carnegie Mellon University) includes joint

The Skorokhod Problem – Multidimensional Version

Given (G,d(·)), for any continuous ψ, find a continuous φ such that

G

1 φ(t) = ψ(t) + η(t) ∈ G2 |η|(t) <∞ for every t ∈ [0,∞)

3 |η|(t) =∫ ∞

0 I{φ(s)∈∂G}d |η|(s)i.e., total variation |η| increasesonly when φ lies in ∂G

K. Ramanan (Carnegie Mellon University) includes joint work with Weining Kang and Martin Reiman (Carnegie Mellon UnivReflected Brownian Motions, Dirichlet Processes and Queueing Networks 4 / 23

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The Skorokhod Problem – Multidimensional Version

Given (G,d(·)), for any continuous ψ, find a continuous φ such that

G

1 φ(t) = ψ(t) + η(t) ∈ G2 |η|(t) <∞ for every t ∈ [0,∞)

3 |η|(t) =∫ ∞

0 I{φ(s)∈∂G}d |η|(s)i.e., total variation |η| increasesonly when φ lies in ∂G

4 η(t) =∫ t

0 γ(s)d |η|(s), with γ(s) ∈ d(φ(s)) d |η| a.e.

K. Ramanan (Carnegie Mellon University) includes joint work with Weining Kang and Martin Reiman (Carnegie Mellon UnivReflected Brownian Motions, Dirichlet Processes and Queueing Networks 4 / 23

Page 13: Reflected Brownian Motions, Dirichlet Processes and ...Reflected Brownian Motions, Dirichlet Processes and Queueing Networks K. Ramanan (Carnegie Mellon University) includes joint

The Skorokhod Problem – Multidimensional Version

Given (G,d(·)), for any continuous ψ, find a continuous φ such that

G

1 φ(t) = ψ(t) + η(t) ∈ G2 |η|(t) <∞ for every t ∈ [0,∞)

3 |η|(t) =∫ ∞

0 I{φ(s)∈∂G}d |η|(s)i.e., total variation |η| increasesonly when φ lies in ∂G

4 η(t) =∫ t

0 γ(s)d |η|(s), with γ(s) ∈ d(φ(s)) d |η| a.e.

Γ denotes the map that takes ψ to φ (when well-defined).

K. Ramanan (Carnegie Mellon University) includes joint work with Weining Kang and Martin Reiman (Carnegie Mellon UnivReflected Brownian Motions, Dirichlet Processes and Queueing Networks 4 / 23

Page 14: Reflected Brownian Motions, Dirichlet Processes and ...Reflected Brownian Motions, Dirichlet Processes and Queueing Networks K. Ramanan (Carnegie Mellon University) includes joint

The Skorokhod Problem – Multidimensional Version

Given (G,d(·)), for any continuous ψ, find a continuous φ such that

G

1 φ(t) = ψ(t) + η(t) ∈ G2 |η|(t) <∞ for every t ∈ [0,∞)

3 |η|(t) =∫ ∞

0 I{φ(s)∈∂G}d |η|(s)i.e., total variation |η| increasesonly when φ lies in ∂G

4 η(t) =∫ t

0 γ(s)d |η|(s), with γ(s) ∈ d(φ(s)) d |η| a.e.

Γ denotes the map that takes ψ to φ (when well-defined).

Note: If X is a martingale, then Z = Γ(X ) is a semimartingale.

semimartingale = local martingale + bounded variation

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The Harrison-Reiman RBM

General Framework

1 φ(t) = ψ(t) + η(t) ∈ G2 |η|(t) <∞ for every t ∈ [0,∞)

3 |η|(t) =∫ ∞

0 I{φ(s)∈∂G}d |η|(s)i.e., total variation |η| increasesonly when φ lies in ∂G

4 η(t) =∫ t

0 γ(s)d |η|(s), with γ(s) ∈ d(φ(s)) d |η| a.e.

K. Ramanan (Carnegie Mellon University) includes joint work with Weining Kang and Martin Reiman (Carnegie Mellon UnivReflected Brownian Motions, Dirichlet Processes and Queueing Networks 5 / 23

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The Harrison-Reiman RBM

General Framework

1 φ(t) = ψ(t) + η(t) ∈ G2 |η|(t) <∞ for every t ∈ [0,∞)

3 |η|(t) =∫ ∞

0 I{φ(s)∈∂G}d |η|(s)i.e., total variation |η| increasesonly when φ lies in ∂G

4 η(t) =∫ t

0 γ(s)d |η|(s), with γ(s) ∈ d(φ(s)) d |η| a.e.

N-dimensional Nonnegative Orthant Framework

1 φ(t) = ψ(t) + Rθ(t) ∈ G; where R is an N × N matrix

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The Harrison-Reiman RBM

General Framework

1 φ(t) = ψ(t) + η(t) ∈ G2 |η|(t) <∞ for every t ∈ [0,∞)

3 |η|(t) =∫ ∞

0 I{φ(s)∈∂G}d |η|(s)i.e., total variation |η| increasesonly when φ lies in ∂G

4 η(t) =∫ t

0 γ(s)d |η|(s), with γ(s) ∈ d(φ(s)) d |η| a.e.

N-dimensional Nonnegative Orthant Framework

1 φ(t) = ψ(t) + Rθ(t) ∈ G; where R is an N × N matrix2 θi non-decreasing function, i = 1, . . . ,N

K. Ramanan (Carnegie Mellon University) includes joint work with Weining Kang and Martin Reiman (Carnegie Mellon UnivReflected Brownian Motions, Dirichlet Processes and Queueing Networks 5 / 23

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The Harrison-Reiman RBM

General Framework

1 φ(t) = ψ(t) + η(t) ∈ G2 |η|(t) <∞ for every t ∈ [0,∞)

3 |η|(t) =∫ ∞

0 I{φ(s)∈∂G}d |η|(s)i.e., total variation |η| increasesonly when φ lies in ∂G

4 η(t) =∫ t

0 γ(s)d |η|(s), with γ(s) ∈ d(φ(s)) d |η| a.e.

N-dimensional Nonnegative Orthant Framework

1 φ(t) = ψ(t) + Rθ(t) ∈ G; where R is an N × N matrix2 θi non-decreasing function, i = 1, . . . ,N3

∫ ∞0 φi(t)dθi(t) = 0, i = 1, . . .N;

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Diffusion approximations for open single-classnetworks

Observation: Open single-class networks are modelled byreflection matrices R that are Minkowski

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Diffusion approximations for open single-classnetworks

Observation: Open single-class networks are modelled byreflection matrices R that are Minkowski

Theorem (Harrison-Reiman, ’81)R Minkowski matrix ⇒ Γ well-defined and continuous

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Diffusion approximations for open single-classnetworks

Observation: Open single-class networks are modelled byreflection matrices R that are Minkowski

Theorem (Harrison-Reiman, ’81)R Minkowski matrix ⇒ Γ well-defined and continuous

Theorems (Reiman ’84, Chen-Mandelbaum ’91)Rigorously shown that diffusion limits of open single-classnetworks are RBMs in R

N+ with Minkowski reflection matrices R

K. Ramanan (Carnegie Mellon University) includes joint work with Weining Kang and Martin Reiman (Carnegie Mellon UnivReflected Brownian Motions, Dirichlet Processes and Queueing Networks 6 / 23

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Diffusion approximations for open single-classnetworks

Observation: Open single-class networks are modelled byreflection matrices R that are Minkowski

Theorem (Harrison-Reiman, ’81)R Minkowski matrix ⇒ Γ well-defined and continuous

Theorems (Reiman ’84, Chen-Mandelbaum ’91)Rigorously shown that diffusion limits of open single-classnetworks are RBMs in R

N+ with Minkowski reflection matrices R

What about multi-class networks?

Peterson (’91) established diffusion approximations of multiclassfeedforward networks; associated Γ is continuous

K. Ramanan (Carnegie Mellon University) includes joint work with Weining Kang and Martin Reiman (Carnegie Mellon UnivReflected Brownian Motions, Dirichlet Processes and Queueing Networks 6 / 23

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Diffusion approximations for open single-classnetworks

Observation: Open single-class networks are modelled byreflection matrices R that are Minkowski

Theorem (Harrison-Reiman, ’81)R Minkowski matrix ⇒ Γ well-defined and continuous

Theorems (Reiman ’84, Chen-Mandelbaum ’91)Rigorously shown that diffusion limits of open single-classnetworks are RBMs in R

N+ with Minkowski reflection matrices R

What about multi-class networks?

Peterson (’91) established diffusion approximations of multiclassfeedforward networks; associated Γ is continuousAll multi-class queueing networks need not be modelled by Γ thatare continuous

K. Ramanan (Carnegie Mellon University) includes joint work with Weining Kang and Martin Reiman (Carnegie Mellon UnivReflected Brownian Motions, Dirichlet Processes and Queueing Networks 6 / 23

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Semimartingale Reflected Brownian Motions (SRBM)

When Γ well-defined, RBM can be defined pathwise;

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Semimartingale Reflected Brownian Motions (SRBM)

When Γ well-defined, RBM can be defined pathwise;

Otherwise, can define RBMs in the weak sense (using thesubmartingale formulation)

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Semimartingale Reflected Brownian Motions (SRBM)

When Γ well-defined, RBM can be defined pathwise;

Otherwise, can define RBMs in the weak sense (using thesubmartingale formulation)

In the orthant framework, R completely-S is necessary andsufficient for existence of solutions to the SP;(Bernard-El Kharroubi, ’91)

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Semimartingale Reflected Brownian Motions (SRBM)

When Γ well-defined, RBM can be defined pathwise;

Otherwise, can define RBMs in the weak sense (using thesubmartingale formulation)

In the orthant framework, R completely-S is necessary andsufficient for existence of solutions to the SP;(Bernard-El Kharroubi, ’91)

But R completely-S does not guarantee uniqueness;

K. Ramanan (Carnegie Mellon University) includes joint work with Weining Kang and Martin Reiman (Carnegie Mellon UnivReflected Brownian Motions, Dirichlet Processes and Queueing Networks 7 / 23

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Semimartingale Reflected Brownian Motions (SRBM)

When Γ well-defined, RBM can be defined pathwise;

Otherwise, can define RBMs in the weak sense (using thesubmartingale formulation)

In the orthant framework, R completely-S is necessary andsufficient for existence of solutions to the SP;(Bernard-El Kharroubi, ’91)

But R completely-S does not guarantee uniqueness;

Completely-S condition is necessary and sufficient for existenceand uniqueness (in distribution) of SRBM(Reiman-Williams ’88, Taylor-Williams)

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Multiclass Queueing Networks to SRBM

The Bramson-Williams Framework (’98)State Space Collapse + Completely-S

⇓Heavy traffic limit theorem for the multi-class queueing network

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Examples Outside this Framework

Example 1: Generalized Processor Sharing

1

2

c

N

α

α1

2

(−1,1)

(1,−1)

(1,1)

α λ − α( ),,1 2 21,

1

2

λ − 1, 0)

λ −

(λ +1 2

ESP describing mapping from inputs to the queue content

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Examples Outside this Framework

Example 1: Generalized Processor SharingThe 3-dimensional GPS Model

d

d

d

1

2

3

n

n

n

1

3

2

d 0

Solutions to the SP do not exist for all (right) continuous paths

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Examples Outside this Framework

Example 1: Generalized Processor SharingThe 3-dimensional GPS Model

d

d

d

1

2

3

n

n

n

1

3

2

d 0

Solutions to the SP do not exist for all (right) continuous pathsDoes not satisfy Completely-S condition

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Examples Outside this Framework

Example 2: FIFO Tandem Queue with Deadlines (Reed)

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Examples Outside this Framework

Example 2: FIFO Tandem Queue with Deadlines (Reed)

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Submartingale Formulation vs. Skorokhod ProblemApproach

Submartingale ProblemSkorokhod Problem

Pros

Cons existence and uniqueness

Can be used to analyze arbitrary processesyields pathwise uniqueness

Constructs strong solutions;

Can only be used to analyze semimartingales

Provides only weak

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The Extended Skorokhod Map

Definition of the ESP on (G,d(·)) (R ’06)

For any continuous ψ find a continuous φ such that1 φ(t) = ψ(t) + η(t) ∈ G;

2 η(t) − η(s) ∈ co[

u∈(s,t] d(φ(u))]

∀0 ≤ s < t <∞,

where co(A) = closure of the convex hull of A andd(x)

.= {0} for x ∈ G◦;

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The Extended Skorokhod Map

Definition of the ESP on (G,d(·)) (R ’06)

For any continuous ψ find a continuous φ such that1 φ(t) = ψ(t) + η(t) ∈ G;

2 η(t) − η(s) ∈ co[

u∈(s,t] d(φ(u))]

∀0 ≤ s < t <∞,

where co(A) = closure of the convex hull of A andd(x)

.= {0} for x ∈ G◦;

Γ : ψ 7→ φ called the Extended Skorokhod Map (ESM)

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The Extended Skorokhod Map

Definition of the ESP on (G,d(·)) (R ’06)

For any continuous ψ find a continuous φ such that1 φ(t) = ψ(t) + η(t) ∈ G;

2 η(t) − η(s) ∈ co[

u∈(s,t] d(φ(u))]

∀0 ≤ s < t <∞,

where co(A) = closure of the convex hull of A andd(x)

.= {0} for x ∈ G◦;

Γ : ψ 7→ φ called the Extended Skorokhod Map (ESM)

Note: If X is a martingale, then Z = Γ(X ) is not necessarily asemimartingale

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Properties of the Extended Skorokhod Problem (ESP)

Theorem(R. ’00, ’06)

If (φ, η) solve the SP for ψ, then (φ, η) solve the ESP for ψ

If (φ, η) solve the ESP for ψ and |η|(t) <∞ ∀t , then (φ, η) solvethe SP

The graph of the ESM Γ is closed.

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Properties of the Limiting Diffusion

Theorem(’R ’06 and Kang-’R ’08)

The reflected diffusion Z associated with the GPS ESP is asemimartingale on the interval [0, τ0], where

τ0 = inf{t ≥ 0 : Z (t) = 0}

but Z is not a semimartingale on [0,∞)

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Properties of the Limiting Diffusion

Theorem(’R ’06 and Kang-’R ’08)

The reflected diffusion Z associated with the GPS ESP is asemimartingale on the interval [0, τ0], where

τ0 = inf{t ≥ 0 : Z (t) = 0}

but Z is not a semimartingale on [0,∞)

2-d + BM case: follows from Williams (’85)

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Dirichlet Processes

DefinitionA process Z is a Dirichlet process if it admits the decomposition

Z = M + A

M local martingale and A a continuous process with A(0) = 0 that haszero quadratic variation,

i.e., for any sequence of partitions {Πn} of [0, t ],

limn→∞

|Πn| → 0 ⇒∑

ti∈Πn

|A(ti+1) − A(ti)|2 (P)→ 0.

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Dirichlet Processes

DefinitionA process Z is a Dirichlet process if it admits the decomposition

Z = M + A

M local martingale and A a continuous process with A(0) = 0 that haszero quadratic variation,

i.e., for any sequence of partitions {Πn} of [0, t ],

limn→∞

|Πn| → 0 ⇒∑

ti∈Πn

|A(ti+1) − A(ti)|2 (P)→ 0.

Note:If A is a process of a.s. finite variation on bounded intervals, Z is a

continuous semimartingales.

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A Dirichlet Process Characterization of RBMs

SetupGiven (G,d(·)), b, σ Lip. cont and σ uniformly elliptic.

Suppose there exists a Markov, weak solution (Z ,B),Ft to theassociated SDER and let Y = Z − X :

X (t) = z +

∫ t

0b(Z (s)) ds +

∫ t

0σ(Z (s)) dB(s).

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A Dirichlet Process Characterization of RBMs

SetupGiven (G,d(·)), b, σ Lip. cont and σ uniformly elliptic.

Suppose there exists a Markov, weak solution (Z ,B),Ft to theassociated SDER and let Y = Z − X :

X (t) = z +

∫ t

0b(Z (s)) ds +

∫ t

0σ(Z (s)) dB(s).

Theorem W. Kang and ’R, (’08)If there exist p ≥ 2 and q ≥ 2 such that for every 0 ≤ s, t ≤ T ,

E [|Y (t) − Y (s)|p|Fs] ≤ E

[

supu∈s,t]

|X (u) − X (s)|qFs

]

,

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A Dirichlet Process Characterization of RBMs

SetupGiven (G,d(·)), b, σ Lip. cont and σ uniformly elliptic.

Suppose there exists a Markov, weak solution (Z ,B),Ft to theassociated SDER and let Y = Z − X :

X (t) = z +

∫ t

0b(Z (s)) ds +

∫ t

0σ(Z (s)) dB(s).

Theorem W. Kang and ’R, (’08)If there exist p ≥ 2 and q ≥ 2 such that for every 0 ≤ s, t ≤ T ,

E [|Y (t) − Y (s)|p|Fs] ≤ E

[

supu∈s,t]

|X (u) − X (s)|qFs

]

,

then Z is a Dirichlet process.

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A Dirichlet Process Characterization of RBMs

SetupGiven (G,d(·)), b, σ Lip. cont and σ uniformly elliptic.

Suppose there exists a Markov, weak solution (Z ,B),Ft to theassociated SDER and let Y = Z − X :

X (t) = z +

∫ t

0b(Z (s)) ds +

∫ t

0σ(Z (s)) dB(s).

Theorem W. Kang and ’R, (’08)If there exist p ≥ 2 and q ≥ 2 such that for every 0 ≤ s, t ≤ T ,

E [|Y (t) − Y (s)|p|Fs] ≤ E

[

supu∈s,t]

|X (u) − X (s)|qFs

]

,

then Z is a Dirichlet process.In particular, this holds when the ESM is Hölder continuous or if thedirections satisfy the so-called generalized completely-S condition.

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Sketch of the Proof

Z (t) = Z (0) + B(t) + Y (t).

B standard Brownian motion, Y is the regulating processNeed to show

ti∈Πn

|Y (ti) − Y (ti−1)|p P→ 0, as||Πn|| → 0.

Defineζm = inf{t > 0 : |Z (t)| ≥ m}.

By localization suffices to show that

ti∈Πn

|Y (ti ∧ ζm) − Y (ti−1 ∧ ζ

m)|p P→ 0, as||Πn|| → 0.

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Sketch of the Proof (contd.)

ε

ε/21τ

τ2σ1

0σ = 0

• control p-variation on [τi , σi) using semimartingale property awayfrom origin; show summable;• obtain estimates on p-variation on [σi , τi+1) in terms of time spent inε/2-nbhd of 0; show it disappears, on sending ε→ 0, by instantaneousreflection property

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Properties of the GPS ESP

Theorem(R ’06, Dupuis-’R ’98)

The GPS ESM is Lipschitz continuous

Proof Involves Constructing an Associated Norm;

Combines convex duality and algebra; vertices of B form the rootsystem for the Lie albegra An−1 of the Lie group sℓn

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Revisiting the GPS Model

1

2

c

N

α

α1

2

Order sources so that

λ1

α1≥λ2

α2≥ . . . ≥

λN

αN,

and define

J .= max

{

j ≤ N :λj

αj=λ1

α1

}

.

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A Heavy Traffic Limit Theorem for the GPS Model

Theorem(R.-Reiman ’03, R.-Reiman ’06)

Suppose the heavy traffic condition holds:

J∑

j=1

λj =J

j=1

αj = 1.

The appropriately scaled workload process in the GPS modelconverges weakly to the pathwise unique solution of a reflected

diffusion in RJ+ associated with the GPS ESP with weights

α̃i =αi

i≤J αi.

and modified covariance (identified explicitly).

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A Heavy Traffic Limit Theorem for the GPS Model

Theorem(R.-Reiman ’03, R.-Reiman ’06)

Suppose the heavy traffic condition holds:

J∑

j=1

λj =J

j=1

αj = 1.

The appropriately scaled workload process in the GPS modelconverges weakly to the pathwise unique solution of a reflected

diffusion in RJ+ associated with the GPS ESP with weights

α̃i =αi

i≤J αi.

and modified covariance (identified explicitly).The reflected diffusion is a Dirichlet process.

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A Heavy Traffic Limit Theorem for the GPS Model

Theorem(R.-Reiman ’03, R.-Reiman ’06)

Suppose the heavy traffic condition holds:

J∑

j=1

λj =J

j=1

αj = 1.

The appropriately scaled workload process in the GPS modelconverges weakly to the pathwise unique solution of a reflected

diffusion in RJ+ associated with the GPS ESP with weights

α̃i =αi

i≤J αi.

and modified covariance (identified explicitly).The reflected diffusion is a Dirichlet process.

Lies outside the Bramson+Williams and cont. mapping frameworksK. Ramanan (Carnegie Mellon University) includes joint work with Weining Kang and Martin Reiman (Carnegie Mellon UnivReflected Brownian Motions, Dirichlet Processes and Queueing Networks 22 / 23

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References

1 Kang and ’R. A Dirichlet process characterization of a class ofmultidimensional reflected diffusions, 2008, to appear in Ann.Probab.

2 ’R. Reflected diffusions defined via the extended Skorokhodproblem. EJP, 2006.

3 ’R and Reiman. The heavy traffic limit of an unbalancedgeneralized processor sharing model, Ann. Appl. Probab., 2008.

4 ’R and Reiman. Fluid and heavy traffic limits for a generalizedprocessor sharing model Ann. Appl. Probab., 2003.

5 Burdzy, Kang and ’R. The Skorokhod problem in atime-dependent interval, Stoch. Proc. Appl., 2009.

6 Kang and ’R. Stationary distributions for Dirichlet reflectedprocesses (in preparation), 2010.

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