# Reflected Brownian Motions, Dirichlet Processes and ... Re¯¬â€ected Brownian...

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Reflected Brownian Motions, Dirichlet Processes and Queueing Networks

K. Ramanan (Carnegie Mellon University)

includes joint work with Weining 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

Queueing Networks and Diffusion Approximations

Some Early Influential Papers

1 M. Harrison. The diffusion approximation for tandem queues in heavy 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

Queueing Networks and Diffusion Approximations

Some Early Influential Papers

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

2 M. Harrison and M. Reiman. Reflected Brownian motion on an orthant. 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

Queueing Networks and Diffusion Approximations

Some Early Influential Papers

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

2 M. Harrison and M. Reiman. Reflected Brownian motion on an orthant. 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

Queueing Networks and Diffusion Approximations

Some Early Influential Papers

2 M. Harrison and M. Reiman. Reflected Brownian motion on an orthant. 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 of multidimensional reflected Brownian motion. SIAM J. Appl. Math., 41 (1981) 345–361.

Queueing Networks and Diffusion Approximations

Some Early Influential Papers

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 of multidimensional reflected Brownian motion. SIAM J. Appl. Math., 41 (1981) 345–361.

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

Queueing Networks and Diffusion Approximations

Some Early Influential Papers

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 of multidimensional reflected Brownian motion. SIAM J. Appl. Math., 41 (1981) 345–361.

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

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

Reflected Processes – what are they?

G d(x) φ

G is the closure of some connected domain in Rn

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

φ satisfies some specified interior dynamics Want φ(t) ∈ G for all 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 3 / 23

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

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

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 2 |η|(t)

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 2 |η|(t)

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 2 |η|(t)

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 2 |η|(t)

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 2 |η|(t)

The Harrison-Reiman RBM

General Framework

1 φ(t) = ψ(t) + η(t) ∈ G 2 |η|(t)

The Harrison-Reiman RBM

General Framework

1 φ(t) = ψ(t) + η(t) ∈ G 2 |η|(t)

The Harrison-Reiman RBM

General Framework

1 φ(t) = ψ(t) + η(t) ∈ G 2 |η|(t)

The Harrison-Reiman RBM

General Framework

1 φ(t) = ψ(t) + η(t) ∈ G 2 |η|(t)

Diffusion approximations for open single-class networks

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

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

Diffusion approximations for open single-class networks

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

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

Diffusion approximations for open single-class networks

Observation: Open single-class networks are modelled by reflection 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-class networks are RBMs in RN+ 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

Diffusion approximations for open single-class networks

Observation: Open single-class networks are modelled by reflection 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-class networks are RBMs in RN+ with Minkowski reflection matrices R

What about multi-class networks?

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

Diffusion approximations for open single-class networks

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

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

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