Mohammadreza Aghajani reza@brown.edu Mohammadreza Aghajani - - PowerPoint PPT Presentation

Outline Ergodicity Theorems for Markov Chains: Classical Results Markov Chains in Infinite-Dimensions: Asymptotic Coupling Application: Many-Server Queuing Systems

About Me

Mohammadreza Aghajani

reza@brown.edu

Mohammadreza Aghajani Asymptotic Coupling with Application in Queuing Systems

Outline Ergodicity Theorems for Markov Chains: Classical Results Markov Chains in Infinite-Dimensions: Asymptotic Coupling Application: Many-Server Queuing Systems

About Me

Mohammadreza Aghajani

reza@brown.edu

Mohammadreza Aghajani Asymptotic Coupling with Application in Queuing Systems

Outline Ergodicity Theorems for Markov Chains: Classical Results Markov Chains in Infinite-Dimensions: Asymptotic Coupling Application: Many-Server Queuing Systems

About Me

* Born Mashhad, Iran 1986

Mohammadreza Aghajani

reza@brown.edu

Mohammadreza Aghajani Asymptotic Coupling with Application in Queuing Systems

Outline Ergodicity Theorems for Markov Chains: Classical Results Markov Chains in Infinite-Dimensions: Asymptotic Coupling Application: Many-Server Queuing Systems

About Me

* * Born Mashhad, Iran 1986 College Sharif Univ. of Tech Tehran, Iran. 2003

Mohammadreza Aghajani

reza@brown.edu

Mohammadreza Aghajani Asymptotic Coupling with Application in Queuing Systems

Outline Ergodicity Theorems for Markov Chains: Classical Results Markov Chains in Infinite-Dimensions: Asymptotic Coupling Application: Many-Server Queuing Systems

About Me

* * * Born Mashhad, Iran 1986 M.S in Electrical Eng. Instituto Superior Tecnico Lisbon, Portugal 2008 College Sharif Univ. of Tech Tehran, Iran. 2003

Mohammadreza Aghajani

reza@brown.edu

Mohammadreza Aghajani Asymptotic Coupling with Application in Queuing Systems

Outline Ergodicity Theorems for Markov Chains: Classical Results Markov Chains in Infinite-Dimensions: Asymptotic Coupling Application: Many-Server Queuing Systems

About Me

* * * * Born Mashhad, Iran 1986 M.S in Electrical Eng. Instituto Superior Tecnico Lisbon, Portugal 2008 M.S. in Electrical Eng Carnegie-Mellon Univ. Pittsburgh, PA, USA. 2009 College Sharif Univ. of Tech Tehran, Iran. 2003

Mohammadreza Aghajani

reza@brown.edu

Mohammadreza Aghajani Asymptotic Coupling with Application in Queuing Systems

Outline Ergodicity Theorems for Markov Chains: Classical Results Markov Chains in Infinite-Dimensions: Asymptotic Coupling Application: Many-Server Queuing Systems

About Me

* * * * * Born Mashhad, Iran 1986 M.S in Electrical Eng. Instituto Superior Tecnico Lisbon, Portugal 2008 PhD in Applied Math Brown University Providence, RI, USA 2010 M.S. in Electrical Eng Carnegie-Mellon Univ. Pittsburgh, PA, USA. 2009 College Sharif Univ. of Tech Tehran, Iran. 2003

Mohammadreza Aghajani

reza@brown.edu

Mohammadreza Aghajani Asymptotic Coupling with Application in Queuing Systems

Outline Ergodicity Theorems for Markov Chains: Classical Results Markov Chains in Infinite-Dimensions: Asymptotic Coupling Application: Many-Server Queuing Systems

Asymptotic Coupling with Application in Queuing Systems

Mohammadreza Aghajani

Brown University

September 6 2012

Mohammadreza Aghajani Asymptotic Coupling with Application in Queuing Systems

Outline Ergodicity Theorems for Markov Chains: Classical Results Markov Chains in Infinite-Dimensions: Asymptotic Coupling Application: Many-Server Queuing Systems

1 Ergodicity Theorems for Markov Chains: Classical Results

Mohammadreza Aghajani Asymptotic Coupling with Application in Queuing Systems

Outline Ergodicity Theorems for Markov Chains: Classical Results Markov Chains in Infinite-Dimensions: Asymptotic Coupling Application: Many-Server Queuing Systems

1 Ergodicity Theorems for Markov Chains: Classical Results 2 Markov Chains in Infinite-Dimensions: Asymptotic Coupling

Mohammadreza Aghajani Asymptotic Coupling with Application in Queuing Systems

Outline Ergodicity Theorems for Markov Chains: Classical Results Markov Chains in Infinite-Dimensions: Asymptotic Coupling Application: Many-Server Queuing Systems

1 Ergodicity Theorems for Markov Chains: Classical Results 2 Markov Chains in Infinite-Dimensions: Asymptotic Coupling 3 Application: Many-Server Queuing Systems

Mohammadreza Aghajani Asymptotic Coupling with Application in Queuing Systems

Survey

1 Ergodicity Theorems for Markov Chains: Classical Results 2 Markov Chains in Infinite-Dimensions: Asymptotic Coupling 3 Application: Many-Server Queuing Systems

Outline Ergodicity Theorems for Markov Chains: Classical Results Markov Chains in Infinite-Dimensions: Asymptotic Coupling Application: Many-Server Queuing Systems

Stability of Markov Chains

Markov Chain on general space (E, E) Given Initial distribution λ Transition Kernel P(x, ·) We have X ∼ Pλ on E ∞. X(n) ∼ λPn. Notions of Stability Invariant Distribution: π = πP. Ergodicity λPN − π → 0.

Mohammadreza Aghajani Asymptotic Coupling with Application in Queuing Systems

Outline Ergodicity Theorems for Markov Chains: Classical Results Markov Chains in Infinite-Dimensions: Asymptotic Coupling Application: Many-Server Queuing Systems

Coupling

X, Y : Two random variables on (E, E) Definition (Coupling) Z = (˜ X, ˜ Y ) on E × E is a coupling of X and Y if ˜ X d = X, ˜ Y

d

= Y . Coupling Inequality L{X} − L{Y } ≤ 2P( ˜ X = ˜ Y )

Mohammadreza Aghajani Asymptotic Coupling with Application in Queuing Systems

Outline Ergodicity Theorems for Markov Chains: Classical Results Markov Chains in Infinite-Dimensions: Asymptotic Coupling Application: Many-Server Queuing Systems

Coupling of Markov Chains

Two independent copies of a the chain P(x, ·) on E ⊂ Z: T: Coupling Time Yn . = ˜ Xn if n ≤ T Xn if n > T Y ∼ ˜ X

x T

~

x y

By Coupling Inequality: Pλ(Xn ∈ ·) − P˜

λ( ˜

Xn ∈ ·) ≤ 2Pλ˜

λ(T > n)

When coupling is ‘successful’, ergodicity holds.

Mohammadreza Aghajani Asymptotic Coupling with Application in Queuing Systems

Outline Ergodicity Theorems for Markov Chains: Classical Results Markov Chains in Infinite-Dimensions: Asymptotic Coupling Application: Many-Server Queuing Systems

Ergodicity for Harris Chains

Definition (Harris Chain) (i) Px (Xn ∈ A; for some n) = 1, ∀x ∈ E (recurrence) (ii) Px (Xn0 ∈ B) ≥ βϕ(B), ∀x ∈ A, ∀B ∈ E (small set) A

S S

1 2

XS1 ~ ϕ XS2 ~ ϕ x

X

Mohammadreza Aghajani Asymptotic Coupling with Application in Queuing Systems

Outline Ergodicity Theorems for Markov Chains: Classical Results Markov Chains in Infinite-Dimensions: Asymptotic Coupling Application: Many-Server Queuing Systems

Ergodicity for Harris Chains

Assume an invariant distribution π exists Two independent copies of the chain: X is initialized at arbitrary λ → Corresponding {Sj} ˜ X is initialized at π → Corresponding {˜ Sj} A ‘successful’ coupling: Coupling time T = Sn = ˜ Sm Renewal Theory ⇒ T is almost surely finite. Coupling inequality gives ergodicity Pλ(Xn ∈ ·) − π ≤ Pλ˜

λ(T > n) → 0

Mohammadreza Aghajani Asymptotic Coupling with Application in Queuing Systems

Survey

1 Ergodicity Theorems for Markov Chains: Classical Results 2 Markov Chains in Infinite-Dimensions: Asymptotic Coupling 3 Application: Many-Server Queuing Systems

Outline Ergodicity Theorems for Markov Chains: Classical Results Markov Chains in Infinite-Dimensions: Asymptotic Coupling Application: Many-Server Queuing Systems

Infinite-Dimensional State Spaces: Example

Example: Stochastic Delay Differential Equation (SDDE) dX(t) = −cX(t)dt + g (X(t − r)) dWt

t t-r

X (s) = X(t+s)

t

{Xt; t ≥ 0} is a Markov Process on C ([−r, 0]) Invariant Distribution Exists for large c. Given the solution Xt for any t > 0, X0 can be recovered using Law of Iterated Logarithms

Mohammadreza Aghajani Asymptotic Coupling with Application in Queuing Systems

Outline Ergodicity Theorems for Markov Chains: Classical Results Markov Chains in Infinite-Dimensions: Asymptotic Coupling Application: Many-Server Queuing Systems

What Goes Wrong?

For SDDE and for typical inf-dim Markov chains: P(x, ·) and P(y, ·) are mutually singular for x = y Consequences: Only small sets are singletons Generally, singletons are not recurrent sets. And therefore, Not Harris chains No successful coupling

Mohammadreza Aghajani Asymptotic Coupling with Application in Queuing Systems

Outline Ergodicity Theorems for Markov Chains: Classical Results Markov Chains in Infinite-Dimensions: Asymptotic Coupling Application: Many-Server Queuing Systems

Asymptotic Coupling

Definition (Asymptotic Coupling) A measure Γ on E ∞ × E ∞ is an ‘Asymptotic Coupling’ for two initial distributions λ,µ on E, if

1 Γ1 ∼ Pλ and Γ2 ∼ Pµ. 2 Γ ({(x, y) ∈ E ∞ × E ∞; limn→∞ d(xn, yn) = 0}) > 0

Theorem (Hairer, Mattingly, Scheutzow) If there exists a ‘large enough’ set A ⊂ E such that for every x, y ∈ A there exists an asymptotic coupling Γx,y of δx and δy, then P has at most one invariant distribution.

Mohammadreza Aghajani Asymptotic Coupling with Application in Queuing Systems

Survey

1 Ergodicity Theorems for Markov Chains: Classical Results 2 Markov Chains in Infinite-Dimensions: Asymptotic Coupling 3 Application: Many-Server Queuing Systems

Outline Ergodicity Theorems for Markov Chains: Classical Results Markov Chains in Infinite-Dimensions: Asymptotic Coupling Application: Many-Server Queuing Systems

Many-Server Queues

μ

λ

N

μ μ μ

1 2 3 N

Where do they arise? Call Centers

Health Care Data Centers

Mohammadreza Aghajani Asymptotic Coupling with Application in Queuing Systems

Outline Ergodicity Theorems for Markov Chains: Classical Results Markov Chains in Infinite-Dimensions: Asymptotic Coupling Application: Many-Server Queuing Systems

A Markovian Representation

Analysis of GI/G/N systems: Usual representation is not Markovian A measure-valued (infinite-dimensional) Markovian representation[Kaspi, Ramanan]: Y N(t) =

• X N, νN, Z N

∈ R × H−2 × W1,1 Interested in invariant distribution πN to assess Quality of Service. πN is hard to characterize.

Mohammadreza Aghajani Asymptotic Coupling with Application in Queuing Systems

Outline Ergodicity Theorems for Markov Chains: Classical Results Markov Chains in Infinite-Dimensions: Asymptotic Coupling Application: Many-Server Queuing Systems

An Approximation Scheme

• I. Process Level

Convergence?

• III. Limit

Interchange?

• II. Limiting Process Characteristics?

π π

N

π: invariant distribution of the limit process Y Hope: πN ⇒ π A crucial question: Uniqueness of π

Mohammadreza Aghajani Asymptotic Coupling with Application in Queuing Systems

Outline Ergodicity Theorems for Markov Chains: Classical Results Markov Chains in Infinite-Dimensions: Asymptotic Coupling Application: Many-Server Queuing Systems

Asymptotic Coupling for Y

Theorem (Aghajani) Y has a unique stationary distribution. An asymptotic coupling scheme: X(t) = X(0) + √ 2B(t) − βt − t h, νs ds Define ˜ X(t) = ˜ X(0) + √ 2˜ B(t) − βt − t h, ˜ νs ds where ˜ Bt = Bt + t

0 ζ(s)ds. Choose ζ such that

∆X = X − ˜ X has a simpler from Girsanov Theorem holds

Mohammadreza Aghajani Asymptotic Coupling with Application in Queuing Systems