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 Mohammadreza Aghajani reza@brown.edu * Born Mashhad, Iran 1986 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 College Sharif Univ. of Tech Tehran, Iran. 2003 * * Born Mashhad, Iran 1986 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 College Sharif Univ. of Tech Tehran, Iran. 2003 * * * Born Mashhad, Iran 1986 M.S in Electrical Eng. Instituto Superior Tecnico Lisbon, Portugal 2008 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 College Sharif Univ. of Tech Tehran, Iran. 2003 * * * * M.S. in Electrical Eng Born Carnegie-Mellon Univ. Mashhad, Iran Pittsburgh, PA, USA. 1986 2009 M.S in Electrical Eng. Instituto Superior Tecnico Lisbon, Portugal 2008 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 College PhD in Applied Math Sharif Univ. of Tech Brown University Providence, RI, USA Tehran, Iran. 2003 2010 * * * * * M.S. in Electrical Eng Born Carnegie-Mellon Univ. Mashhad, Iran Pittsburgh, PA, USA. 1986 2009 M.S in Electrical Eng. Instituto Superior Tecnico Lisbon, Portugal 2008 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 We have Initial distribution λ X ∼ P λ on E ∞ . X ( n ) ∼ λ P n . Transition Kernel P ( x , · ) Notions of Stability Invariant Distribution: π = π P . Ergodicity � λ P N − 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 d ˜ ˜ = X , Y = Y . Coupling Inequality �L{ X } − L{ Y }� ≤ 2 P ( ˜ 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 x � ˜ if n ≤ T Y n . X n y = X n if n > T ~ Y ∼ ˜ x X T By Coupling Inequality: λ ( ˜ � P λ ( X n ∈ · ) − P ˜ X n ∈ · ) � ≤ 2 P λ ˜ 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) P x ( X n ∈ A ; for some n ) = 1 , ∀ x ∈ E (recurrence) (ii) P x ( X n 0 ∈ B ) ≥ βϕ ( B ) , ∀ x ∈ A , ∀ B ∈ E (small set) x 0 X A X S 1 ~ ϕ X S 2 ~ ϕ S S 2 1 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 { S j } X is initialized at π → Corresponding { ˜ ˜ S j } A ‘successful’ coupling: Coupling time T = S n = ˜ S m Renewal Theory ⇒ T is almost surely finite. Coupling inequality gives ergodicity � P λ ( X n ∈ · ) − π � ≤ P λ ˜ 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 )) dW t X (s) = X(t+s) t t-r t { X t ; t ≥ 0 } is a Markov Process on C ([ − r , 0]) Invariant Distribution Exists for large c . Given the solution X t for any t > 0, X 0 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 Γ 1 ∼ P λ and Γ 2 ∼ P µ . 2 Γ ( { ( x , y ) ∈ E ∞ × E ∞ ; lim n →∞ d ( x n , y n ) = 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 Γ x , y of δ x and δ y , 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
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