beamer - Vrije Universiteit Amsterdam

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Existence and Uniqueness of Equilibrium Distribution . Theorem 3.3.2 & Theorem 3.5.9 . Assume the unichain condition with a finite transient set and a positive recurrent set (cf. Assumption 3.3.1), then the probabilistic long-run averages . (πj) (see slide 15) form the unique equilibrium distribution. Proof: for finite recurrent sets: ▶ Existence follows from slides 22 and 18; ▶ Uniqueness: suppose that (xj)j satisfies xj = ∑ i∈I xipij. See page 128 (book) for concluding that xj = cπj; i.e., π is the only distribution. Infinite recurrent sets: more involved. c⃝ Ad Ridder (VU) SOR– Fall 2012 24 / 36
Limiting Probabilities . Equation (3.5.11) . A. For transient states j and any initial state i ∈ I . lim n→∞ p(n) ij = 0 B. For recurrent aperiodic states j and any initial state i ∈ I lim n→∞ p(n) ij = fijπj Proof: see lecture 3 for A. Part B is more advanced; outside scope of this course. c⃝ Ad Ridder (VU) SOR– Fall 2012 25 / 36
Page 1 and 2: Stochastic Operations Research Lect
Page 3 and 4: Example ⎛ ⎜ P = ⎜ ⎝ 0.6 0.4
Page 5 and 6: Recap ▶ State j transient ⇔ fjj
Page 7 and 8: Proper First-Passage Times . Coroll
Page 9 and 10: Compute Powers of P ⎛ P 129 ⎜ =
Page 11 and 12: Observations We might conclude 1. l
Page 13 and 14: Mean Return Times and Probabilities
Page 15 and 16: Probabilistic Averages II . Theorem
Page 17 and 18: Probabilistic Averages IV . Corolla
Page 19 and 20: Infinite Recurrent Sets In the situ
Page 21 and 22: Infinite Transient or Nonrecurrent
Page 23: Equilibrium Distribution . Definiti
Page 27 and 28: Summary Suppose Assumption 3.3.1 (e
Page 29 and 30: Finite Unichain Case . Ergodic Theo
Page 31 and 32: Expected Reward ▶ See Remark 3.3.
Page 33 and 34: Rewrite ▶ Rewrite to classic form
Page 35 and 36: Gauss-Seidel Method ▶ Construct a

beamer - Vrije Universiteit Amsterdam

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