beamer - Vrije Universiteit Amsterdam

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Topics 1. §3.3 Equilibrium Probabilities 2. §3.3.3 Markov Reward Theorem 3. §3.4.1 Computation by an Iterative Method 4. §3.5 Theoretical Considerations c⃝ Ad Ridder (VU) SOR– Fall 2012 2 / 36
Example ⎛ ⎜ P = ⎜ ⎝ 0.6 0.4 0 0 0 0 0 0 0 0 0 0.3 0.2 0 0.5 0 0 0 0 0.1 0.1 0.4 0 0.2 0.2 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0.5 0.5 0 0 0 0 0 0.3 0.7 0 0 Transient states T = {1, 2, 3}; irreducible recurrent set R1 = {4, 5} with period 2; irreducible aperiodic recurrent set R2 = {6, 7, 8, 9} (see picture next page). c⃝ Ad Ridder (VU) SOR– Fall 2012 3 / 36 ⎞ ⎟ ⎠
Page 1: Stochastic Operations Research Lect
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 and 24: Equilibrium Distribution . Definiti
Page 25 and 26: Limiting Probabilities . Equation (
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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