Lecture9

23 vij pij 0 1 i fl i absorbing class transient class

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Unformatted text preview: tates (0.1, 2) Absorbing State (3) Markov Chains Classifications Classifications (cont’d) Periodic Markov Chain Transient Class l States may only be visited periodically 23) 6‘ 1 0 1 1 2 1 Transient States (0, 1, 2) (period=3 epochs) Recurrent Sxes (3,4) Classifications l Definition Aperiodic Markov Chain Ergodk Markov Chain All states may be visited at any epoch A markov chain is called Ergodic if l lt is Irreducible - its states are all in one class l All states are Aperiodic l All states are Recurrent Atter a (large) number of transitions, an Ergodic markov chain will eventually reach a steady state 35-1 ’ CT? 2 1 .- EM-602 / QM-710 (NJ) Lecture 9 Page 9-6 Steady State l l l Steady State Long-Run Probability Long-Run Probability We can predict the long run behavior of a system described as an Ergodic markov chain As more transitions occur, the system becomes less dependent on initial conditions State of system will eventually be a function only of the transition probabilities Given an ergodk markov chain pfl) pcnj (where n is very large) becomes a matrix with identkai rows A stationary distribution. long-run, Or steady-state vector can be found x - A(“), n+ao These nth step (n+oo) absolute probabilities are unaffected by the initial state vector Ato) l l l Steady-State Vector Steady State Vector Independent of I.C. (proof) Computations A’“’ = Calculating high powers of PC”) can be computa~ionally intensive - instead . . . Consider a system at equilibrium in x = A(“), n-r-: which The steady state vector is given by x The one-step transition matrix is P One more transition at equilibrium results in vector A@*lb A@)P n+c This is equivalent to x = x P /pp(“) 1.4286 57141 “” = [I ‘1,J286 l 5714 1 l A”’ = [ A286 /I”’ = [0 l] A”’ = [ .S 5714 l A”’ = [A286 5714 l .5] A”’ = [ A286 .5714] l .. Example l l l l l...
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This document was uploaded on 03/31/2014 for the course MS 602 at NJIT.

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