Lecture9

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Unformatted text preview: Solution Gambler Problem Gambler Problem Two piayers have $3 each In each game, each player bets $1 and the winner takes It all Player A Is better than 6 and has a 60% chance of winning The match ends when either player has won all the money (the other is broke) Develop a transition matrix and classify ail states and classes l l l l Define the epoch as one game Detlne the states of the system as the amount of money that Player A has Valid states are 0 (player A is broke) through 6 (Player A has all the money) The transition probabilities will be based on the winning percentage of A EM-602 I QM-710 (NJ) Lecture 9 Page 9-7 Gambler Problem Gambler Problem Transition matrix (i) Transition matrix (2) 0 1 2 3 4 5 6 8 1 2 3 4 5 6 1 0 0 0 0 0 0 .4 0 .6 0 0 0 0 Gambler Problem Gambler Problem Transition matrix (3) Transition matrix (comptete) 1 0 0 .4 2 0 .6 0 3 0 0 .6 4 0 0 0 5 0 0 0 0 1 2 0 1 0 0 1 .4 0 .6 2 0 .J 0 6 0 0 0 Gambler Problem 4 0 0 0 5 0 0 0 6 0 0 0 0 .4 0 0 .6 0 0 .6 .4 0 0 0 0 0 .6 1 0 0 0 0 0 0 0 0 .4 0 0 0 Mean Recurrence Time Transitton matrix (complete) 0 1 2 3 4 5 6 3 0 0 .6 3 4 5 6 0 1 .4 0 Classiftcation States 0 and 6 are Recurrent 6 Absorbing l l States 1 thru 5 are Transient Definition - The expected value of the number of epochs between successive visits to a state j A relationship exists between steady state vector xi and mean recurrence time !kij accordtng to: p, =+ J EM-602 I QM-710 (NJ) Lecture 9 Page 9-8 Problem 18-l 7 Problem 18-l 7 (page 7121 Find the mean recurrence times for each state From the relationship: RP = II Problem 18-l 7 Problem 18-17 Solution Fern the relationship: Rp = R 1 1 1 -x,+--n*+--x,=x, Eliminate one 3 2 5 1 ~~,+fl~+$c,=~~~ RedundantEqn Solving Simultaneously.. . = r27 28 $36 3 7 271 75 74 * . - =175 pj = Mean Recurrence Time = L R/ r75 75 751 Jo =~jy u, ~~=[278 2 . 6 8 3.751 1 1 1 -X,f--Xz+-X,=X, . . . and add the 3 -I 5 Normalizing Equation: x,+x,+x,=1 Problem 18-17 Interpretation l l l From the steady-state vector n, and the mean recurrence time &L, we conclude: System will (In the long run) spend 36% of the time in state 1, 37% in state 2, and 27% in state 3 The mean time between re-visits is 2.78 epochs for state 1, 2.68 epochs for state 2, and for 3.75 epochs for state 3 EM-602 I QM-710 (NJ) Lecture 9 Page 9-9...
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This document was uploaded on 03/31/2014 for the course MS 602 at NJIT.

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