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Lecture10

# Period 2 ww em 602 i qm 710 nj lecture 10 page 1 o 2

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Unformatted text preview: .. EM-602 I QM-710 (NJ) Lecture 10 Page 1 O-2 First visit was in period 1 -- First Recurrence Time First Passage Time Expected ttrst Passage time is defined as the expected value of the number of time periods required to: l Start at state i, and end at state j in the nlh period, and l The visit in the nth period to state j was the ttrst visit In that case, the expected ttrst passage time will be given the symbol &ii(“) l l p’ii < c*) if the state is Transient First Passage Time Graphical Analogy Solution State 2 can be reached from all other states First passage probability is computed according to: k=J if the state is Recurrent CL1 =m First Passage Time l A special case of first passage time is Expccted First Return or Expected Recurrence Tlme p\$” For a given state: . For example, if there are 4 states kz = 1 f PA p42 + PI, CL,2 + PI, k2 .. Example (solution) Numerical Example First Passage Probability First Passage Probabiltty l Lkvclop thr equation from me f;s’ = p;J’_ f~"p\$l _ f;"PA" A stochastic system (with states 1,2, S 3) behaves according to the one-step transition matrtx shown below: I.2 .3 .q In order to solve, first compurc: JO] A3 .20 i 30 ‘=G :: :I Find the probability of first passage from state 1 to state 2 in the third epoch A9 P’?) = .32 l /.,I A5 .25 I EM-602 I QM-710 (NJ) Lecture IO Page IO-3 \$?cncml romda Example (solution) Numerical Example First Passage Probability Expected First Passage Time l A stochastic system (with states 1, 2, 6 3) behaves according to the one-step transition matrix shown below: f:,” = (47) - (.2)(.48) - f,:“(.6) r.2 .3 ?'=/A 51 f,‘;’ = p:,’ - f,‘;‘P;;’ = (49) - (.3X.6) l Substituting and solving: 0; L.5 .5 In order to solve, we need to find /I:” .6 OJ Find the expected first passage time from state 1 to state 2 /,:” = 0.185 Example (solution) Homework Problem #l Expected First Passage Time Lecture 10 Use the transition matrix from Problem 18-17 to find l (a) the probability of first passage from state 1 to 3 in the third time period l (b) the expected first passage time from state 1 to all other states k-1 Pll = 1 + 4, PI2 + P,, P32 pu P’n = 1 + Pa + P,, PI2 After substituting, combine and solve ~,*-o.2~,z = 1...
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