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TM 732 Markov Chains

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First Passage Time Consider (s,S) inventory system: first passage from 3 to 1 = 2 weeks recurrence time (3 to 3) = 5 weeks

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First Passage Time Consider (s,S) inventory system: first passage from 3 to 1 = 2 weeks Let fPfirstpassagetimeitojn ij n () {}

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Recursion fPXjXi P () {|} 1 10

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Recursion fPXjXi P () {|} 1 10 fPXjXi () {|} 2 20

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Recursion fPXjXi P () {|} 1 10 fPXjXi () {|} 2 20 PXjXkPXkXi kj {|}{|} 2110

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Recursion fPXjXi P () {|} 1 10 fPXjXi PXjXkPXkXi PXjXkPXkXi PXjXjPXjXi kj k () {|} {|}{|} {|}{|} {|}{|}

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Recursion fPXjXi P () {|} 1 10

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Recursion f P ()1 f ()2 PfP jj ()()21 fPfPfP fP ij n n jj n ijjj n ij n jj ()()()()()() ()()...

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Expected First Passage Let ij ectedfirstpassagetime fromstateitostatej exp

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Expected First Passage Proposition: ijikkj kj P 1 ij n n jistransient nfjisrecurrent ,, () 1 {

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Example; Inventory ijikkj kj P 1

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Example; Inventory PPP ijikkj kj P 1

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Example; Inventory PPP PPP ijikkj kj P 1

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Example; Inventory PPP PPP PPP ijikkj kj P 1

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Example; Inventory ijikkj kj P 1 P

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Example; Inventory ... ijikkj kj P 1 P

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Example; Inventory ... ... ijikkj kj P 1 P

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Example; Inventory ... ... ... ijikkj kj P 1 P

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Example; Inventory ... .. . . (.) ..wks

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Example; Inventory (1.58) 0368 .. ... .. 10 1.58

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Example; Inventory (1.58) 0368 .. ... .. 10 1.58 (.).(.)

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Example; Inventory (1.58) 0368 .. ... .. 10 1.58 (.).(.).(.) (.). wks

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Example; Inventory ... 10 1.58 (1.58) 0368 (2.51)0368 ...

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Example; Inventory ... 10 1.58 (1.58) 0368 (2.51)0368 ... .(.).(.)..wks

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Example; Inventory 10 weeks to stock out

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Example; Inventory Find: 00

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Example; Inventory Find: 00 PPP.(.).(.).(.).

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Example; Inventory Find: 00 PPP.(.).(.).(.). ..

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