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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 () {|} {|}{|} {|}{|} {|}{|} 2 20 2110 2110 2110
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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 ()()()()()() ()()... 1122 11
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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 30311032203330 1 PPP ijikkj kj P 1
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Example; Inventory 30311032203330 1 PPP 20211022202330 1 PPP ijikkj kj P 1
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Example; Inventory 30311032203330 1 PPP 20211022202330 1 PPP 10111012201330 1 PPP ijikkj kj P 1
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Example; Inventory ijikkj kj P 1 P 0080018403680 063203680000 026403680 00 0080018403680................
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Example; Inventory 30102030 1018403680 ... ijikkj kj P 1 P 0080018403680 063203680000 026403680 00 0080018403680................
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Example; Inventory 30102030 1018403680 ... 20102030 103680 00 ... ijikkj kj P 1 P 0080018403680 063203680000 026403680 00 0080018403680................
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Example; Inventory 30102030 1018403680 ... 20102030 103680 00 ... 10 2030 103680000 ... ijikkj kj P 1 P 0080018403680 063203680000 026403680 00 0080018403680................
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Example; Inventory 30102030 1018403680 ... 201020 103680 .. 10 10368 . 10 10368 . 10 103681 (.) 10 1 10368 158 ..wks
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Example; Inventory 20 10368(1.58) 0368 .. 30102030 1018403680 ... 201020 103680 .. 10 1.58
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Example; Inventory 20 10368(1.58) 0368 .. 30102030 1018403680 ... 201020 103680 .. 10 1.58 20 1036810 158(.).(.)
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Example; Inventory 20 10368(1.58) 0368 .. 30102030 1018403680 ... 201020 103680 .. 10 1.58 20 1036810 158 10368158 10368 251 (.).(.).(.) (.). wks
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Example; Inventory 30102030 1018403680 ... 20 2.51 10 1.58 30 10184 (1.58) 0368 (2.51)0368 ...
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Example; Inventory 30102030 1018403680 ... 20 2.51 10 1.58 30 10184 (1.58) 0368 (2.51)0368 ... 30 101841580368251 10368 350 .(.).(.)..wks
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Example; Inventory 30 3.50 20 2.51 10 1.58 3.5 weeks to stock out
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Example; Inventory Find: 00
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Example; Inventory Find: 00 00011002200330 1 1018415803682510368350 3 PPP.(.).(.).(.).
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Example; Inventory Find: 00 00011002200330 1 1018415803682510368350 3 PPP.(.).(.).(.). 0 00 11 35 0286 ..
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