# TM 732 Markov Chains. 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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TM 732 Markov Chains

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

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

Recursion fPXjXi P () {|} 1 10  

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

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

Recursion fPXjXi P () {|} 1 10   fPXjXi PXjXkPXkXi PXjXkPXkXi PXjXjPXjXi kj k () {|} {|}{|} {|}{|} {|}{|} 2 20 2110 2110 2110       

Recursion fPXjXi P () {|} 1 10  

Recursion f P ()1  f ()2  PfP jj ()()21  fPfPfP fP ij n n jj n ijjj n ij n jj ()()()()()() ()()...     1122 11

Expected First Passage Let  ij ectedfirstpassagetime fromstateitostatej  exp

Expected First Passage Proposition:  ijikkj kj P    1  ij n n jistransient nfjisrecurrent     ,, () 1 {

Example; Inventory  ijikkj kj P    1

Example; Inventory  30311032203330 1  PPP  ijikkj kj P    1

Example; Inventory  30311032203330 1  PPP  20211022202330 1  PPP  ijikkj kj P    1

Example; Inventory  30311032203330 1  PPP  20211022202330 1  PPP  10111012201330 1  PPP  ijikkj kj P    1

Example; Inventory  ijikkj kj P    1 P  0080018403680 063203680000 026403680 00 0080018403680................

Example; Inventory  30102030 1018403680 ...  ijikkj kj P    1 P  0080018403680 063203680000 026403680 00 0080018403680................

Example; Inventory  30102030 1018403680 ...  20102030 103680 00 ...  ijikkj kj P    1 P  0080018403680 063203680000 026403680 00 0080018403680................

Example; Inventory  30102030 1018403680 ...  20102030 103680 00 ...  10 2030 103680000 ...  ijikkj kj P    1 P  0080018403680 063203680000 026403680 00 0080018403680................

Example; Inventory  30102030 1018403680 ...  201020 103680 ..  10 10368 .  10 10368 .  10 103681  (.)  10 1 10368 158   ..wks

Example; Inventory   20 10368(1.58) 0368  ..  30102030 1018403680 ...  201020 103680 ..  10  1.58

Example; Inventory   20 10368(1.58) 0368  ..  30102030 1018403680 ...  201020 103680 ..  10  1.58  20 1036810 158(.).(.) 

Example; Inventory   20 10368(1.58) 0368  ..  30102030 1018403680 ...  201020 103680 ..  10  1.58   20 1036810 158 10368158 10368 251 (.).(.).(.) (.).      wks

Example; Inventory  30102030 1018403680 ...  20 2.51   10  1.58   30 10184 (1.58) 0368 (2.51)0368  ...

Example; Inventory  30102030 1018403680 ...  20 2.51   10  1.58   30 10184 (1.58) 0368 (2.51)0368  ...  30 101841580368251 10368 350    .(.).(.)..wks

Example; Inventory  30 3.50   20 2.51   10  1.58 3.5 weeks to stock out

Example; Inventory Find:  00

Example; Inventory Find:  00  00011002200330 1 1018415803682510368350 3    PPP.(.).(.).(.).

Example; Inventory Find:  00  00011002200330 1 1018415803682510368350 3    PPP.(.).(.).(.).   0 00 11 35 0286 ..

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