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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.

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Presentation on theme: "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."— Presentation transcript:

1 TM 732 Markov Chains

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

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

4 Recursion fPXjXi P () {|} 1 10  

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

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

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

8 Recursion fPXjXi P () {|} 1 10  

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

10 Expected First Passage Let  ij ectedfirstpassagetime fromstateitostatej  exp

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

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

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

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

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

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

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

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

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

20 Example; Inventory  ...  ..  .  .   (.)    ..wks

21 Example; Inventory   (1.58) 0368  ..  ...  ..  10  1.58

22 Example; Inventory   (1.58) 0368  ..  ...  ..  10  1.58  (.).(.) 

23 Example; Inventory   (1.58) 0368  ..  ...  ..  10  1.58   (.).(.).(.) (.).      wks

24 Example; Inventory  ...    10  1.58   (1.58) 0368 (2.51)0368  ...

25 Example; Inventory  ...    10  1.58   (1.58) 0368 (2.51)0368  ...     .(.).(.)..wks

26 Example; Inventory      10  weeks to stock out

27 Example; Inventory Find:  00

28 Example; Inventory Find:  00     PPP.(.).(.).(.).

29 Example; Inventory Find:  00     PPP.(.).(.).(.).   ..


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