Note for Quantitative Techniques II - QT II by Jitendra Pal
- Quantitative Techniques II - QT II
- Note
- Biju Patnaik University of Technology BPUT - BPUT
- Computer Science Engineering
- B.Tech
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BIJU PATNAIK UNIVERSITY OF TECHNOLOGY, ODISHA Lecture Notes On QUANTITATIVE TECHNIQUE-II Prepared by, Dr. Subhendu Kumar Rath, BPUT, Odisha.
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QUANTITATIVE TECHNIQUE-II A stochastic process is a indexed collection of random variables {X } = t { X , X , X , … } for describing the behavior of a system operating 0 1 2 over some period of time. For example : X 0 = 3, X 1 = 2, X 2 = 1, X 3 = 0, X 4 = 3, X 5 = 1 An inventory example: A camera store stocks a particular model camera. D represents the demand for this camera during week t. t D has a Poisson distribution with a mean of 1. t X represents the number of cameras on hand at the end of week t. ( t X =3) 0 If there are no cameras in stock on Saturday night, the store orders three cameras. { X } is a stochastic process. X t t+1 = max{ 3 – D max{ X - D t t+1 t+1 ,0} if X = 0 t if X ≥ 0 ,0} t A stochastic process {X } is a Markov chain if it has Markovian t property. Markovian property: P{ X = j | X 0 = k 0 , X 1 = k 1 , ..., X t-1 = k t-1 , X t = i } t+1 = P{ X t+1 = j | X t = i } P{ X t+1 = j | X = i } is called the transition probability. t Stationary transition probability: If ,for each i and j, P{ X t+1 = j | X t = i } = P{ X 1 = j | X 0 = i }, for all t, then the transition probability are said to be stationary. Formulating the inventory example: Transition matrix: Prepared by Dr. Subhendu Kumar Rath, Dy. Registrar, BPUT
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X t+1 = max{ 3 – D max{ X - D p p p p 03 02 01 00 = P{ D = P{ D = P{ D = P{ D t t+1 t+1 t+1 t+1 t+1 t+1 ,0} ,0} if X = 0 t if X ≥ 1 = 0 } = 0.368 t = 1 } = 0.368 = 2 } = 0.184 ≥ 3 } = 0.080 The state transition diagram: n-step transition probability : p ij (n) = P{ X t+n = j | X t = i } n-step transition matrix : Prepared by Dr. Subhendu Kumar Rath, Dy. Registrar, BPUT
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Chapman-Kolmogorove Equation : M (n) ij p = ∑ pik( m ) pkj( n −m ) k =0 The special cases of m = 1 leads to : M (n) ij p = ∑ pik(1) pkj( n −1) k =0 Thus the n-step transition probability can be obtained from one-step transition probability recursively. Conclusion : (n) (n-1) (n-2) n = PPP = ... = P P = PP n-step transition matrix for the inventory example : Prepared by Dr. Subhendu Kumar Rath, Dy. Registrar, BPUT
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