Download presentation

Presentation is loading. Please wait.

Published byJaylin Ludgate Modified over 2 years ago

1
1 Introduction to Discrete-Time Markov Chain

2
2 Motivation many dependent systems, e.g., inventory across periods state of a machine customers unserved in a distribution system time excellent good fair bad

3
3 Motivation any nice limiting results for dependent X n ’s? no such result for general dependent X n ’s nice results when X n ’s form a discrete-time Markov Chain

4
4 Discrete-Time, Discrete-State Stochastic Process a stochastic process: a sequence of indexed random variables, e.g., {X n }, {X(t)} a discrete-time stochastic process: {X n } a discrete-state stochastic process, e.g., state {excellent, good, fair, bad} set of states {e, g, f, b} {1, 2, 3, 4} {0, 1, 2, 3} state to describe weather {windy, rainy, cloudy, sunny}

5
5 Markov Property a discrete-time, discrete-state stochastic process possesses the Markov property if P{X n+1 = j|X n = i, X n−1 = i n−1,..., X 1 = i 1, X 0 = i 0 } = p ij, for all i 0, i 1, …, i n 1, i n, i, j, n 0 time frame: presence n, future n+1, past {i 0, i 1, …, i n 1 } meaning of the statement: given presence, the past and the future are conditionally independent the past and the future are certainly dependent

6
6 One-Step Transition Probability Matrix p ij 0, i, j 0,

7
7 Example 4-1 Forecasting the Weather state {rain, not rain} dynamics of the system rains today rains tomorrow w.p. does not rain today rains tomorrow w.p. weather of the system across the days, {X n }

8
8 Example 4-3 The Mood of a Person mood {cheerful (C), so-so (S), or glum (G)} cheerful today C, S, or G tomorrow w.p. 0.5, 0.4, 0.1 so-so today C, S, or G tomorrow w.p. 0.3, 0.4, 0.3 glum today C, S, or G tomorrow w.p. 0.2, 0.3, 0.5 X n : mood on the nth day, such that mood {C, S, G} {X n }: a 3-state Markov chain (state 0 = C, state 1 = S, state 2 = G)

9
9 Example 4.5 A Random Walk Model a discrete-time Markov chain of number of states {…, -2, -1, 0, 1, 2, …} random walk: for 0 < p < 1, p i,i+1 = p = 1 − p i,i−1, i = 0, 1,...

10
10 Example 4.6 A Gambling Model each play of a game a gambler gaining $1 w.p. p, and losing $1 o.w. end of the game: a gambler either broken or accumulating $N transition probabilities: p i,i+1 = p = 1 − p i,i−1, i = 1, 2,..., N − 1; p 00 = p NN = 1 example for N = 4 state: X n, the gambler’s fortune after the n play {0, 1, 2, 3, 4}

11
11 Limiting Behavior of Irreducible Chains

12
12 Limiting Behavior of a Positive Irreducible Chain cost of a visit state 1 = $5 state 2 = $8 what is the long-run cost of the above DTMC? 1 2 0.8 0.1 0.9 0.2

13
13 Limiting Behavior of a Positive Irreducible Chain j = fraction of time at state j N: a very large positive integer # of periods at state j j N balance of flow j N i ( i N)p ij j = i i p ij

14
14 Limiting Behavior of a Positive Irreducible Chain j = fraction of time at state j j = i i p ij 1 = 0.9 1 + 0.2 2 2 = 0.1 1 + 0.8 2 linearly dependent normalization equation: 1 + 2 = 1 solving: 1 = 2/3, 2 = 1/3 1 2 0.8 0.1 0.9 C 0.2

15
15 Limiting Behavior of a Positive Irreducible Chain 1 = 0.75 2 + 0.01 3 3 = 0.25 2 1 + 2 + 3 = 1 1 = 301/801, 2 = 400/801, 3 = 100/801 1 2 3 0.25 0.99 1 0.75 0.01

16
16 Limiting Behavior of a Positive Irreducible Chain an irreducible DTMC {X n } is positive there exists a unique nonnegative solution to j : stationary (steady-state) distribution of {X n }

17
17 Limiting Behavior of a Positive Irreducible Chain j = fraction of time at state j j = fraction of expected time at state j average cost c j for each visit at state j random i.i.d. C j for each visit at state j for aperiodic chain:

18
18 Limiting Behavior of a Positive Irreducible Chain 1 = 301/801, 2 = 400/801, 3 = 100/801 profit per state: c 1 = 4, c 2 = 8, c 3 = -2 average profit 1 2 3 0.25 0.99 1 0.75 0.01

19
19 Limiting Behavior of a Positive Irreducible Chain 1 = 301/801, 2 = 400/801, 3 = 100/801 C 1 ~ unif[0, 8], C 2 ~ Geo(1/8), C 3 = -4 w.p. 0.5; and = 0 w.p. 0.5 E(C 1 ) = 4, E(C 2 ) = 8, E(C 3 ) = -2 average profit 1 2 3 0.25 0.99 1 0.75 0.01

Similar presentations

Presentation is loading. Please wait....

OK

IERG5300 Tutorial 1 Discrete-time Markov Chain

IERG5300 Tutorial 1 Discrete-time Markov Chain

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on ideal gas law worksheet Ppt on biodegradable and non biodegradable materials list Ppt on palm island in dubai Ppt on rational numbers for class 7 Ppt on monopolistic business model Ppt on db2 mainframes ibm Ppt on forward rate agreement risk Ppt on dda line drawing algorithm in computer Ppt on chapter management of natural resources Ppt on programmable logic array programmer