2008 Operation Research 2
Assignment 8
Due day:
2008/04/28
Problems:
1) Consider he Markov chain that has the following (one-step)
transition matrix.
0 1 2 3 4
0 ┌ 0 4/5 0 1/5 0 ┐
1 │1/4 0 1/2 1/4 0 │
P= 2 │ 0 1/2 0 1/10 2/5│
3 │ 0 0 0 1 0 │
4 └1/3 0 1/3 1/3 0 ┘
(a) Determine the classes of this Markov chain and, for each
class, determine whether it is recurrent or transient.
(b) For each of the classes identified in part (a), determine
the period of the states in that class.
2) A transition matrix P is said to be doubly stochastic if
the sum over each column equals 1; that is
sum(i=0 to i=M)Pij=1 for all j
If such a chain is irreducible, aperiodic, and consists of M+1
states, show that
πj=1/(M+1) for j=0,1,...,M
(n)
(postscript: Lim{p }=πj)
n->inf ij
注意: 請用A4紙張作答，不用A4作答不予計分,並在作業的最上方標明
「學號」及「姓名」。
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