3 state; 1,2,3

v1=2, v2=3,v3=5

embedded markov chain has transition matrix P

0 ; 1/4 ; 3/4

2/5 ; 0 ; 3/5

1/3 ; 2/3 ; 0

a. determine the long run distribution of this chain

b. determine the mean first passage time from state 2 to state 3

my solution

v1P1=q21P2+q31P3

v2P2=q12P1+q32P3

v3p3=q13P1+q23P2

P1+P2+P3=1

with q21=v2P21=3.2/5=6/5

similarly q31=v3P31=5/3

q12=v1P12=.5

q32=vpP32=10/3

q13=v1P13=1.5

q23=v2P23=9/5

I solve P1=30/73, P2=25/73, P3=18/73

b. mean time from state 2 to 3

m23=1+P21m13+P22m23

=1+2/5 m13

m13=1+P12m23+P11m13

= 1+1/4 m23

I solve m23=14/9

I would like to know whether or not my solution is corrrect. if it's not correct, hope someoone give me some comments to correct it

thank you