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