Finding stationary distribution on markov chain

Question

We have the state space {0,1,2,3,4,5,...,}

we got the following transition probabilities $p_{00}=q+r$

$p_{01}=p$

$p_{i,i-1}=q$

$p_{i,i}=r$

$p_{i,i+1}=p$

for $i>=1$

where we have $p+q+r=1$

FInd the stationary distribuion where $p=.2,q=.4,r=.4$ Is the stationary distribution unique

For this one I am kind of stuck I tried using flow balance equations and got in=out and the following

$t_0=(.2t_1+.8t_0)$ and I get $t_0=t_1$ then I did

$t_1=.2t_0+.4t_1+.4t_2$ and using $t_0=t_1$ I get $t_2=t_1$ and $t_2=t_0$

then for state 2 I did

$t_2=.2t_1+.4*t_2+.4*t_3$ and I get $t_2=t_3$ and so $t_3=t_0$

So then I keep getting $t_i=t_0$ but I do not think this is right because

$\sum t_i=\sum t_0$ is supposed to add up to 1.

I am kind of stuck.

Answer 1

Your first balance should be $t_0 = 0.4t_1 +0.4t_0$ so we get $t_0 = \frac{2}{3}t_1$, your other flow balances are correct.