## Markov Chains Comparison

I have two very simple continuous-time Markov chains $M^a$ and $M^b$ defined on states $(i,j)\in\{(0,0),(1,0),(0,1)\}$ by the following transition rates: For $M^a$

$
p_{10,01}=\mu, ~~
p_{01,10}=\mu (1-z),~~
p_{01,00}=\mu z,
$

and for $M^b$

$
p_{10,01}=\mu z, ~~
p_{01,00}=\mu z
$

where 0<z<1 and the omitted transition rates are zero.

Assume that both chains start in state (0,1). Now, consider the random variable $X^a(t)=X_1^a(t)+X_2^a(t)$, i.e., the sum of the state indices i and j over time, for $M^a$, and $X^b(t)=X_1^b(t)+X_2^b(t)$ analogously.

Is it true that $X^a(t)$ has the same distribution of $X^b(t)$ for all t?

Is it true that $X^a(t)\le_{st}X^b(t)$? (or $X^a(t)\ge_{st}X^b(t)$?)

Why?

Thanks