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

<br />
p_{10,01}=\mu, ~~<br />
p_{01,10}=\mu (1-z),~~<br />
p_{01,00}=\mu z,<br />

and for M^b

<br />
p_{10,01}=\mu z, ~~<br />
p_{01,00}=\mu z<br />

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