I am considering a Markov chain $\lbrace X(t) \rbrace_{t≥0}$ in continunous time on the countable state space $S=\lbrace 0 \rbrace\cup \lbrace (i,j) \mid i \in \mathcal{A} , j \in \mathbb{N} \rbrace, \mathcal{A}= \lbrace A,B,C,D,E,F \rbrace$ . The transistion intensities are gived by
\begin{align} q_{(i,j),(i,j+1)} & =j\beta_i,& j\in \mathbb{N},\ i\in\mathcal{A}\\ q_{(i,j),(i,j-1)} & =j\delta_i,& j\geq 2,\ i \in\mathcal{A}\\ q_{(i,1),0} & =\delta_i,& i\in\mathcal{A}\\ q_{0,(i,1)} & =1,& i\in\mathcal{A} \end{align} Now I have to construct a pure birth process where the time of the n-th jump happens earlier than for $\lbrace X(t) \rbrace_{t≥0}$. I must have that $q_{(i,j),(i,j-1)}=j\delta_i=0$. But how do I chose $q_{(i,j),(i,j+1)}$ such that the jump happens earlier? Is it correct if I put $q_{(i,j),(i,j+1)}=2j\beta_i$?