Let $\lambda_k,\mu_k\in\mathbb R_{\ge0}$ $(k\ge1)$ be nonnegative real numbers, let $S=\mathbb Z_{\ge0}$ be the nonnegative integers, let $T=\mathbb R_{\ge0}$ be the nonnegative real numbers and consider the continuous-time Markov chain $X=(X_t)_{t\in T}$ on $S$ with rates $$Q(n,n+k)=(n+1)\lambda_k\quad(k\ge1),\qquad Q(n,n-k)=(n+1-k)\mu_k\quad(1\le k\le n).$$ (This Markov chain appears in biology as a model of the length of an evolving DNA sequence (Miklós et. al. 2004). I have also examined some properties of this process in a Math.StackExchange post.)

For example, if $0=\lambda_k=u_k$ for all integers $k\ge2,$ then we recover the *linear birth-death process with immigration* with birth rate $\lambda_1,$ death rate $\mu_1$ and immigration rate $\lambda_1,$ whose nonzero rates are
$$Q(n,n+1)=(n+1)\lambda_1\quad(n\ge0,k\ge1),\qquad Q(n,n-1)=n\mu_1\quad(n\ge1).$$
Or, for example, given parameters $\mu\in\mathbb R_{>0},\gamma,r\in(0,1),$ we can let $\mu_k=\mu(1-r)^2r^{k-1}$ and $\lambda_k=\mu(1-r)^2\gamma^kr^{k-1}$ for all $k\ge1.$ Both these examples have been used in, and are of interest in, computational biology.

Now, **for which parameters $\lambda_k,\mu_k$ does such a Markov chain exist?** I have heard that the Hille-Yoshida theorem may be helpful and we need a ``dissipative" condition on the growth rate of the terms $Q(n,m).$ However, I don't know how to apply such a theorem here.

In addition, **for which parameters $\lambda_k,\mu_k$ does there exist such a Markov chain with all the regularity properties that are important for applications?** (E.g., conservative, standard (maybe?), maybe more....)

I have leafed through parts of Anderson's *Continuous-Time Markov Chains*, Karlin & Taylor's *Second Course* and Ethier & Kurtz's *Markov Processes*, but none of these books contains anything directly helpful.

Miklós, I., Lunter, G. A., & Holmes, I. (2004). A “long indel” model for evolutionary sequence alignment. Molecular Biology and Evolution, 21(3), 529-540.