Let $\{M(t):t\geqslant 0\}$ be a continuous-time Markov chain on $\{0,1\}$ with generator $$Q=\pmatrix{-\alpha&\alpha\\\beta&-\beta}.$$ Let $\lambda_0,\lambda_1>0$ and define $$ \lambda(t) = \lambda_{M(t)},\ t\geqslant 0. $$ Then the Cox process $\{N(t):t\geqslant 0\}$ with intensity function $\lambda(t)$ is a Markov modulated Poisson process. Suppose customers arrive to a single-server queue according to $N(t)$ with the service times i.i.d. $\mathrm{Exp}(\mu)$ (assume that $\lambda_0,\lambda_1<\mu$ so the process is not explosive). Let $\{T_n:n\geqslant 0\}$ be the arrival times, $Z(t)$ the number of customers in the system at time $t$ (with $Z(0)=0$ a.s.), and define $$ X_n = \lim_{t\to T_n^-}Z(t). $$ That is, $X_n$ is the number of customers in the system just before the $n^{\mathrm{th}}$ arrival. Then $\{X_n:n=0,1,\ldots\}$ is a Markov chain - indeed, this is a standard way to analyze the $G/M/1$ queue.
Given this particular type of arrival process, can we analyze the limiting behavior of the system (e.g. long-run average number of customers in system, sojourn times) in a simpler or more intuitive way than the general analysis of a $G/M/1$ queue? I'm also interested in the transient behavior, if it isn't particularly intractable.