= Heterogeneous two-server queue
{title2=$\mu=\mu_1+\mu_2$}
A queue with a <Poisson process> of arrivals and two independent <service times> with <exponential distributions> of different rates must distinguish the two possible single-customer states. If an arrival chooses either idle server with probability $1/2$, the <continuous-time Markov chain> is <reversible>. Write $a,b$ for the states with only server one or two busy, and $n\geq2$ for total population. Its <detailed balance equations> give $\pi_a=\pi_0\nu/(2\mu_1)$, $\pi_b=\pi_0\nu/(2\mu_2)$ and $\pi_n=\pi_0\nu^2(\nu/\mu)^{n-2}/(2\mu_1\mu_2)$. These weights normalize exactly when $\nu<\mu$. In equilibrium, reversed upward transitions have total rate $\nu$ in every state, so <time reversal of a continuous-time Markov chain> makes the departure stream a <Poisson process> of rate $\nu$. This output conclusion requires stationarity.
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