= Solution
The stochastic quasi-steady-state simulation evolves only $X$:
1. At the current integer $x\geq1$, compute $q(x)=\alpha_3V^2/(\alpha_4x)$ and the averaged rates $\lambda_1(x)=\alpha_1q(x)^2/V$ and $\lambda_2(x)=\alpha_2x(x-1)/V$.
2. Draw a waiting time $\Delta t\sim\operatorname{Exp}(\lambda_1+\lambda_2)$.
3. Set $x\leftarrow x+1$ with probability $\lambda_1/(\lambda_1+\lambda_2)$; otherwise set $x\leftarrow x-1$.
4. Advance time by $\Delta t$ and repeat.
This is a <Gillespie algorithm> for the averaged slow master equation. It samples the fast conditional equilibrium analytically through its factorial moment and never simulates individual fast $Y$ births or deaths.
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