Consider a $M/M/c/c$ queue, e.g. arrivals to a shop form a Poisson process rate $\lambda$, each customer spends an iid exponential time mean $1/\mu$ in the shop. The shop has maximum capacity of $c$, so any person arriving when the shop is full doesn't enter and leaves. Show that this process has the PASTA property (Poisson arrivals see time averages): i.e. long run proportion of arriving people seeing the shop with $n$ customers is equal to the equilibrium probability of $n$.
We can model this as a CTMC with transitions $q(i-1,i)=\lambda$ and $q(i,i-1)=i\mu$ for $1\leq i\leq c$. Then it's fairly straightforward to show that the equilibrium distribution $\pi_i$ is given by $$\pi_i=\frac{\nu^i}{i!}\pi_0,$$ where $\nu=\lambda/\mu$ and $$\pi_0=\left(\sum_{k=0}^c\frac{\nu^k}{k!}\right)^{-1}.$$ So if we let $Y_n$ be the number of customers in the shop just before the arrival of the $n$-th customer in the arrival process (including those that leave immediately because the shop is full), would it be enough to show that $Y_n$ is a Markov chain with equilibrium $\pi$? But I'm not sure how to do this.