= Multiclass single-server queue
An ordered state $x=(c_1,\ldots,c_n)$ records the classes and positions of customers at a single server. Class-$r$ arrivals have rate $\alpha_r$ and exponential service parameter $\mu_r$. Under a <symmetric service discipline>, its <stationary distribution> is $(1-\rho)\prod_i(\alpha_{c_i}/\mu_{c_i})$, where $\rho=\sum_r\alpha_r/\mu_r<1$. If every $\mu_r=\mu$, class-independent insertion and service position rules need not be equal: <time reversal of a continuous-time Markov chain> interchanges them and proves the same law. Aggregating words gives $(1-\rho)n!\prod_r[(\alpha_r/\mu_r)^{n_r}/n_r!]$ by the <multinomial coefficient>. With class-dependent service rates, <first come first served> need not have this product law.
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