= Blocking probability in a finite-line sequential-service network
{title2=$P_{\mathrm{loss}}=H_c(N)/\sum_{n=0}^NH_c(n)$}
Represent $N$ telephone lines by a <closed migration process> with free-line, operator and automated-service colonies. Accepted arrivals have rate $\nu$ when a free line exists, $c$ operators complete at rate $\lambda\min(k,c)$, and $i$ automated calls complete at rate $\mu i$. Define $D_c(k)=\prod_{h=1}^k\min(h,c)$ and
$$
H_c(n)=\sum_{i=0}^n\frac{(\nu/\lambda)^{n-i}(\nu/\mu)^i}{D_c(n-i)i!}.
$$
The <product-form stationary distribution of a closed migration process> and <Poisson arrivals see time averages> give the displayed blocking probability. This extends a single-operator model without treating the independent operators as a server with rate $c\lambda$ when fewer than $c$ calls are present.
// Target: probability-and-statistics.bigb
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