Blocking probability in a finite-line sequential-service network ($P_{\mathrm{loss}}=H_c(N)/{\sum }_{n=0}^N{H}_c(n)$)

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 $D_c(k)={\prod }_{h=1}^k\min (h,c)$ andThe 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

In a one-operator closed migration process representing a finite-line switchboard, let $K$ count calls waiting for or receiving the operator's connection service and let $I$ count connected calls. At a stationary distribution, the expected drift of $K$ is $\nu \mathrm{Pr}\{K+I<N\}-\lambda \mathrm{Pr}\{K>0\}=0$, and that of $I$ is $\lambda \mathrm{Pr}\{K>0\}-\mu \mathbb{E}[I]=0$. Poisson arrivals see time averages identifies the full-state probability with $P_{\mathrm{loss}}$. Thus the accepted arrival rate, operator throughput and completed-call rate coincide. This is a useful independent check on the product-form stationary distribution of a closed migration process.