= Stationary flow conservation in a finite-line switchboard
{title2=$\nu(1-P_{\rm loss})=\lambda\Pr\{K>0\}=\mu\mathbb E[I]$}
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 $\nu\Pr\{K+I<N\}-\lambda\Pr\{K>0\}=0$, and that of $I$ is $\lambda\Pr\{K>0\}-\mu\mathbb E[I]=0$. <Poisson arrivals see time averages> identifies the full-state probability with $P_{\rm loss}$. 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>.
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