= Erlang loss formula
{c}
{title2=$E(a,C)$}
= Erlang B formula
{c}
{synonym}
The <Erlang loss formula> gives the blocking probability for a single resource with $C$ identical circuits, no waiting room, independent <Poisson> arrivals of rate $\lambda$ and independent exponential holding times of mean $1/\mu$. Each admitted call occupies one circuit. Write <offered traffic> $a=\lambda/\mu$.
The occupied-circuit count is a <birth-death process> on $0,\ldots,C$, with upward rate $\lambda$ below capacity and downward rate $n\mu$. <Detailed balance> gives $\lambda\pi(n)=(n+1)\mu\pi(n+1)$, hence $\pi(n)\propto a^n/n!$. By <Poisson arrivals see time averages>, the arrival blocking probability is
$$
E(a,C)=\pi(C)=\frac{a^C/C!}{\sum_{k=0}^Ca^k/k!}.
$$
The resource has a finite state space even when $a>C$; excess calls are rejected rather than accumulating.<Stochastic Networks by Frank Kelly and Elena Yudovina>[Section 1.3, equation 1.5, printed pp. 18–19]{ref}
For $C=2$, $a=1$, the weights are $1,1,1/2$, giving blocking $1/5$ and <carried load of an Erlang loss resource> $a[1-E(a,C)]=4/5$. A useful recursion avoids large factorials:
$$
E(a,0)=1,\qquad E(a,k)=\frac{aE(a,k-1)}{k+aE(a,k-1)}.
$$
The <insensitivity of loss networks> extends the occupancy formula to independent general holding times with the same mean under the standard loss-model assumptions.
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