 Erlang loss formula

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 $0,\dots ,C$, 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 isThe resource has a finite state space even when $a>C$; excess calls are rejected rather than accumulating.^[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:The insensitivity of loss networks extends the occupancy formula to independent general holding times with the same mean under the standard loss-model assumptions.