What is the relation of the Erlang B and C formulas to the Erlang distribution?

Question

I am confused as to whether the Erlang formulas in telecommunications are related to the Erlang distribution or whether they share the same name simply because they are coined after the same Mathematician. I know for example that the Erlang B formula deals with a system that has Poisson arrivals and Exponential departures, I'm curious as to whether Erlang distributions are explicitly applied when deriving the Erlang B formula for such a system

Answer 1

You certainly do not need to know the Erlang distribution to derive the Erlang B formula.

Under the Poisson arrivals and exponential service assumptions, the state of an $M/M/k/k$ (exponential arrivals, exponential service, $k$ servers, at most $k$ customers in system at any time) queueing system is a simple birth-death Markov chain with state space $\{0,1,2,\ldots,k\}$. If arrivals have rate $\lambda$ and service has rate $\mu$, the chain transitions from $i$ to $i+1$ at rate $\lambda$ ($i=0,1,\ldots,k-1$), and it transitions from $i$ to $i-1$ at rate $\mu$ ($i=1,2,\ldots,k$). With this information, you can derive the stationary distribution \begin{align*} \pi_i = \frac{(\lambda/\mu)^i/i!}{\sum_{j=0}^k (\lambda/\mu)^j/j!}, \quad i=0,1,2\ldots,k. \quad (\ast) \end{align*} You can also derive this distribution by first finding the stationary distribution for an $M/M/\infty/\infty$ system (infinitely many servers); the state space in that case is $\{0,1,2,\ldots\}$, the stationary distribution is $\pi_i = e^{-\lambda}(\lambda/\mu)^i/i!$ (Poisson distribution), and you can get $(\ast)$ by reducing the state space to $\{0,1,2,\ldots,k\}$ and renormalizing the distribution.

Anyway, once you know the stationary distribution in $(\ast)$, you can derive the Erlang formula by using the PASTA property (Poisson arrivals see time averages). This property means that the probability that a Poisson arrival finds the system in state $i$ is $\pi_i$. So the probability that a Poisson arrival arrives to a full system is $$ \pi_k = \frac{(\lambda/\mu)^k/k!}{\sum_{j=0}^k (\lambda/\mu)^j/j!}, $$ i.e. the long-run probability that there are $k$ customers in the system.

Summary: The Erlang B formula only requires knowledge of how to obtain stationary distributions for simple birth-death Markov chains and knowledge of the PASTA property. You do not need to know anything about the Erlang distribution.