I am currently working on a small optimization problem in which I need to find an optimal number of servers for an objective function that incorporates the Erlang Loss formula. To this end, I have been searching for an expression for the first order difference of the Erlang Loss formula with respect to the number of servers, i.e. $B(E,m+1)-B(E,m)$, where m is the number of servers and $B(E,m)$ is given by:
$B(E,m)={\dfrac {{\dfrac {E^{m}}{m!}}}{\sum _{{i=0}}^{m}{\dfrac {E^{i}}{i!}}}}$
Unfortunately, until now I can't derive or find such an expression (if it exists) and was wondering whether one on this forum could help me out?
Many thanks in advance!
Consider the $M/G/c/c$ system consisting of $c$ servers, no queue and load $\rho := \lambda \, \mathbb{E}[B]$, where $\lambda$ is the arrival rate and $B$ the service time. The probability that an arriving job finds all servers busy is called the Erlang loss probability and is denoted by $B(c,\rho)$.
The Erlang loss probability satisfies a recursion, see Section 11.3 (page 116) of these lecture notes,
\begin{equation} B(c,\rho) = \frac{\rho B(c - 1,\rho)}{c + \rho B(c - 1,\rho)}. \end{equation}
This recursion can be rewritten to
\begin{equation} \frac{c}{\rho} B(c,\rho) - B(c - 1,\rho) = B(c,\rho) B(c - 1,\rho), \end{equation}
but I do not think that this is what you were hoping for.