Consider the Erlang loss B formula $E(N, x) = \frac{\frac{x^N}{x!}}{\sum\limits_{i=0}^N \frac{x^i}{i!}}$, where $0<x<\infty$ and $N$ is a positive integer.
As $N \to \infty$, $E(N, x) \to 0$ and as $x \to \infty$, $E(N, x) \to 1$.
I want to know the convergence rate of $E(N, x)$ in $N$ and $x$. I guess they are both exponential but do not if there is any theorem to ensure it.
The denominator $\sum_{k=0}^n x^k/k!$ is an incomplete gamma function and there are many approximations available for it. The problem as stated is not quite well-defined. There is a transition point according to whether $x$ is approximately as large as $n.$ One way to approach the problem is to parameterize $x=t\ n$ where $t$ is 'small,' say, $0<t<10.$ Then the denominator is $$ \tag{0} S_n(nt) = \sum_{k=0}^n \frac{(nt)^k}{k!}$$
In ref [1] there are two approximations for this Szego sum where I've left off the order of the approximation. $$\tag{1} e^{-nt}S_{n-1}(nt)=1- \frac{(te^{1-t})^n}{\sqrt{2 \pi \, n}(1-t)} $$
and
$$\tag{2} e^{-nt}S_{n}(nt)=\delta(t)+ \frac{t \, \xi(t)}{\sqrt{2}(t-1)} \text{erfc}\big(\sqrt{n}\, \xi(t) \big)$$ where $\delta(t) = 1 \text{ if } 0\le t <1,$ and $\delta(t) = 0 \text{ if } t\ge 1,$ $\xi(t) = \sqrt{|t-1-\log{t}|} $ and erfc is the complementary error function (in Mathematica notation; the reference uses an atypical notation.)
Formula (1) is good for $t$~0 and has the advantage of being parameterized in terms of 'simple functions.' Formula (2) is good for all t except really close to $t=1.$ There are expansions for $t$ really close to 1, but I'll skip it, assuming your application is one of these two cases. To end, here is a table that compares the true ratio $(0)$ for various $n,t$ with that when the denominator is approximated by formula (1) or (2) $$ R^{0}_n(t)=\frac{(nt)^n}{n! S_n(nt) } $$ Recall that $R^1_n$ is not meant to be good for $t>1.$ Both formulas were derived in the large $n$ approximation, and $n=20$ or $n=200$ is not very large.
$$ \begin{array}{c|lcr} (n,t) & R^0_n & R^1_n & R^2_n \quad \\ \hline (20, 0.2) & 8.27746 \,\cdot\, 10^{-6} & 8.27746\,\cdot\, 10^{-6} & 8.27746\,\cdot\, 10^{-6} \\ (20, 0.8) & 6.44109 \,\cdot\, 10^{-2}& 7.63646\,\cdot\, 10^{-2} & 6.53697\,\cdot\, 10^{-2} \\ (20, 0.95) &0.1338 & -0.125 & 0.1434 \\ (20, 1.05) &0.1841 & 0.0321 & 0.2033 \\ (20, 2.0) &0.5213 & 0.00019 & 0.5336 \\ (200, 0.2) & 1.39101 \,\cdot\, 10^{-72} & 1.39101\,\cdot\, 10^{-72} & 1.39101\,\cdot\, 10^{-72} \\ (200, 0.8) & 2.756911 \,\cdot\, 10^{-4}& 2.75787\,\cdot\, 10^{-4} & 2.756953\,\cdot\, 10^{-4} \\ (200, 0.95) &0.027968 & 0.03846 & 0.028203 \\ (200, 1.05) &0.085749 & 0.01536 & 0.08807 \\ (200, 2.0) &0.50245 & 6.271 \,\cdot\, 10^{-29} & 0.50380 \\ \end{array} $$
[1] On the Zero Attractor of the Euler Polynomials by W.M.Y.Goh & R Boyer, Propositions 1-3, https://arxiv.org/abs/math/0409062