# Simplify a equation containing factorial, summation, and fraction

I really need some help on simplifying this math equation. Please help me reduce it as simple as possible! Thanks in advance!

$$\large P_0=\frac{1}{\left[\sum_{i=0}^{M-1} \frac{1}{i!}\left(\frac{\lambda}{\mu}\right)^i\right]+\frac{1}{M!}\left(\frac{\lambda}{\mu}\right)^M\frac{M\mu}{M\mu-\lambda}}$$

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## 1 Answer

You are attempting to simplify normalization factor in derivation of ErlangC formula. It can not really be simplified any further, using elemenetary function.

The sum can be summed in terms of incomplete gamma-function $\Gamma_x(s) = \int_{x}^\infty y^{s-1} \mathrm{e}^{-y} \mathrm{d} y$: $$\sum_{i=0}^{M-1} \frac{1}{i!} \frac{\lambda^i}{\mu^i} = \exp(\frac{\lambda}{\mu}) \frac{\Gamma_{\frac{\lambda}{\mu}}\left(M \right)}{\Gamma(M)}$$

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I see.. awh, i see integrals.. sigh. Anyways, thank you for the help :) Have a nice day! –  Dino55 Sep 28 '11 at 22:23
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