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\begin{aligned} L_{s} &=\sum_{N=0}^{c-1} N \cdot P_{N}+\sum_{N=c}^{k} N \cdot P_{N} \\&=\frac{P_{0}}{c !}\left(\sum_{N=0}^{c-1} N \cdot \rho^{N}+\sum_{N=c}^{k} N \cdot \frac{\rho^{N}}{c^{N-c}}\right) \end{aligned}