In Stochastic Simulation: Algorithms and Analysis by Søren Asmussen, on Page 38
A Poisson r.v. $N$ with rate $\lambda$ ($P(X = n) = e^{−\lambda} \frac{\lambda^n}{n!}$) can be constructed using the relation between the exponential distribution and the Poisson process: generate $X_1 , X_2 , ...$ as i.i.d. exponentials with rate $\lambda$ one at a time until the random time at which the sum exceeds 1, say $X_1 + · · ·+ X_N < 1 < X_1 + · · ·+ X_{N +1}$ . Then N is the desired Poisson r.v. To reduce the number of evaluations of special functions, use $N = max \lbrace n \geq 0 : \prod_{i=1}^n U_i > e^{-\lambda} \rbrace$
where $U_i$ is r.v. uniform in $[0,1]$.
I was wondering why to "reduce the number of evaluations of special functions", the $N$ in the last equation has Poisson distribution with rate λ?
Thanks and regards!