Suppose There are n severs each of which has a service time exponentially distributed with mean m. What is the number of servers which complete their task till time t, nm m$\gg$t.
I am expecting it to have a Poisson distribution and the process to be a Poisson process but can't seem to prove it.
Let $Z$ denote the number of servers which have completed their task. We are interested in finding $P(Z=k), \; k=0,\ldots,n$.
Let $X_i$ denote the time taken by the $i^{th}$ server, where $X_i$ are $\exp(1/m)$
$P(Z=0) = P(X_1>t, X_2>t, \ldots, X_n>t) = P(X_1>t)^n = (e^{-t/m})^n = e^{-nt/m}$
$P(Z=1) = {n \choose 1}e^{-t/m}(1-e^{-t/m})^{n-1}$
$P(Z=k) = {n \choose k}e^{-kt/m}(1-e^{-t/m})^{n-k}$
So $Z$ follows a binomial distribution with the parameters $(n, e^{-t/m})$