$\newcommand{\length}{\textrm{length}}$I have queues of packets of different sizes. The probability of choosing queue $i$ where I pop the first packet is
$p(i) = \frac{\length(i)}{ \sum_j \length(j)}$
The probability that I don't choose queue $i$ is $1-p(i)$. It should thus follow a Bernoulli distribution. I calculate these probabilities at the beginning and stick to them until the end.
My goal is have for each queue an approximation of a Poisson Process. Is this possible?
Example:
I have queues $a$,$b$,$c$ each with with packets 1, 2, 3, etc. Say that I get the following output:
$a_1, b_1, b_2, c_1, a_2, b_3, a_3, c_2, b_4, a_4, b_5,\dots$
I want to approximate the arrival times for elements from queue $a$ by some exponential random variable with a certain intensity. Same thing for queues $b$ and $c$. Is this possible?