I need to simulate sending messages from $A$ to $B$ as follows:
Each message is sent $N$ times from $A$ on the same time, passes through a certain route $R_n$ and arrives at $B$. Travel time of $R_n$ is a random number with a mean of $X_n$ and standard deviation of $Y_n$.
I want to write an algorithm that calculates the possible arrival time of a stream of messages $(t_1, t_2, \dotsc, t_n)$ of messages $1,\dotsc,n$ that are generated with a random rate with a mean of $X_a$ and standard deviation of $Y_a$. I think I should use Poisson distribution here somehow, but having learned this math too long ago I would appreciate if someone can help me to figure out how to calculate this.
For reference - I can easily generate random numbers in Java with Poisson distribution or normal distribution, etc.