# Poisson Process¶

(c) Tobias Hossfeld (Aug 2021)

This script and the figures are part of the following book. The book is to be cited whenever the script is used (copyright CC BY-SA 4.0):
Tran-Gia, P. & Hossfeld, T. (2021). Performance Modeling and Analysis of Communication Networks - A Lecture Note. Würzburg University Press. https://doi.org/10.25972/WUP-978-3-95826-153-2

A Poisson process with rate $\lambda >0$ is a renewal process in which the interarrival time distribution follows a negative exponential distribution with rate $\lambda$.

We examine some basic properties of the Poisson properties and use the package scipy.stats

## Simulation of Poisson Process¶

We randomly generate interarrival times $A_i$. The arrival times are $t_i = \sum_{k=0}^i A_k$.

## Thinning a Poisson Process¶

### Random Thinning¶

We consider random thinning: With a probability $p$ an arrival event is taken in the thinned process. The empirical CDFs are derived using numpy.histogram

### Cyclic or Deterministic Thinning¶

Now, we are skipping every second arrival event. The resulting distribution is, thus, the sum of two exponential phases which is an Erlang $E_2$ distribution with rate $\lambda$.

## Number of Arrivals within Fixed Interval¶

The number $N$ of Poisson arrivals within an fixed interval of length $T$ follows a Poisson distribution with parameter $\lambda\cdot T$:

$N \sim \mathrm{POIS}(\lambda\cdot T)$

To get smoother results, we need to simulate more arrival events. Then, the curves will overlap.

## Uniform Distribution of Poisson Arrivals¶

Consider a Poisson process with rate $\lambda$. There are $n$ arrivals within an fixed interval of duration $T$. Then these arrivals are uniformly distributed within this interval. The $n$ arrivals are uniformly distributed and occur at times $u_1, u_2, u_3, \dots, u_n$. Note that the times $u_i$ are not ordered. The r.v.s $u_1, u_2, \dots, u_n$ follow all the same uniform distribution.

The arrivals correspond to a Poisson process and the interarrival times are therefore exponentially distributed with parameter $n/T$.