Chapter 2 — Poisson Processes

This page develops the mathematics of Poisson processes and keeps the mathematical discussion separate from the Python API and worked examples.

1. Introduction

A Poisson process is introduced as a continuous-time process with non-negative integer values, used to describe the times at which random events occur: calls, customer arrivals, failures, particle emissions, and similar phenomena.

2. Counting process

Let \(N(t)\) be the number of occurrences observed in \(]0,t]\). A process \((N(t))_{t\ge0}\) is a counting process if

\[ N(t)\in\mathbb N, \qquad s<t\Rightarrow N(s)\le N(t), \]

and if \(N(t)-N(s)\) represents the number of occurrences on \(]s,t]\).

3. First definition of the Poisson process

A counting process is a Poisson process with rate \(\lambda>0\) if:

  • \(N(0)=0\) almost surely;
  • the increments are independent;
  • the increments are stationary;
  • \(P(N(h)=1)=\lambda h+o(h)\);
  • \(P(N(h)\ge2)=o(h)\).

Here \(o(h)/h\to0\) as \(h\to0\).

4. Second definition

An equivalent formulation is: \(N(0)=0\), the increments are independent, and for every interval length \(t\),

\[ P(N(t+s)-N(s)=n)=e^{-\lambda t}\frac{(\lambda t)^n}{n!}, \qquad n\in\mathbb N. \]

This law does not depend on \(s\), which gives stationary increments.

5. Equivalence of the definitions

Let

\[ p_n(t)=P(N(t)=n). \]

The analysis over \([t,t+h]\) gives

\[ p_0'(t)=-\lambda p_0(t),\qquad p_0(0)=1, \]

and, for \(n\ge1\),

\[ p_n'(t)=-\lambda p_n(t)+\lambda p_{n-1}(t), \qquad p_n(0)=0. \]

Solving these equations gives

\[ \boxed{P(N(t)=n)=e^{-\lambda t}\frac{(\lambda t)^n}{n!}}. \]

Therefore \(N(t)\sim\mathcal P(\lambda t)\) and

\[ N(t+s)-N(s)\sim\mathcal P(\lambda t). \]

6. Inter-occurrence times

If

\[ 0=\tau_0<\tau_1<\tau_2<\cdots \]

are the occurrence times, the inter-arrival times

\[ T_n=\tau_n-\tau_{n-1} \]

form an i.i.d. sequence with exponential distribution of parameter \(\lambda\):

\[ T_n\sim\mathrm{Exp}(\lambda). \]

7. Conditional occurrence times

Conditionally on \(N(s)=1\), the time of the unique occurrence in \([0,s]\) is uniform:

\[ T_1\mid\{N(s)=1\}\sim\mathcal U([0,s]). \]

More generally, conditionally on \(N(s)=k\), the \(k\) occurrence times in \([0,s]\) have the distribution of the order statistics of a sample of \(k\) independent uniform random variables on \([0,s]\).

8. Superposition

If \(N^{(1)}\) and \(N^{(2)}\) are independent Poisson processes with rates \(\lambda_1\) and \(\lambda_2\), then

\[ N(t)=N^{(1)}(t)+N^{(2)}(t) \]

is again a Poisson process, with rate

\[ \boxed{\lambda_1+\lambda_2}. \]

9. Splitting / thinning

Starting from a Poisson process of rate \(\lambda\), assign each occurrence independently to type 1 with probability \(p\), or to type 2 with probability \(1-p\). The two resulting processes are independent and have rates

\[ \lambda p \qquad\text{and}\qquad \lambda(1-p). \]

10. Non-homogeneous Poisson process

In the non-homogeneous case, the rate depends on time: \(\lambda=\lambda(t)\). Increments are therefore no longer stationary. Introduce the cumulative mean function

\[ \boxed{m(t)=\int_0^t\lambda(u)\,du}. \]

Locally,

\[ P(\text{one occurrence in }[t,t+h[)=\lambda(t)h+o(h). \]

This model is suited to phenomena whose occurrence rate varies over time.

11. Key results

\[ N(t)\sim\mathcal P(\lambda t), \qquad T_n\sim\mathrm{Exp}(\lambda). \]

Increments over disjoint intervals are independent; in the homogeneous case, their law depends only on the interval lengths. Conditionally on a fixed number of occurrences, the occurrence times are distributed as uniform order statistics.

12. Summary

\[ \text{counting} \rightarrow \text{Poisson definition} \rightarrow \text{Poisson law} \rightarrow \text{inter-arrival times} \]
\[ \rightarrow \text{conditional times} \rightarrow \text{superposition} \rightarrow \text{splitting} \rightarrow \text{non-homogeneous Poisson process}. \]