Section 17.1

Arrivals, Thinning and Superposition

Split and merged Poisson streams, covariances of counts, conditioning on arrivals.

Toolkit

A Poisson process $N(t)$ of rate $\lambda$ has independent, stationary increments: $N(t)-N(s)\sim\mathrm{Poisson}(\lambda(t-s))$ independent of $N(s)$ for $s<t$ . This lets you write $N(t)=N(s)+(N(t)-N(s))$ to compute covariances or condition on arrivals.

Key facts:

\operatorname{Var}(N(t))=\lambda t,\qquad \operatorname{Cov}(N(s),N(t))=\lambda\min(s,t).

Thinning: if each arrival is independently marked type $i$ with probability $p_i$ ( $\sum p_i=1$ ), the type- $i$ counts form independent Poisson processes of rate $\lambda p_i$ . Superposition: independent Poisson processes of rates $\lambda_1,\dots,\lambda_k$ merge into a Poisson process of rate $\sum\lambda_i$ . Random number of trials: if $N\sim\mathrm{Poisson}(\lambda)$ and each of the $N$ items independently “succeeds” with probability $p$ , the success count is $\mathrm{Poisson}(\lambda p)$ , giving $\mathbb{P}(\text{no success})=e^{-\lambda p}$ . Marks/labels attached to arrivals are independent of the arrival times/counts, so probabilities about labels can be computed by conditioning on the count and using its distribution.

Problems