Transformations: Non-Homogeneous and Spatial Poisson Processes

This note closes the Poisson-process toolkit with two further generalisations: thinning with a position-dependent keep probability (which turns a homogeneous process into a non-homogeneous one, and gives a strikingly clean formula for the number of "calls in progress" in a simple queueing model), the order-statistics property (conditional on $n$ arrivals by time $t$, their locations are exactly as if $n$ points were thrown uniformly at random), and the spatial Poisson process (replacing the time axis with $\mathbb{R}^d$ entirely). Each is a genuinely reusable tool: the M/G/1 result is the ancestor of every "number of items currently in service" queueing formula in the next topic, and the order-statistics property underlies the standard technique for simulating a Poisson process on a computer.

Learning objectives. By the end of this note you should be able to: