# Marked Poisson point process with multiple marks?

In most of the literature, I come across marked Poisson Poisson process with just one marks. Can someone give me details on Marked Poisson process with multiple marks?

First denote with $\lambda$ the exponential rate of the process and with $\varphi$ the probability distribution of marks (we assume that marks can only be positive). Consider the marked Poisson process on the line and define the random variable $\mathcal{N}(a,b)$, that counts the number of marks in the interval $(a,b)$. The probability law of $\mathcal{N}(a,b)$ is $$\mathbb{P}(\mathcal{N}(a,b)=N)=\sum_{\mu \vdash N} \sum_{\eta \sim \mu} e^{-\lambda(b-a)} \frac{\lambda^{l(\mu)}(b-a)^{l(\mu)}}{l(\mu)!}\varphi(\eta_1) \cdots \varphi(\eta_{l(\mu)}),$$ where the symbol $\mu \vdash N$ means that $\mu$ is a partition of $N$, the symbol $\eta \sim \mu$ means that $\eta$ is a permutation of $\mu$ and $l(\mu)$ is the length of the partition $\mu$.
We can also compute the generating function of $\mathcal{N}(a,b)$ as $$\mathbb{E}(x^{\mathcal{N}(b-a)})=\exp\{ \lambda(b-a) (\mathbb{E}_\varphi(x^{\mathfrak{m}}) -1) \},$$ where $\mathfrak{m}$ is the number of marks at a specific location.