Bartlett's theorem

Suppose that customers arrive according to a non-stationary Poisson process with rate ${\displaystyle A(t)}$, and that subsequently they move independently around a system of nodes. Write ${\displaystyle E}$ for some particular part of the system and ${\displaystyle p(s,t)}$ the probability that a customer who arrives at time ${\displaystyle s}$ is in ${\displaystyle E}$ at time ${\displaystyle t}$. Then the number of customers in ${\displaystyle E}$ at time ${\displaystyle t}$ has a Poisson distribution with mean^[1]

${\displaystyle \mu (t)=\int _{-\infty }^{t}A(s)p(s,t)\,\mathrm {d} s.}$