Bartlett's theorem

This article is about the theorem in probability. For the theorem in electricity, see Bartlett's bisection theorem.

In queueing theory, Bartlett's theorem gives the distribution of the number of customers in a given part of a system at a fixed time.

Theorem

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

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