# Bartlett's 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[1]

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

## References

1. ^ Kingman, John (1993). Poisson Processes. Oxford University Press. p. 49. ISBN 0198536933.