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]

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

References

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