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 , and that subsequently they move independently around a system of nodes. Write for some particular part of the system and the probability that a customer who arrives at time ' is in at time '. Then the number of customers in at time ' has a Poisson distribution with mean