Bartlett's theorem explained

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 ^{t} _{-inf ty} A(s) p(s,t) d t.

Notes and References

1. Book: Kingman, John . John Kingman

   . Poisson Processes . limited . 49 . John Kingman . 1993 . 0198536933 . Oxford University Press .