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