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 ) {\displaystyle A(t)} , and that subsequently they move independently around a system of nodes. Write E {\displaystyle E} for some particular part of the system and p ( s , t ) {\displaystyle p(s,t)} the probability that a customer who arrives at time s {\displaystyle s} is in E {\displaystyle E} at time t {\displaystyle t} . Then the number of customers in E {\displaystyle E} at time t {\displaystyle t} has a Poisson distribution with mean
μ ( t ) = ∫ − ∞ t A ( s ) p ( s , t ) d s . {\displaystyle \mu (t)=\int _{-\infty }^{t}A(s)p(s,t)\,\mathrm {d} s.}
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