= Burke's theorem
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= Burke theorem
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In a stationary <M M 1 queue> with <independent> Poisson arrivals of rate $\lambda$ and exponential <service time>[service times] of rate $\mu>\lambda$, the departure <counting process> is a <Poisson process> of rate $\lambda$. Moreover, the departures before a fixed time $t$ are <independent> of the queue population $N_t$. This <independence> concerns the past departures: future departures depend on whether the queue is currently empty.
The proof uses time reversal. The stationary <Geometric distribution> satisfies <Detailed balance>, $\pi_n\lambda=\pi_{n+1}\mu$, so the stationary queue population is a <Reversible Markov chain>. Departures in forward time are arrivals in reversed time. In the reversed chain the upward rate is again the constant $\lambda$, so those arrivals are an <independent> <Poisson process>, including <independence> of their future increments from the present queue population. Reversing time gives both assertions. Consequently, feeding the output of a stationary <M M 1 queue> into another queue supplies Poisson input to that second queue. The claim depends on equilibrium; a queue started empty has no immediate departures even though arrivals have positive rate.
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