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Departures from many queues in series. (English) Zbl 0749.60090
Consider a series of $$n$$ queues with a common service time distribution, and let $$D(k,n)$$ denote the departure time of customer $$k$$, assuming that initially $$k$$ customers are placed in the first queue. The limiting behaviour of $$D(k,n)$$ as $$n\to\infty$$ is studied in various regimes for $$k$$: if $$k$$ is fixed, $$(D(k,n)-n)/\sqrt n$$ converges in distribution to a certain functional of $$k$$-dimensional Brownian motion; if $$k=k_ n\approx xn^{1-\varepsilon}$$, $$(D(k,n)-n)/\sqrt {nk}$$ converges in probability to a constant independent of $$x$$; and if $$k_ n\approx xn$$, then $$D(k,n)/n$$ converges in probability to a constant dependent on $$x$$ (this result may be interpreted as a hydrodynamic limit). A basic observation is a duality between $$k$$ and $$n$$, which is obtained by expressing $$D(k,n)$$ as the maximum partial sum of service times over paths of length $$k+n-1$$ in a $$k\times n$$ lattice of service times. The techniques involve weak convergence in function spaces, strong approximations and the subadditive ergodic theorem.

##### MSC:
 60K25 Queueing theory (aspects of probability theory) 60F17 Functional limit theorems; invariance principles 90B22 Queues and service in operations research 60J60 Diffusion processes 60F15 Strong limit theorems
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