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November, 1991 Departures from Many Queues in Series
Peter W. Glynn, Ward Whitt
Ann. Appl. Probab. 1(4): 546-572 (November, 1991). DOI: 10.1214/aoap/1177005838

Abstract

We consider a series of $n$ single-server queues, each with unlimited waiting space and the first-in first-out service discipline. Initially, the system is empty; then $k$ customers are placed in the first queue. The service times of all the customers at all the queues are i.i.d. with a general distribution. We are interested in the time $D(k,n)$ required for all $k$ customers to complete service from all $n$ queues. In particular, we investigate the limiting behavior of $D(k,n)$ as $n \rightarrow \infty$ and/or $k \rightarrow \infty$. There is a duality implying that $D(k,n)$ is distributed the same as $D(n,k)$ so that results for large $n$ are equivalent to results for large $k$. A previous heavy-traffic limit theorem implies that $D(k,n)$ satisfies an invariance principle as $n \rightarrow \infty$, converging after normalization to a functional of $k$-dimensional Brownian motion. We use the subadditive ergodic theorem and a strong approximation to describe the limiting behavior of $D(k_n,n)$, where $k_n \rightarrow \infty$ as $n \rightarrow \infty$. The case of $k_n = \lbrack xn \rbrack$ corresponds to a hydrodynamic limit.

Citation

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Peter W. Glynn. Ward Whitt. "Departures from Many Queues in Series." Ann. Appl. Probab. 1 (4) 546 - 572, November, 1991. https://doi.org/10.1214/aoap/1177005838

Information

Published: November, 1991
First available in Project Euclid: 19 April 2007

zbMATH: 0749.60090
MathSciNet: MR1129774
Digital Object Identifier: 10.1214/aoap/1177005838

Subjects:
Primary: 60K25
Secondary: 60F15 , 60F17 , 60J60 , 90B22

Keywords: departure process , Hydrodynamic limit , interacting particle systems , invariance principle , large deviations , limit theorems , Queueing networks , queues in series , reflected Brownian motion , strong approximation , subadditive ergodic theorem , Tandem queues , transient behavior

Rights: Copyright © 1991 Institute of Mathematical Statistics

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Vol.1 • No. 4 • November, 1991
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