The Annals of Applied Probability

Departures from Many Queues in Series

Peter W. Glynn and Ward Whitt

Full-text: Open access


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.

Article information

Ann. Appl. Probab., Volume 1, Number 4 (1991), 546-572.

First available in Project Euclid: 19 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60K25: Queueing theory [See also 68M20, 90B22]
Secondary: 60F17: Functional limit theorems; invariance principles 90B22: Queues and service [See also 60K25, 68M20] 60J60: Diffusion processes [See also 58J65] 60F15: Strong theorems

Tandem queues queues in series queueing networks departure process transient behavior reflected Brownian motion limit theorems invariance principle strong approximation subadditive ergodic theorem large deviations interacting particle systems hydrodynamic limit


Glynn, Peter W.; Whitt, Ward. Departures from Many Queues in Series. Ann. Appl. Probab. 1 (1991), no. 4, 546--572. doi:10.1214/aoap/1177005838.

Export citation