## The Annals of Applied Probability

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

### Departures from Many Queues in Series

#### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1177005838

**Digital Object Identifier**

doi:10.1214/aoap/1177005838

**Mathematical Reviews number (MathSciNet)**

MR1129774

**Zentralblatt MATH identifier**

0749.60090

**JSTOR**

links.jstor.org

**Subjects**

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

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.aoap/1177005838