## The Annals of Applied Probability

### 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

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