# Bits and Queues

Introduction :: Codes :: Bounds

## Introduction

With all the developments in networking, there have been relatively few "scientific" studies of networking and information theory. The union is, to quote one researcher, as yet "uncosummated." However, some recent work begins to point the way. It deals with passing information through queues. In this lecture we discuss this paper, which is important for a variety of reasons. (1) It serves to illustrate again the concept of capacity. (2) It is of recent research interest; (3) it has application to future technology. (4) It will put a lot of our tools to use and (5) it allows us to introduce (or review) the concepts of queuing, which are important engineering models. Our primary reference is "Bits through queues" by Venkat Anantharam and Segio Verdu (IEEE Trans. Info. Theory, v. 42, no. 1, pp. 4-18), which won the award for the best paper of 1996.

## Queues

A queue: input distribution; service distribution; number of queues. We denote a queue by
*
X
*
/Y/1: is is input distribution,
*
Y
*
is service distribution. We characterize the service time by μ , the service rate, so the average service time is 1 / μ. To begin with, we represent information using codes consisting of
*
M
*
codewords, where each codeword constists of
*
n
*
interarrival times. We thus code information by the amount of delay between transmissions (not in the actual information in the packet transmitted). Fig. 2 of paper:
*
A
*
_{
i
}
is the interarrival time, so the
*
k
*
th arrival (at the queue) is at time

*
D
*
_{
i
}
= interdeparture time, so the departures occur at times

*
S
*
_{
i
}
= service time.
*
W
*
_{
i
}
= idling time, the time between the (
*
i
*
-1)st departure and the
*
i
*
th arrival (if the
*
i
*
th arrival occurs before the (
*
i
*
-1)st departure, then
*
W
*
_{
i
}
= 0. We have

*D*

_{ i }=

*W*

_{ i }+

*S*

_{ i }.

Then

Discuss how the service time affects the transmission.