UNIVERSITY OF SOUTHERN CALIFORNIA, SPRING 2008
1
Lecture Notes 1
EE 549 — Queueing Theory
Instructor: Michael Neely
I. I
NTRODUCTION TO
Q
UEUEING
S
YSTEMS
A modern data communication system consists of a
network
of information processing stations or
nodes
intercon
nected by data transmission channels or
links
. Information flows from node to node in packetized data units. These
packets
can originate from a variety of sources, and the paths over which they traverse the network can intersect, so
that multiple packets are often intended for delivery over the same transmission link. When the number of packets
in a particular node exceeds the service capabilities of the outgoing links of the node, some packets must either
be dropped (resulting in a loss of information) or placed in a storage buffer or
queue
for future service. Queueing
theory is the study of congestion and delay in such systems.
The dynamics of queueing systems are
event driven
, that is, the state of a queue changes based on discrete
events such as packet arrivals or departures. Such systems are fundamentally different from the linear or nonlinear
systems studied in classical control theory, where inputs are continuous functions of time (such as voltage or
current waveforms) and where state dynamics are described by differential equations. Queueing systems are also
very different from the pointtopoint transmission models of physical layer communication, where inputs consist
of bandlimited signals that are corrupted by continuous background processes such as additive Gaussian noise.
Thus, queueing systems require a different and unique set of mathematical tools. An understanding of these tools
is essential in the analysis, design, and control of data networks.
In this course, we study queueing systems from both a deterministic and probabilistic perspective. In our
deterministic analysis, we illuminate several simple properties and conservation laws exhibited by queueing systems
with any arbitrary set of arrival times and packet sizes. We also develop several fundamental bounds on the worst
case congestion and delay in networks when traffic is deterministically controlled so that the rate and burstiness
of packet arrivals conforms to specified constraints. This deterministic analysis sets the stage for a probabilistic
description of queueing systems. In this context, the arrival streams and service processes are assumed to be driven
by simple probability laws, so that exact performance metrics can be calculated, such as the average delay in the
network, or the probability distribution for the number of packets in a particular queue. The probabilistic study
of event driven systems involves the theory of
discrete stochastic processes
. We shall find that several of the key
results of discrete stochastic process theory, such as the ergodicity properties of Markovian networks as well as
Little’s Theorem for average delay, establish a fundamental relationship between deterministic and probabilistic
analysis of queueing networks.
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 Spring '08
 Neely
 Probability theory, Stochastic process, Unfinished work

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