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Big Queues (Lecture Notes in Mathematics)

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Queuing theory is of enormous importance in applications and the nature of a queue permeates everyday life. Therefore the understanding of the time dependence of queues, i.e. how they fill up and empty (if ever), has been the topic of intense research that spans many decades. Mathematics, especially the theory of stochastic processes and the theory of large deviations, has been the main tool used in the understanding of queues, and this book, written by a few of the major contributors to these branches of mathematics, is a specialized overview of some of the more contemporary developments. The study of packet networks in telecommunications is the main target of the book, and it could be read by anyone who has a background in topology and probability theory. The mathematics and concepts in this book have been used extensively in the understanding of packet networks, and there have been commercial products, used primarily for network management and quality of service, developed in the last five years that are based on the contents of the book.

One of the most interesting developments coming about because of the rise of the Internet has been the claim that traffic in the Internet has the property of long range dependence. There has been a rather large amount of research on this claim, both theoretical and empirical, and in recent years some counterexamples have been made to this claim by a few researchers. The authors discuss some of the necessary background needed to understand these developments in this book. These discussions, along with others, are not done in a definition-theorem-proof format, as is usually done in so many books on modern mathematics. Instead, the authors have chosen to explain and motivate the main results. This alone increases dramatically the didactic quality of the book. Central to the book is the theory of large deviations, which is basically a theory of rare events. Packet drops in queuing networks are an example of rare events, and the calculation of the probabilities of these events are of great interest to those who are trying to provide quality of service in these networks.

The authors motivate their discussion in the first chapter by examining the case of a single server queue and the usual (Lindley) recursion relation that connects the customer waiting time before service begins to the service time and the interarrival time between customers. Using a discrete-time analog of the M/M/1 queue they write the solution for the distribution of the equilibrium queue length in terms of the logarithm of the probability that the queue exceeds a certain quantity q. Their claim that an approximate version of this equation holds in general motivates the use of the theory of large deviations. This theory attempts to understand large fluctuations around the mean of a random variable, based on the main observation that the probability of these fluctuations has an exponential decay in the sample size. In one dimension, the case that they first consider, the proofs of the bounds on the logarithms of the probabilities involve the use of the log moment generation function of the random variable and its `convex conjugate', the Fenchel-Legendre transform. More importantly, the log moment generation function and its convex conjugate appear in the statement of `Cramer's Theorem', which is essentially a set of inequalities called the large deviations lower bound and the large deviations upper bound. The convex conjugate appears as a rate function in this theorem. Cramer's theorem is proved in chapter 2, along with some of its generalizations.

To set up later constructions the authors discuss a large deviations principle (LDP) in a more abstract setting in chapter 4. Of main focus is the finding of a space of continuous function that represent the collection of input flows at a queue. Also important in this setting is the notion of `good' rate function, which is a non-negative function from a Hausdorff space to the extended real numbers that is lower semicontinuous and which has compact level sets. The large deviations principle that they discuss involves sequences of Borel measures on Borel sigma-algebras, and the inequalities relate the smallest values of the rate function in the interior to the smallest values of the rate function in the closure. Of greatest importance in this chapter though is the `contraction principle', which allows one to begin with a LDP for a sequence of random variables and obtain another LDP for another sequence of random variables using a continuous function. The contraction principle is used to study large queues and queues subjected to multiple flows. The asymptotic behavior of large-buffers is of course very important for networking applications, and the authors construct the appropriate continuous functions using an appropriate topology generated by the (scaled) uniform norm.

At least for this reviewer, the most interesting discussions in the book occur in the last chapters on long range dependence, particularly in the notion of `effective bandwidth' and how it can be viewed as sort of an interpolation between the average and peak rates. The effective bandwidth is something that can actually be calculated for real traffic patterns and can be used as the authors point out for admission control. Such practical uses justify the time needed to master the contents of this book, and since network managers and service providers are under extreme pressure at the present time to squeeze every drop out of performance out of their networks, one can expect these ideas to continue to flourish and be extended to the even more complex networks of the future.

The focus on the queueing networks is novel and meaningful. But the book needs to be polished--reading with errata is anyway unpleasant. Applications to queues are good, but the underlying LDP theory are scratchy. Use it as a reference book, not a textbook.