Communication networks underpin our modern world, and provide fascinating and challenging examples of large-scale stochastic systems. Randomness arises in communication systems at many levels: for example, the initiation and termination times of calls in a telephone network, or the statistical structure of the arrival streams of packets at routers in the Internet. How can routing, flow control, and connection acceptance algorithms be designed to work well in uncertain and random environments? And can we design these algorithms using simple local rules so that they produce coherent and purposeful behaviour at the macroscopic level?

The first two parts of the book will describe a variety of classical models that can be used to help understand the behaviour of large-scale stochastic networks. Queueing and loss networks will be studied, as well as random access schemes and the concept of an effective bandwidth. Parallels will be drawn with models from physics, and with models of traffic in road networks.

The third part of the book will study more recently developed models of packet traffic and of congestion control algorithms in the Internet. This is an area of some practical importance, with network operators, content providers, hardware and software vendors, and regulators actively seeking ways of delivering new services reliably and effectively. The complex interplay between end-systems and the network has attracted the attention of economists as well as mathematicians and engineers.

We describe enough of the technological background to communication networks to motivate our models, but no more. Some of the ideas described in the book are finding application in financial, energy, and economic networks as computing and communication technologies transform these areas.

This book is about stochastic networks and their applications. Large-scale systems of interacting components have long been of interest to physicists. For example, the behaviour of the air in a room can be described at the microscopic level in terms of the position and velocity of each molecule. At this level of detail a molecule's velocity appears as a random process. Consistent with this detailed microscopic description of the system is macroscopic behaviour best described by quantities such as temperature and pressure. Thus the pressure on the wall of the room is an average over an area and over time of many small momentum transfers as molecules bounce off the wall, and the relationship between temperature and pressure for a gas in a confined volume can be deduced from the microscopic behaviour of molecules.

Economists, as well as physicists, are interested in large-scale systems, driven by the interactions of agents with preferences rather than inanimate particles. For example, from a market with many heterogeneous buyers and sellers there may emerge the notion of a price at which the market clears.

Over the last 100 years, some of the most striking examples of largescale systems have been technological in nature and constructed by us, from the telephony network through to the Internet. Can we relate the microscopic description of these systems in terms of calls or packets to macroscopic consequences such as blocking probabilities or throughput rates? And can we design the microscopic rules governing calls or packets to produce more desirable macroscopic behaviour? These are some of the questions we address in this book.

A major practical and theoretical issue in the design of communication networks concerns the extent to which control can be decentralized. Over a period of time the form of the network or the demands placed on it may change, and routings may need to respond accordingly. It is rarely the case, however, that there should be a central decision-making processor, deciding upon these responses. Such a centralized processor, even if it were itself completely reliable and could cope with the complexity of the computational task involved, would have its lines of communication through the network vulnerable to delays and failures. Rather, control should be decentralized and of a simple form: the challenge is to understand how such decentralized control can be organized so that the network as a whole reacts sensibly to fluctuating demands and failures.

The behaviour of large-scale systems has been of great interest to mathematicians for over a century, with many examples coming from physics. For example, the behaviour of a gas can be described at the microscopic level in terms of the position and velocity of each molecule. At this level of detail a molecule's velocity appears as a random process, with a stationary distribution as found by Maxwell. Consistent with this detailed microscopic description of the system is macroscopic behaviour, best described by quantities such as temperature and pressure. Similarly, the behaviour of electrons in an electrical network can be described in terms of random walks, and yet this simple description at the microscopic level leads to rather sophisticated behaviour at the macroscopic level: the pattern of potentials in a network of resistors is just such that it minimizes heat dissipation for a given level of current flow.

In the previous chapter, we considered a network with a fixed number of users sending packets. In this chapter, we look over longer time scales, where the users may leave because their files have been transferred, and new users may arrive into the system. We shall develop a stochastic model to represent the randomly varying number of flows present in a network where bandwidth is dynamically shared between flows, where each flow corresponds to the continuous transfer of an individual file or document. We assume that the rate control mechanisms we discussed in Chapter 7 work on a much faster time scale than these changes occur, so that the system reaches its equilibrium rate allocation very quickly.

Evolution of flows

We suppose that a flow is transferring a file. For example, when Elena on her home computer is downloading files from her office computer, each file corresponds to a separate flow. In this chapter, we allow the number of flows using a given route to fluctuate. Let nr be the number of active flows along route r. Let xr be the rate allocated to each flow along route r (we assume that it is the same for each flow on the same route); then the capacity allocated to route r is nr xr at each resource j ∈ r. The vector x = (xr, r ∈ ℜ) will be a function of n = (nr, r ∈ ℜ); for example, it may be the equilibrium rate allocated by TCP, when there are nr users on route r.

We assume that new flows arrive on route r as a Poisson process of rate νr, and that files transferred over route r have a size that is exponentially distributed with parameter μr.

This chapter will serve as a quick review of the essential results in the theory of Poisson point processes (PPPs) that we shall use for our analysis later in the book. However, we shall first take a step back and discuss point processes in general and their applicability to our wireless deployments of interest. Then we shall provide arguments in support of the use of Poisson point processes before summarizing the most important mathematical results. This chapter is not meant to be an exhaustive treatment of the theory of PPPs – for that, see Kingman (1993). The principal theorems will be quoted with citations only to appropriate references, but heuristic arguments will be provided to help the reader understand them.

Stochastic models for BS locations

In this book, we restrict ourselves to BS deployments on a plane, i.e. in two dimensions. This models many scenarios of interest, but not all, e.g. low-power BSs (“access points”) mounted indoors on several floors of an office building. Nonetheless, the theory in two dimensions is rich enough to yield useful results for a number of practical scenarios, and the techniques we employ to derive these results in this book can be extended straightforwardly to three dimensions.

A plot of the locations of the BSs in such a deployment is a so-called “spatial point pattern,” or “point pattern” for short. In practice, network operators spend enormous amounts of time and resources on finding the best locations for their BS sites, taking into account such factors as the traffic they expect to support, geographical obstacles and features, municipal laws and zoning permissions, etc.