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.