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.