Some Observations on Equation-Based Rate Control

We consider one aspect of the general problem of unicast equation based rate control in the Internet, which we formulate as follows. When a so called ``loss-event" occurs, a data source updates its sending rate by setting it to $f(\hat{p_n})$, where $\hat{p}_n$ is an estimate of $\overline{p}$, the rate of loss-events. Function $f$ (the target loss-throughput function) defines the objective of the control method: we would like that the throughput $\overline{x}$, attained by the source, satisfies the equation $\overline{x}\leq f(\overline{p})$. If so, we say that the control is conservative. In the Internet, function $f$ is obtained by analyzing the dependency of throughput versus the rate of loss-events for a real TCP source. A non-TCP source which implements a control system as we describe is said to be TCP-friendly if the control is conservative. In this paper, we examine whether such a control system is conservative. We first consider a simple stochastic model which assumes that the intensity of the loss-events is proportional to the current sending rate. We show that, for this model, the control is always conservative if $f(p)$ is a concave function of $1/p$; otherwise this may not be true. Then we consider a second model where the loss-event inter-arrival times is an exogeneous stationary random process. We show that, for this second model, there exist statistics of the loss-event inter-arrival times such that the control is non-conservative, even if $f(p)$ is a concave function of $1/p$. We validate our analytical results with simulations. Another aspect of unicast equation-based rate control in the Internet is the influence of the variability of round-trip times, which is not analyzed in this paper. KEYWORDS: Equation-based, Rate control, TCP-friendliness, Internet, Stochastic recurrence, Autoregressive process, Markov modulated process, Non-linear system, Estimation, Palm expectation