Analysis of a feedback fluid model for heterogeneous TCP sources

In this paper we consider a bottleneck link and buffer used by one or two fluid sources subject to feedback. During overflow, the buffer sends negative feedback signals to the sources to reduce the sending rate. Otherwise the buffer sends positive signals to indicate that the sources can increase the rate. The analysis is targeted at the Transport Control Protocol of the Internet. In this specific context we find closed form expressions for the eigenvalues and eigenvectors of the solution for the single-source case. The case of two sources extends the single-source analysis considerably so that now source parameters, e.g. feedback rate, can be individually specified. By deriving a sufficient number of boundary conditions for the solution values and derivatives we prove that the related two-point boundary value problem is uniquely solvable in the stationary case. We establish also a numerically efficient procedure to compute the coefficients of the solution of the differential equations. The numerical results of this model are presented in an accompanying paper.