Exercise: Congestion Control Stability Analysis

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Note: This problem is now solved in the text, so it has been removed from the list of problems for the Dynamic Behavior chapter.

A simple model for congestion control between computers connected by a router is given by the differential equation

=
=

where , are the transmission rates for the sources of data, is the current buffer size of the router, is the maximum buffer size, and is the capacity of the link connecting the router to the computers. The equation represents the control law that the individual computers use to determine how fast to send data across the network (this version is motivated by a protocol called "Reno") and the equation represents the rate at which the buffer on the router fills up.

Consider the case where (so that we have three states, , , and ), and take Mb and Mb/sec. Locate the equilibrium points for the system and indicate whether each is asymptotically stable, stable (but not asymptotically stable), or unstable.

Instructors note: for an easy version of this problem, allow the proof to be via representative simulation.