On the use of packet classes in communication networks to enhance congestion pricing based on marks

Next, consider the system of differential equations d dt y r (t) = r w r (t) 0 y r (t) j2r j (t) (11) for r 2 R, where j(t) = pj r:j2r yr(t) (12) and p j (1) is a positive continuous increasing function, for j 2 J. We interpret the relations (11)–(12) as follows. Suppose that resource j marks a proportion p j (z) of packets with a feedback signal when the total flow through resource j is z; and that user r views each feedback signal as a congestion indicator requiring some reduction in the flow x r. Then, (11) corresponds to a response by user r that comprises two components: a steady increase at rate proportional to w r (t), and a steady decrease at rate proportional to the stream of feedback signals received. It is shown in [13] that if w r (t) = w r for r 2 R then the system of differential equations (11)–(12) has a stable point, to which all trajec-tories converge. The variable j (t) can be viewed as the shadow price per unit of flow through resource j at time t, and at the stable point y r = wr j2r j : (13) The rates y determined by (13) have an interpretation as a set of rates that are proportionally fair per unit charge, as discussed in [12] and [13]. Next, suppose that user s is able to monitor the rates yr(t); r 2 s, continuously, and to vary smoothly the parameters wr(t); r 2 s, so as to satisfy wr(t) = yr(t)U r s (xs(t)): (14) This would correspond to a user who observes a charge per unit flow of r = w r (t)=y r (t) on routes r 2 s, and chooses w r = w r (t); r 2 s, to solve the optimization problem USER s (U s ;). Then, with Cj(y) = y 0 pj(z)dz (15) the objective function of the problem SYSTEM(U; C) is a Lyapunov function for the system of differential equations (11)–(12), (14), and the vector y maximizing the objective function is a stable point of the system, to which all trajectories converge. ACKNOWLEDGMENT The authors wish to thank the editor and referees for their thorough review of earlier drafts of this note. REFERENCES [1] J. Alvarez and B. Hajek, " On the use of packet classes in …