On Delay-Independent Diagonal Stability of Max-Min Congestion Control

Network feedback in a congestion-control system is subject to delay, which can significantly affect stability and performance of the entire system. While most existing stability conditions explicitly depend on delay D_i of individual flow i, a recent study in Zhang, Y. et al, (2004) shows that the combination of a symmetric Jacobian A and condition rho(A) < 1 guarantees local stability of the system regardless of D_i. However, the requirement of symmetry is very conservative and no further results have been obtained beyond this point. In this paper, we proceed in this direction and gain a better understanding of conditions under which congestion-control systems can achieve delay-independent stability. Towards this end, we first prove that if Jacobian matrix A satisfies parApar < 1 for any monotonic induced matrix norm parmiddotpar, the system is locally stable under arbitrary diagonal delay D_i. We then derive a more generic result and prove that delay-independent stability is guaranteed as long as A is Schur diagonally stable in Kaszkurewicz, E. and Bhaya, A. (2000), which is also observed to be a necessary condition in simulations. Utilizing these results, we identify several classes of well-known matrices that are stable under diagonal delays if rho(A) < 1 and prove stability of MKC in Zhang, Y. et al, (2004) with arbitrary parameters alpha_i and beta_i