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 shows that the combination of a symmetric Jacobian
A
and condition p(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 technical note, 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 ||A|| < 1 for any monotonic induced matrix norm ||.||, 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 , 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 with arbitrary parameters alpha
_i
and beta
_i
.