  This paper is a comprehensive study of a long observed phenomenon of increase
in the stability margin and so the rate of convergence of a class of linear
systems due to time delay. We use Lambert W function to determine (a) in what
systems the delay can lead to increase in the rate of convergence, (b) the
exact range of time delay for which the rate of convergence is greater than
that of the delay free system, and (c) an estimate on the value of the delay
that leads to the maximum rate of convergence. For the special case when the
system matrix eigenvalues are all negative real numbers, we expand our results
to show that the rate of convergence in the presence of delay depends only on
the eigenvalues with minimum and maximum real parts. Moreover, we determine the
exact value of the maximum rate of convergence and the corresponding maximizing
time delay. We demonstrate our results through a numerical example on the
practical application in accelerating an agreement algorithm for
networked~systems by use of a delayed feedback.
