  It has been observed in recent works that, for several classes of linear
time-invariant time-delay systems of retarded or neutral type with a single
delay, if a root of its characteristic equation attains its maximal
multiplicity, then this root is the rightmost spectral value, and hence it
determines the exponential behavior of the system, a property usually referred
to as multiplicity-induced-dominancy (MID). In this paper, we investigate the
MID property for one of the simplest cases of systems with two delays, a scalar
delay-differential equation of first order with two delayed terms of order
zero. We discuss the standard approach based on the argument principle for
establishing the MID property for single-delay systems and some of its
limitations in the case of our simple system with two delays, before proposing
a technique based on crossing imaginary roots that allows to conclude that the
MID property holds in our setting.
