  In this paper we introduce the notion of a patient first-mean stable system.
Such systems are switched systems that are first-mean stable meaning that they
converge to a globally attracting fixed point on average. They are also patient
so that they do not lose their first-mean stability when time-delays are
introduced into the system. As time-delays are, in general, a source of
instability and poor performance patient first-mean stability is a much
stronger condition than first-mean stability. This notion of patient stability
allows one to design systems that cannot be destabilized via time-delays. It
also significantly reduces the difficulty of modeling such systems since in
patient systems time-delays can, to a large extent, be safely ignored. The
paper's main focus is on giving a sufficient criteria under which a system is
patient first-mean stable and we give a number of examples that demonstrate the
simplicity of this criteria.
