  The main subject of the study in this paper is the simultaneous renewal time
for two time-inhomogeneous Markov chains which start with arbitrary initial
distributions. By a simultaneous renewal we mean the first time of joint
hitting the specific set $C$ by both processes. Under the condition of
existence a dominating sequence for both renewal sequences generated by the
chains and non-lattice condition for renewal probabilities an upper bound for
the expectation of the simultaneous renewal time is obtained.
