We examine the question whether the memoryless property associated with the geometric and exponential distributions remains true for the overshoot functionals of almost upper semicontinuous integer- and real-valued processes on a Markov chain. It is established that under the condition of a level attainability and knowing the environment state at the moment of reaching the level this property only holds for overshoot through the level x?0 and its distribution depends neither on the overshoot moment noron x. The undershoot distribution of a level x is determined in terms of the zero-level undershoot distribution. A similar dependence is established for a jump that covers the level x.
