We consider Markov additive processes that are skip-free in one direction at least. M/G/1-type Markov chains and spectrally-positive Markov-modulated Lévy processes are skip-free in the negative direction. These processes only allow jumps in the positive direction and either continuous downward movement or discrete downward steps of size one.

In contrast, GI/M/1-type queues and spectrally-negative Markov-modulated Lévy processes are skip-free in the positive direction. These processes allow only downward jumps, with continuous upward movement or discrete upward steps of size one.

Meanwhile, quasi-birth-and-death (QBD) processes present a structure where transitions are limited to adjacent levels, which allows them to be skip-free in both directions under certain conditions. This structure makes QBDs a versatile tool in modeling systems with layered or hierarchical dynamics.

Other related examples include fluid queues and Markov-modulated Brownian motion. These models also exhibit skip-free properties and are widely used to represent systems with continuous flow dynamics modulated by a Markov process.