A Markov-additive process (MAP) of arrivals is a process (X,J) on the state space ℝr×E such that the increments in X correspond to arrivals. A typical example is that of different classes of arrivals at a queueing system. In this chapter, the authors investigate the lack of memory property, interarrival times and moments of the number of counts. They then consider transformations of the process that preserve the Markov-additive property, such as linear transformations, patching of independent processes, and linear combinations. Random time transformations are also investigated. Finally, the authors consider secondary recordings that generate new arrival processes from the original process. These include, in particular, marking, coloring, and thinning. For Markov-Bernoulli recording, the secondary process, in each case, turns out to be an MAP or arrivals.
