Descriptors for point processses based on runs: The Markovian arrival process

This paper is part of a broader investigation of properties of a point process that can be identified by imagining that the process is involved in a competition for the generation of runs of events. The general purpose of that methodology is to quantify the prevalence of gaps and bursts in realizations of the process. The Markovian arrival process (MAP) is highly versatile in qualitative behavior and its analysis is numerically tractable by matrix-analytic methods. It can therefore serve as well as a benchmark process in that investigation. This paper considers the MAP and a regular grid competing for runs of lengths at least r1 and rÅ2, respectively. A run of length r in one of the processes is defined as a string of r successive events occurring without an intervening event in the other process.