Abstract
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 well as a benchmark process in that investigation. In this paper, we consider the MAP and a regular grid competing for runs of lengths at least