  It is the central goal of our studies to describe parton fragmentation in the
hot and dense medium of a quark gluon plasma (QGP). Under the assumption that
the medium is not static and homogeneous, knowledge about the temporal
evolution of the processes involved can be of essential importance. Therefore,
parton fragmentation has been studied with a Monte-Carlo algorithm that
approximates the DGLAP-evolution of fragmentation functions via a set of parton
cascades. The presented work consists mainly of implementations of this kind of
algorithm and the application of a simple approximation, which gives the time
development of partonic cascades. As approximations to the parton-splitting
processes of gluons and massless quarks in the vacuum two different schemes
with the same leading-log contributions are investigated. Furthermore, the
temporal evolutions of quantities related to partonic cascades have been
examined: To our understanding, especially variables (like, e.g.: parton
multiplicities or virtualities) that give insight, when parton splittings take
place, are of great interest, as they allow identifying the parts of the
fragmentation process that will be most affected by medium interactions.
