From Markov Jump Processes to Spatial Queues aims to develop a
unified theory of spatial queues that yields concrete results for the
performance analysis of mobile communication networks. A particular
objective is to develop the most natural generalization of existing
concepts (e.g. the BMAP) toward the needs of mobile communication
networks. To these belong the spatial distribution of batch arrivals and
users in the system as well as time-inhomogeneous (e.g. periodic)
arrival intensities and user movements.
One of the major recent challenges for the stochastic modelling of
communication systems is the emergence of wireless networks, which are
used by more and more subscribers today. The main new feature of those,
which is not covered by classical queuing theory, clearly is the
importance of the user location within the area that is served by the
base stations of the network.
In the framework of queuing theory, this opens up the natural extension
of classical queuing models towards queues with a structured space in
which users are served. The present book is intended to introduce this
extension under the name of spatial queues. The main point of view and
the general approach will be that of Markov jump processes. We start
with a closer look into the theory. Then we present new results for the
theory of stochastic processes as well as for classical queuing theory.
Finally we introduce the new concepts of spatial Markovian arrival
processes and spatial queues.
The main text is divided into three parts. The first part provides a new
presentation of the theory of Markov jump processes. We derive a number
of new results, especially for time-inhomogeneous processes, which have
been neglected too much in the current textbooks on stochastic
processes. For the first time, the class of Markov-additive jump
processes is analysed in detail. This extends and unifies all Markovian
arrival processes that have been proposed up to now (including arrivals
for fluid queues) and provides a foundation for the subsequent
introduction of spatial Markovian arrival processes.
The second part contains new results for classical queues with BMAP
input. These include the first explicit formulae for the distribution of
periodic queues. The class of fluid Markovian arrival processes is
introduced, and we give statistical estimates for the parameters of a
BMAP.
In the third part, the concepts of spatial Markovian arrival processes
(abbreviated: SMAPs) and spatial queues are introduced. After that,
periodic spatial Markovian queues are analysed as a model for the cells
of a wireless communication network.
From Markov Jump Processes to Spatial Queues is intended to reach
queuing theorists, researchers in the field of communication systems, as
well as engineers with some background in probability theory.
Furthermore, it is suitable as a textbook for advanced queuing theory on
the graduate or post-graduate level.
