I A Closer Look into Theory.- 1. Markov Jump Processes.- 1. Definition and Construction.- 2. Kolmogorov Differential Equations.- 3. The Homogeneous Case and an Approximation.- 4. Quasi-Commutability of Generators.- 5. Periodic Markov Jump Processes.- 5.1 Transient distributions.- 5.2 Asymptotic distributions.- 2. Markov-Additive Jump Processes.- 1. Definition and Transition Probabilities.- 2. Elementary Properties.- 3. MAJPs on a Real Vector Space.- 4. Discrete Markovian Arrival Processes.- 5. Laws of Large Numbers.- II Classical Queues.- 3. Examples of Markovian Arrival Processes.- 1. Batch Markovian Arrival Processes.- 1.1 The homogeneous case.- 1.2 The general case.- 2. Fluid Markovian Arrival Processes.- 2.1 The homogeneous case.- 2.2 The general case.- 4. The Periodic BMAP/PH/C Queue.- 1. Transition Probabilities.- 2. Stability and Asymptotic Distributions.- 5. The BMAP/G/? Queue.- 1. The Homogeneous Case.- 2. Asymptotic Stability.- 3. The General Case.- 4. An Approximation.- 5. Bounds for the BMAP/G/c/c Loss System.- 6. Model Fitting For Homogeneous BMAPs.- 1. An EM algorithm for MAPs and BMAPs.- 1.1 Complete sample case for MAPs.- 1.2 EM for MAPs.- 1.3 EM for BMAPs.- 2. A simpler estimation procedure.- 2.1 Estimating the Matrix D0.- 2.2 Phases at Arrival Instants.- 2.3 Estimating the Matrices Dn for n ? 1.- 3. Numerical results.- III Spatial Queues.- 7. Spatial Markovian Arrival Processes.- 1. Stochastic Point Fields.- 1.1 Definition and construction.- 1.2 Examples.- 2. Definition and Construction of SMAPs.- 3. The homogeneous case.- 4. The general case.- 5. Examples.- 8. The SMAP/MT/C/C Queue.- 1. Transition Probabilities.- 2. Asymptotic Distributions.- 3. A Loss Formula.- 9. Spatial Queues with Infinitely Many Servers.- 1. Homogeneous Arrival Rates without User Movements.- 1.1 One-dimensional marginal distributions.- 1.2 Joint distribution in finitely many subsets.- 1.3 Asymptotic Stability.- 2. General Arrival Rates without User Movements.- 2.1 Transient Distribution.- 2.2 An Approximation.- 3. General Arrival Rates with User Movements.- 4. Application: Planning a Mobile Communication Network.- 10. Model Fitting for a Class of SMAPs.- 1. Arrival Rates.- 2. Phases at Arrival Instants.- 3. Generator Matrix.- References.
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.