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Spectral properties of the tandem Jackson network, seen as a quasi-birth-and-death process. (English) Zbl 1078.60078
Matrix geometrical methods for quasi-birth-death processes and more general systems usually rely on having a structured state space of essentially two dimensions. One dimension is the level of the process, the other dimension represents the phase of the system usually considered as a subclassification of the levels. Standard is to have the level space infinite while the phase space is usually finite. For this situation the steady state analysis and even the speed of convergence to equilibrium are (more or less) well understood.
It is shown that there occur strange phenomena when the phase space is infinite as well. These are discussed mostly by exploiting the behavior of a two-stage (exponential) ergodic tandem system, where the queue length of the second server is the level of the system, while the queue length of the first server is considered to be the phase. Although this a queueing network which is thought to be well understood, the authors show that there are details of the behavior, that become appearent only when considering the system in the light of the quasi-birth-death formalism. E.g., truncating the first queue length (loss system) and defining in a natural way an infinite sequence of approximating systems poses problems with the interchange of limiting behavior of the decay rates for the stationary probabilities.

##### MSC:
 60K25 Queueing theory (aspects of probability theory) 60J27 Continuous-time Markov processes on discrete state spaces
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