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In this paper, we consider the occupancy distribution for an open network of infinite server queues with multivariate batch arrivals following a non-homogeneous Poisson process, and general service time distributions. We derive a probability generating function for the transient occupancy distribution of the network and prove that it is necessary and sufficient for ergodicity that the expected occupancy time for each batch be finite. Further, we recover recurrence relations for the transient probability mass function formulated in terms of a distribution obtained by compounding the batch size with a multinomial distribution. [ABSTRACT FROM AUTHOR]

In this paper we introduce a structure called the Markovian tree (MT). We define the MT and explore its alternative representation as a continuous-time Markovian Multitype Branching Process. We then develop two algorithms, the Depth and Order algorithms to determine the probability of eventual extinction of the MT process. We show that both of these algorithms have very natural physically intuitive interpretations and are analogues of the Neuts and U algorithms in Matrix-analytic Methods. Furthermore, we show that a special case of the Depth algorithm sheds new light on the interpretation of the sample paths of the Neuts algorithm. [ABSTRACT FROM AUTHOR]

We consider several known algorithms and introduce some new algorithms that can be used to calculate the probability of return to the initial level in the Markov stochastic fluid flow model. We give the physical interpretations of these algorithim within the fluid flow environment. The rates of convergence are explained in terms of the physical properties of the fluid flow processes. We compare these algorithms with respect to the numbers of iterations required and their complexity. The performance of the algorithms depends on the nature of the process considered in the analysis. We illustrate this with examples and give appropriate recommendations. [ABSTRACT FROM AUTHOR]

The existence of product form solutions in networks of queues has been known for many years. Recently the authors have shown that there is a class of stochastic Petri nets (SPN's), distinct from queueing networks, which also have a product form equilibrium distribution. Most practical SPN's, however, do not fall within this class. In this paper the class of SPN's which can be analyzed using these methods is extended in several directions. Using embedded discrete time processes we study a class of SPN's which have a closed form equilibrium distribution. These SPN's have probabilistic output bags, colored tokens, and alternating periods of arbitrarily distributed enabling and firing times (periods of time between transitions becoming enabled and absorption of tokens and between transitions absorbing tokens and depositing them in output places, respectively.) In addition an aggregation procedure is proposed which, in certain nets, not only reduces a complex SPN to a much simpler skeleton SPN but obtains results for the skeleton SPN which are exact marginal distributions for the original SPN. [ABSTRACT FROM AUTHOR]