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Description:
Queueing networks with product-form steady-state distribution have found many fields of applications, e.g. production systems, telecommunications, and computer system modeling. The success of this class of models and its relatives stems from the simple structure of the steady-state distribution which provides access to easy performance evaluation procedures. Starting from the work of Jackson [Jac57] various generalizations have been developed. In real world queueing systems are not isolated and interact with their environment. Adding a random environment to a model usually makes the model more realistic but also more complex to analyze. Nevertheless, under some conditions it is still possible to obtain analytical results. A branch of research which recently has found interest are queueing networks in a random environment with product form steady-state distributions. The main theoretical contributions of this thesis are twofold: (i) We develope a general theory that comprise models with stationary product-form distribution in inventory theory in [Sch04] and Jackson networks with unreliable nodes with stationary product-form distribution in [Sau06]. An important property of the resulting general model is that the queueing system and the environment interact in both directions: the queues can influence the environment and the environment can influences the queues. (ii) With respect to applications we show how different models known from literature can be interpreted in terms of the general theory, construct new models in various applications, and develope an approximation method. In Part I we analyze single-queue systems. In Section 1 we introduce a loss system. In Section 2 we generalize product form lost-sales inventory models from [Sch04] and several other published papers with related models as a loss system with exponential service time. The term loss means that customers get lost when the environment stays in some special states -- the blocking states. In Section 2.1.4 we develop an approximation method for ...