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Motivated by communication networks, we study an admission control problem for a Markovian loss system comprised of two finite capacity service stations in tandem. Customers arrive to station 1 according to a Poisson process, and a gatekeeper, who has complete knowledge of the number of customers at both stations, decides whether to accept or reject each arriving customer. If a customer is rejected, a rejection cost is incurred. If an admitted customer finds that station 2 is full at the time of his service completion at station 1, he leaves the system and a loss cost is incurred. The goal is to find easy-to-implement policies that minimize long-run average cost per unit time. We formulate two intuitive, extremal policies and provide analytical results on their performances. We also present necessary and/or sufficient conditions under which each of these policies is optimal. Next, we show that for some states of the system it is always optimal to admit new arrivals. We also fully characterize the optimal policy when the capacity of each station is two and discuss some characteristics of optimal policies in general. Finally, we design heuristic admission control policies using these insights. Numerical experiments indicate that these heuristic policies yield near-optimal long-run average cost performance. [ABSTRACT FROM AUTHOR]

A large number of stochastic networks including loss networks and certain queueing networks have product-form steady-state probabilities. However, for most practical networks, evaluating the system performance is a difficult task due to the presence of a normalization constant. We propose a new framework based on probabilistic graphical models to tackle this task. Specifically, we use factor graphs to model the stationary distribution of a network. For networks with arbitrary topology, we can apply efficient message-passing algorithms like the sum-product algorithm to compute the exact or approximate marginal distributions of all state variables and related performance measures such as blocking probabilities. Through extensive numerical experiments, we show that the sum-product algorithm returns very accurate blocking probabilities and greatly outperforms the reduced load approximation for loss networks with a variety of topologies. The factor graph model also provides a promising approach for analyzing product-form queueing networks. [ABSTRACT FROM AUTHOR]

We survey a new approach that the author and his co-workers have developed to formulate stochastic control problems (predominantly queueing systems) as mathematical programming problems. The central idea is to characterize the region of achievable performance in a stochastic control problem, i.e., find linear or nonlinear constraints on the performance vectors that all policies satisfy. We present linear and nonlinear relaxations of the performance space for the following problems: Indexable systems (multiclass single station queues and multiarmed bandit problems), restless bandit problems, polling systems, multiclass queueing and loss networks. These relaxations lead to bounds on the performance of an optimal policy. Using information from the relaxations we construct heuristic nearly optimal policies. The theme in the paper is the thesis that better formulations lead to deeper understanding and better solution methods. Overall the proposed approach for stochastic control problems parallels efforts of the mathematical programming community in the last twenty years to develop sharper formulations (polyhedral combinatorics and more recently nonlinear relaxations) and leads to new insights ranging from a complete characterization and new algorithms for indexable systems to tight lower bounds and nearly optimal algorithms for restless bandit problems, polling systems, multiclass queueing and loss networks. [ABSTRACT FROM AUTHOR]