We study a variational problem (VP) that is related to semimartingale reflecting Brownian motions (SRBMs). Specifically, this VP appears in the large deviations analysis of the stationary distribution of SRBMs in the d-dimensional orthant. When d=2, we provide an explicit analytical solution to the VP. This solution gives an appealing characterization of the optimal path to a given point in the quadrant and also provides an explicit expression for the optimal value of the VP. For each boundary of the quadrant, we construct a ‘‘cone of boundary influence,’’ which determines the nature of optimal paths in different regions of the quadrant. In addition to providing a complete solution in the 2-dimensional case, our analysis provides several results which may be used in analyzing the VP in higher dimensions and more general state spaces.
In this paper we investigate the stability of a class of two-station multiclass fluid networks with proportional routing. We are able to obtain explicit necessary and sufficient conditions for the global stability of such networks. By virtue of a stability theorem of Dai, these results also give sufficient conditions for the stability of a class of related multiclass queueing networks. Our study extends the results of Dai and VandeVate, who provided a similar analysis for fluid models without proportional routing, which arise from queueing networks with deterministic routing. The models we investigate include fluid models which arise from a large class of two-station queueing networks with probabilistic routing. The stability conditions derived turn out to have an appealing intuitive interpretation in terms of virtual stations and push-starts which were introduced in earlier work on multiclass networks.
This paper explores the global stability region of a three-station fluid network. We prove that 1. A piecewise linear Lyapunov function characterizes the monotone global stability region of the three station network; 2. The global stability region of fluid networks with more than two stations may not be monotone in terms of the service time vector; 3. The linear program proposed by Bertsimas, Gamarnik and Tsitsiklis (1995) does not characterize the (monotone) global stability region of our three-station network; 4. The global stability region of our three-station network is not the intersection of its stability regions under the static buffer priority disciplines. Key words : stability, fluid models, multiclass queueing networks, piecewise linear Lyapunov functions, monotone global stability.
In this paper, we obtain necessary conditions for the global stability of a d-station multiclass queueing network. The conditions are given explicitly in terms of the average service and arrival rates of the network. Although these conditions are in general not sufficient for d > 2, they may still highlight hidden bottlenecks in complex manufacturing systems such as wafer fabrication processes. Key words : multiclass queueing networks, stability, virtual station