We prove some new pathwise comparison results for single class stochastic fluid networks. Under fairly general conditions, monotonicity with respect to the (state- and time-dependent) routing matrices is shown. Under more restrictive assumptions, monotonicity with respect to the service rates is shown as well. We conclude by using the comparison results to establish a moment bound, a stability result for stochastic fluid networks with Lévy inputs, and a comparison result for multi-class GPS networks.This research was supported in part by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada (NSERC)

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It has recently been shown that in the heavy traffic limit, the stationary distribution of the scaled queue length process of a Generalized Jackson Network converges to the stationary distribution of its corresponding Reflected Brownian Motion limit. In this paper, we show that this "interchange of limits" is valid for Stochastic Fluid Networks with Lévy inputs. Furthermore, under additional assumptions, we extend the result to show that the interchange is valid for moments of the stationary distribution and for state-dependent routing. The results are obtained using monotonicity and sample-path arguments.

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It has recently been shown [3, 5] that in the heavy traffic limit, the stationary distributions of the scaled queue length process of Generalized Jackson Networks converges to the stationary distribution of its corresponding Reflected Brownian Motion limit. In this paper we show that such an "interchange of limits" is valid for the workload process of Stochastic Fluid Networks with Lévy inputs. Our technique is of independent interest because we do not require the use of any Lyapunov techniques, a method that was used in the previous two papers.