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This paper presents an efficient solution technique for the steady-state analysis of the second-order stochastic fluid model underlying a second-order fluid stochastic Petri net (FSPN) with constant flow and transition rates, and a single bounded fluid place. The new solution technique is an extension of existing solution techniques developed for (first-order) Fluid Models; the solution algorithm uses upwind semi-discretization and Matrix Geometric techniques to efficiently compute the steady-state probabilities of the mixed discrete and continuous state space of the model. The effectiveness of our technique is proven first on a simple producer-consumer second-order FSPN model and then on a complex example taken from the literature where the analysis of the completion time distribution and the packet loss probability of short-lived TCP connections is investigated through the decomposition of a model for the whole system into several (simpler) sub-models; the interaction between the different submodels is handled by iterating their solution until the complete model solution converges according to a fixed point algorithm. The introduction of a second-order FSPN in a fixed point iteration scheme has been made possible thanks to the efficiency of the proposed solution technique.