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We consider fluid models with infinite buffer size. Let \{Z_N(t)\} be the net input rate to the buffer, where \{Z_N(t)\} is a superposition of N homogeneous alternating on-off flows. Under heavy traffic environment \{Z_N (t)\} converges in distribution to a centred Gaussian process with covariance function of a single flow. The aim of this paper is to prove the convergence of the stationary buffer content process \{X^*_N(t)\} in the Nth model to the buffer content process \{X^*(t)\} in the limiting Gaussian model.[PUBLICATION ABSTRACT]