The open Jackson queueing network with \(r<\infty\) nodes, M/M/\(\infty\) (or G/M/\(\infty)\) with Poisson (or renewal) external input (with arrival rate \(\lambda_ i)\) in each node is considered. Let \(\| P_{ij}\|\) be the transition matrix, and \(\mu_ i/n\) be the service rate of one channel in the i-th node (n is the parameter of the series). Let \(X_ i^{(n)}(t)\) be the number of engaged channels at time moment t in i-th node, \[ \xi^{(i)}_ n(t)=n^{-1/2}(X_ i^{(n)}(nt)-\alpha_ in), \] where \(\alpha =(\alpha_ 1,...,\alpha_ r)\) is the solution of the equation \(\alpha \Theta =-\lambda \equiv (\lambda_ 1,...,\lambda_ r)\) \((\Theta =\| q_{ij}\|\), and \(q_{ij}=-\mu_ i\) for \(i=j\) and \(=\mu_ ip_{ij}\) for \(i\neq j).\) The convergence of the process \((\xi_ n^{(1)}(t),...,\xi_ n^{(r)}(t))\) to some diffusion process in the uniform topology (as \(n\to \infty)\) is proved.