We show that the stationary distribution of a two-dimensional stochastic fluid network with (possibly dependent) Lévy inputs does not have product form other than in truly obvious cases. This is in contrast to queueing networks, where product form exists for non-obvious situations in which the inputs are independent, and for Brownian networks, where it typically exists for cases where the driving processes are actually dependent.

We introduce open stochastic fluid networks that can be regarded as continuous analogues or fluid limits of open networks of infinite-server queues. Random exogenous input may come to any of the queues. At each queue, a c.d.f.-valued stochastic process governs the proportion of the input processed by a given time after arrival. The routeing may be deterministic (a specified sequence of successive queue visits) or proportional, i.e. a stochastic transition matrix may govern the proportion of the output routed from one queue to another. This stochastic fluid network with deterministic c.d.f.s governing processing at the queues arises as the limit of normalized networks of infinite-server queues with batch arrival processes where the batch sizes grow. In this limit, one can think of each particle having an evolution through the network, depending on its time and place of arrival, but otherwise independent of all other particles. A key property associated with this independence is the linearity: the workload associated with a superposition of inputs, each possibly having its own pattern of flow through the network, is simply the sum of the component workloads. As with infinite-server queueing models, the tractability makes the linear stochastic fluid network a natural candidate for approximations.