  Motivated by a web-server model, we present a queueing network consisting of
two layers. The first layer incorporates the arrival of customers at a network
of two single-server nodes. We assume that the inter-arrival and the service
times have general distributions. Customers are served according to their
arrival order at each node and after finishing their service they can re-enter
at nodes several times (as new customers) for new services. At the second
layer, active servers act as jobs which are served by a single server working
at speed one in a Processor-Sharing fashion. We further assume that the degree
of resource sharing is limited by choice, leading to a Limited
Processor-Sharing discipline. Our main result is a diffusion approximation for
the process describing the number of customers in the system. Assuming a single
bottleneck node and studying the system as it approaches heavy traffic, we
prove a state-space collapse property. The key to derive this property is to
study the model at the second layer and to prove a diffusion limit theorem,
which yields an explicit approximation for the customers in the system.
