
The use of approximate models of queueing networks has become a common place in the queueing
literature over the last decades. In this context, the use of heavy-traffic diffusion
approximations for multiclass queueing networks has proven to be a practical way to analyze and
study performance measures that are otherwise intractable in the original network.

Early work on the literature focused on the formulation and justification of the diffusion limits
under the classical heavy-traffic regime. Reiman's seminar paper \cite{Reiman:84} prove that, in
the context of open-single-class queueing-networks, as the traffic intensity approaches one, the
normalized vector queue length processes converges in distribution to a Reflected Brownian motion
(RBM) on the positive orthant, $S = \mathbb{R}^d_+$. This diffusion was first constructed in
earlier work of Harrison and Reiman [X]. On the interior of $S$ this process, denoted by $Z
=\set{Z(t),t\geq 0}$, behaves like an ordinary $d$-dimensional Brownian motion with drift $\mu$ and
covariance matrix $\Gamma$, and on each of the boundaries of $S$ it is instantaneously reflected
back to $S$ in an specific direction. Similar heavy-traffic limit results had been proven for
various types of open d-station networks with possibly multiple customer types [X,X,X].

Another type of heavy-traffic regime was introduced by Halfin and Whitt \cite{Halfin-Whitt:81} in
the context of a $M/M/C$ queue when letting the $C$ grow large and concurrently letting the traffic
intensity approach 1 at an appropriate rate. Under this regime, the number of users in the system
can be expressed roughly as $C+\sqrt{C}X(t)$ for $C$ large enough, where $X(t)$ is a simple
dimensional diffusion process. There exists an extensive literature providing similar results for
many-servers multiple customer types systems  [X,X,X]. The interest in the Halfin-Whitt regime
largely stems from its ability to succinctly capture the natural statistical economies scale that
are present in many large capacity service system.

The aim of this early work was to provide approximate models with the hope that the dynamics of the
original network were close to the ones of the limiting process, and therefore been able to
approximate queue length, waiting time and workload processes using the limiting diffusions. In
practice the approach has been to use the stationary distribution of the diffusion model to
approximate the one corresponding to the original network in order to compute performance measures.

The validity of this approach remains an outstanding problem for current research. It is important
to notice that this approach is not rigourously justified, and does not follow from the
heavy-traffic approximation results. To date, the validity of the approach has not been established
in full generality. In this context, the work of Gamarnik and Zeevi \cite{Gamarnik-Zeevi:06} shows
that, under certain conditions on the arrival and service processes, the sequence of stationary
distributions generated under the classical heavy-traffic regime converges to the stationary
distribution of the limiting diffusion for the case of generalized Jackson networks. The work of of
Gurvich and Zeevi \cite{Gurvich-Zeevi} extend this result to the general open networks case. In
general, there is a large amount of evidence pointing out the validity of the approach [X,X,X,X].

The potential benefits coming from using an approximate model should come from the ability to
``easily'' compute interesting performance measures, in comparison to computing them for the
original and more intractable system. However this is not usually the case: for the classical
heavy-traffic regime, only RBMs whose data satisfy a certain skew symmetric condition in Harrison
and Williams [X] have explicit formulas for stationary distributions. In generality, it is very
unlikely that analytical solutions will ever be found. The same holds for other types of
heavy-traffic regime. This arises the problem of computing these stationary distributions.
Conveniently, as we will see, they can be characterized as a solution to a highly structured
partial differential equation problems known as the Basic Adjoint Relationship (BAR). The form of
the BAR condition will depend explicitly on the type of heavy-traffic considered. Dai and Harrison
\cite{Dai:90} had used this BAR condition to propose an algorithm for the RBM case. There, the
authors view the BAR as an orthogonality condition between a infinite dimensional functional space
and the stationary distribution. By consider an increasing sequence of finite dimensional
approximation of the infinite dimensional functional space they obtain sequence of approximating
densities by taking orthogonal projections in to the increasing sequence of finite dimensional
spaces. By making a conjecture they prove this sequence of densities to converge to the proper
stationary density.
