Stochastic processing networks arise commonly from applications in com-
puters, telecommunications, and large manufacturing systems. Study of
stability and control for such networks is an active and important area of
research. In general the networks are too complex for direct analysis and
therefore one seeks tractable approximate models. Heavy traffic limit theory
yields one of the most useful collection of such approximate models. Typical
results in the theory say that, when the network processing resources are
roughly balanced with the system load, one can approximate such systems
by suitable diffusion processes that are constrained to live within certain
polyhedral domains (e.g., positive orthants). Stability and control problems
for such diffusion models are easier to analyze and, once these are resolved,
one can then infer stability properties and construct good control policies
for the original physical networks. In this talk I will consider two related
problems concerning stability and long time control for such networks and
their diffusion approximations.
In the first part of the talk I will consider invariant distributions of an
important subclass of stochastic networks, namely the generalized Jackson
networks (GJN). It is shown that, under natural stability and heavy traffic
conditions, the invariant distributions of GJN converge to unique invariant
probability distribution of the corresponding constrained diusion model.
The result leads to natural methodologies for approximation and simulation
of steady state behavior of such networks. In the second part of the talk I
will consider a rate control problem for stochastic processing networks with
an ergodic cost criterion. It is shown that value functions and near optimal
controls for limit diusion models serve as good approximations for the same
quantities for certain physical networks that are heavily loaded.