  The basic adjoint relationship (BAR) approach is an analysis technique based
on the stationary equation of a Markov process. This approach was introduced to
study heavy-traffic, steady-state convergence of generalized Jackson networks
in which each service station has a single job class. We extend it to
multiclass queueing networks operating under static-buffer-priority (SBP)
service disciplines. Our extension makes a connection with Palm distributions
that allows one to attack a difficulty arising from queue-length truncation,
which appears to be unavoidable in the multiclass setting.
  For multiclass queueing networks operating under SBP service disciplines, our
BAR approach provides an alternative to the "interchange of limits" approach
that has dominated the literature in the last twenty years. The BAR approach
can produce sharp results and allows one to establish steady-state convergence
under three additional conditions: stability, state space collapse (SSC) and a
certain matrix being "tight." These three conditions do not appear to depend on
the interarrival and service-time distributions beyond their means, and their
verification can be studied as three separate modules. In particular, they can
be studied in a simpler, continuous-time Markov chain setting when all
distributions are exponential.
  As an example, these three conditions are shown to hold in reentrant lines
operating under last-buffer-first-serve discipline. In a two-station,
five-class reentrant line, under the heavy-traffic condition, the tight-matrix
condition implies both the stability condition and the SSC condition. Whether
such a relationship holds generally is an open problem.
