We study the distribution of steady-state queue lengths in
multiclass queueing networks under a stable policy. We propose a general
methodology based on Lyapunov functions for the performance analysis of
infinite state Markov chains and apply it specifically to Markovian multiclass
queueing networks. We establish a deeper connection between stability and
performance of such networks by showing that if there exist linear or piecewise
linear Lyapunov functions that show stability, then these Lyapunov functions
can be used to establish geometric-type lower and upper bounds on the tail
probabilities, and thus bounds on the expectation of the queue lengths. As an
example of our results, for a reentrant line queueing network with two
processing stations operating under a work-conserving policy, we show that
$\mathrm{E}[L] = O(\frac{1}{1 - \rho^*)^2})$, where $L$ is the total number of
customers in the system, and $\rho^*$ is the maximal actual or virtual traffic
intensity in the network. In a Markovian setting, this extends a recent result
by Dai and Vande Vate, which states that a reentrant line queueing network with
two stations is globally stable if $\rho^*<1$. We also present several
results on the performance of multiclass queueing networks operating under
general Markovian and,in particular,priority policies. The results in this
paper are the first that establish explicit geometric-type upper and lower
bounds on tail probabilities of queue lengths for networks of such generality.
Previous results provide numerical bounds and only on the expectation, not the
distribution, of queue lengths.