  In this work we study the problem of asymptotically optimal control of a
well-known multi-class queuing network, referred to as the ``crisscross
network,'' in heavy traffic. We consider exponential inter-arrival and service
times, linear holding cost and an infinite horizon discounted cost criterion.
In a suitable parameter regime, this problem has been studied in detail by
Martins, Shreve and Soner [SIAM J. Control Optim. 34 (1996) 2133-2171] using
viscosity solution methods. In this work, using the pathwise solution of the
Brownian control problem, we present an elementary and transparent treatment of
the problem (with the identical parameter regime as in [SIAM J. Control Optim.
34 (1996) 2133-2171]) using large deviation ideas introduced in [Ann. Appl.
Probab. 10 (2000) 75-103, Ann. Appl. Probab. 11 (2001) 608-649]. We obtain an
asymptotically optimal scheduling policy which is of threshold type. The proof
is of independent interest since it is one of the few results which gives the
asymptotic optimality of a control policy for a network with a more than
one-dimensional workload process.
