A large deviations approach to asymptotically optimal control of crisscross network in heavy traffic

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.