In Hajek's arrival routing problem, customers are routed to one of n queues to minimize average holding cost. Interarrival and service times are exponentially distributed. We solve the associated fluid model. The optimal fluid policy tells us the asymptotic slopes of the switching surfaces in the original problem when the queues are large. If these slopes are nonzero, then numerical tests indicate that the fluid policy performs well in the original stochastic network. The fluid policy also indicates the approximate path that will be taken to recover from large queues: Routing only switches to queues with larger holding cost and once a large queue empties it will remain approximately empty.