Parallel machines have the potential to satisfy the large computational demands of emerging real-time applications. These applications require a predictable communication network, where time-constrained traffic requires bounds on latency or throughput while good average performance suffices for best-effort packets. This paper presents a router architecture that tailors low-level routing, switching, arbitration and flow-control policies to the conflicting demands of each traffic class. The router implements deadline-based scheduling, with packet switching and table-driven multicast routing, to bound end-to-end delay for time-constrained traffic, while allowing best-effort traffic to capitalize on the low-latency routing and switching schemes common in modern parallel machines. To limit the cost of servicing time-constrained traffic, the router shares packet buffers and link-scheduling logic between the multiple output ports. Verilog simulations demonstrate that the design meets the performance goals of both traffic classes in a single-chip solution.