Emerging applications in areas such as advanced manufacturing and healthcare systems demand predictable delivery of prioritized message streams over multihop wireless systems. The dominance protocol is an important approach in which transmitting nodes use priority identifiers to dynamically regulate access to the shared medium. This protocol, however, has an unresolved problem called the Multihop Competing Problem (MCP). Despite recognizing that the problem could occur, the current literature offers no characterization of the problem or when it can occur. Because MCP could occur, it was not possible to analytically determine the end-to-end latency bounds of the message streams. When MCP occurred, the end-to-end latency of the message streams was affected adversely. We characterize MCP and show that it is an exposed terminal problem that occurs when the nodes use certain priority identifiers in adjacent two-hop neighborhoods of a transmitting node. Using a random disk graph model, we show that MCP occurs often in such systems. We utilize our characterization of MCP to derive priority identifiers that guarantee that MCP will not occur. The MCP free labels propose enable the predictable delivery of multihop message streams, improve end-to-end latency of the streams, and improve throughput in well-engineered and ad hoc networks.