A bus bridge situated between two buses includes two queues: an outbound request queue and an inbound request queue. Requests originating on the first bus which target a destination on the second bus are placed into the outbound request queue. Requests originating on the second bus which target a destination on the first bus are placed into the inbound request queue. A transaction arbitration unit (TAU) within the bridge maintains transaction ordering and avoids deadlocks. The TAU determines whether requests can be placed in the inbound request queue. The TAU also determines whether requests originating on the first bus can be responded to immediately or whether the agent originating the request must wait for a reply. In addition, the TAU includes logic for determining whether a request in the outbound request queue can be executed on the second bus. The TAU determines whether posting to the inbound request queue is enabled or disabled; whether any posted transactions exist in the inbound request queue; and whether ownership of the second bus is available.