This paper addresses the design of a statistical multiplexer as customer-premises equipment to concentrate data from multiple input ports for access to a virtual-circuit packet network based on the LAPD protocol. It is assumed that the multiplexer will be adjacent to the front-end processor of hosts employing synchronous (and, possibly, asynchronous) data link layer protocols, with emphasis on SDLC. With an overall objective of reducing end-to-end delay, we propose and analyze a priority queue solution with three queues. In this scheme, polls from the synchronous links go to the highest-priority queue (unless messages to the same cluster controller are in the queue); priority between the other two queues is set by the urgency of the application. The priority of asynchronous traffic is based on message size and favors short messages, such as echoplex. Numerical results show that giving polls priority reduces end-to-end delay significantly — even for applications with low urgency. This is because of the reduction in polling overhead, which is valid for all messages.