A necessary and sufficient condition for the construction of 2-to-1 optical FIFO multiplexers by a single crossbar switch and fiber delay lines

Abstract

In this paper, we prove a necessary and sufficient condition for the construction of 2-to-1 optical buffered first-in-first-out (FIFO) multiplexers by a single crossbar switch and fiber delay lines. We consider a feedback system consisting of an (M + 2) × (M + 2) crossbar switch and M fiber delay lines with delays d1,d2,...,dM. These M fiber delay lines are connected from M outputs of the crossbar switch back to M inputs of the switch, leaving two inputs (respectively, two outputs) of the switch for the two inputs (respectively, two outputs) of the 2-to-1 multiplexer. The main contribution of this paper is the formal proof that d1 = 1 and d i ≤ di+1≤ 2di, i = 1,2,..., M - 1, is a necessary and sufficient condition on the delays d1,d 2,..., dM for such a feedback system to be operated as a 2-to-1 FIFO multiplexer with buffer ∑i=1M d i under a simple packet routing policy. Specifically, the routing of a packet is according to a specific decomposition of the packet delay, called the C-transform in this paper. Our result shows that under such a feedback architecture a 2-to-1 FIFO multiplexer can be constructed with M = O(log B), where B is the buffer size. Therefore, our construction improves on a more complicated construction recently proposed by Sarwate and Anantharam that requires M = O(√B) under the same feedback architecture (we note that their design is more general and works for priority queues). © 2006 IEEE.