We present an optimal and scalable permutation routing algorithm for three reconfigurable models based on linear arrays that allow pipelining of information through an optical bus. Specifically, for any $P\le N$, our algorithm routes any permutation of $N$ elements on a $P$-processor model in $O(N/P)$ steps. This algorithm extends naturally to one for routing $h$-relations optimally in $O(h)$ steps. These results hold much significance toward designing scaling simulations for these models.
We also establish the equivalence of the three models, LARPBS, LPB, and POB. This implies an automatic translation of algorithms (without loss of speed or efficiency) among these models. In particular, since the LPB is the same as an LARPBS without the ability to segment its buses, their equivalence establishes reconfigurable delays (rather than segmenting ability) as the key to the power of optically pipelined models.