Constructions of Optical Queues With a Limited Number of Recirculations--Part I: Greedy Constructions

One of the main problems in all-optical packet-switched networks is the lack of optical buffers, and one feasible technology for the constructions of optical buffers is to use optical crossbar Switches and fiber Delay Lines (SDL). In this two-part paper, we consider SDL constructions of optical queues with a limited number of recirculations through the optical switches and the fiber delay lines. Such a problem arises from practical feasibility considerations. We show that the constructions of certain types of optical queues, including linear compressors, linear decompressors, and 2-to-1 FIFO multiplexers, under a simple packet routing scheme and under the constraint of a limited number of recirculations can be transformed into equivalent integer representation problems under a corresponding constraint. Specifically, we show that the effective maximum delay of a linear compressor/decompressor and the effective buffer size of a 2-to-1 FIFO multiplexer in our constructions are equal to the maximum representable integer B(d 1 ; k) with respect to d 1 and k (defined in (4) in Section I), where d 1 = (d1, d2, . . . , dM ) is the sequence of the delays of the M fibers used in our constructions and k is the maximum number of times that a packet can be routed through the M fibers. Given M and k, therefore, the problem of finding an optimal construction, in the sense of maximizing the maximum delay (resp., buffer size), among our constructions of linear compressors/decompressors (resp., 2-to-1 FIFO multiplexers) is equivalent to the problem of finding an optimal sequence d∗ 1 in AM (resp., BM ) such that B(d M 1 ; k) = maxdM 1 ∈AM B(d 1 ; k) (resp., B(d∗ 1 ; k) = maxdM 1 ∈BM B(d 1 ; k)), where AM (resp., BM ) is the set of all sequences of fiber delays allowed in our constructions of linear compressors/decompressors (resp., 2-to-1 FIFO multiplexers). In Part I, we propose a class of greedy constructions of linear compressors/decompressors and 2-to-1 FIFO multiplexers by specifying a class GM,k of sequences such that GM,k ⊆ BM ⊆ AM and each sequence in GM,k is obtained recursively in a greedy manner. For d 1 ∈ GM,k, we obtain an explicit recursive expression for di in terms of d1, d2, . . . , di−1 for i = 1, 2, . . . ,M , and obtain an explicit expression for the maximum representable integer B(d 1 ; k) in terms of d1, d2, . . . , dM . We then use these expressions to show that every optimal construction must be a greedy construction. In Part II, we further show that there are at most two optimal constructions and give a simple algorithm to obtain the optimal construction(s). Index Terms FIFO multiplexers, integer representation, linear compressors, linear decompressors, maximum representable integer, optical buffers, optical queues, packet switching, routing. This paper was presented in part at the IEEE International Conference on Computer Communications (INFOCOM’08), Phoenix, AZ, USA, April 13–18, 2008. The authors are with the Department of Electrical Engineering and the Institute of Communications Engineering, National Tsing Hua University, Hsinchu 30013, Taiwan, R.O.C. (e-mails: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected]).