Path switching over multirate Benes network.

Mui Sze Wai.Thesis (M.Phil.)--Chinese University of Hong Kong, 2003.Includes bibliographical references (leaves 62-65).Abstracts in English and Chinese.Chapter 1. --- Introduction --- p.1Chapter 1.1 --- Evolution of Multirate Networks --- p.2Chapter 1.2 --- Some Results from Previous Work --- p.2Chapter 1.3 --- Multirate Traffic on Benes Network --- p.5Chapter 1.4 --- Organization --- p.7Chapter 2. --- Background Knowledge on Benes Network and Path Switching --- p.8Chapter 2.1 --- Benes Network --- p.9Chapter 2.1.1 --- Construction of Large Switching Fabrics --- p.9Chapter 2.1.2 --- Routing in Benes Network --- p.11Chapter 2.1.3 --- Performance when Operated as a Large Switch Fabric --- p.13Chapter 2.2 --- Path Switching --- p.14Chapter 2.2.1 --- Basic Concept of Path Switching --- p.14Chapter 2.2.2 --- Capacity Allocation and Route Assignment --- p.15Chapter 3. --- Path Switching over Benes Network --- p.20Chapter 3.1 --- The Model of path-switched Benes Network --- p.21Chapter 3.2 --- Module-to-Module Implementation --- p.21Chapter 3.2.1 --- The First Stage (Input Module) --- p.22Chapter 3.2.2 --- The Middle Stage (Central Module) --- p.23Chapter 3.2.3 --- The Last Stage (Output Module) --- p.24Chapter 3.3 --- Port-to-Port Implementation --- p.24Chapter 3.3.1 --- Uniform Traffic --- p.25Chapter 3.3.2 --- Mult irate Traffic --- p.26Chapter 3.4 --- Closing remarks --- p.29Chapter 4. --- Performance Analysis --- p.31Chapter 4.1 --- Traffic Constraints and Perform- ance Guarantees --- p.32Chapter 4.1.1 --- Arrival Curve and Service Curve --- p.33Chapter 4.1.2 --- Delay Bound and Backlog Bound --- p.36Chapter 4.2 --- Service Guarantees --- p.39Chapter 4.3 --- Deterministic Bounds --- p.42Chapter 4.3.1 --- Delay --- p.42Chapter 4.3.2 --- Backlog at Input Module --- p.44Chapter 4.3.3 --- Backlog at Output Module --- p.47Chapter 5. --- Simulation Results --- p.52Chapter 5.1 --- Uniform Traffic --- p.53Chapter 5.2 --- Multirate Traffic --- p.55Chapter 6. --- Conclusions and Future Research --- p.59Chapter 6.1 --- Suggestions for future research --- p.61Bibliography --- p.6

Symmetric rearrangeable networks and algorithms

A class of symmetric rearrangeable nonblocking networks has been considered in this thesis. A particular focus of this thesis is on Benes networks built with 2 x 2 switching elements. Symmetric rearrangeable networks built with larger switching elements have also being considered. New applications of these networks are found in the areas of System on Chip (SoC) and Network on Chip (NoC). Deterministic routing algorithms used in NoC applications suffer low scalability and slow execution time. On the other hand, faster algorithms are blocking and thus limit throughput. This will be an acceptable trade-off for many applications where achieving ”wire speed” on the on-chip network would require extensive optimisation of the attached devices. In this thesis I designed an algorithm that has much lower blocking probabilities than other suboptimal algorithms but a much faster execution time than deterministic routing algorithms. The suboptimal method uses the looping algorithm in its outermost stages and then in the two distinct subnetworks deeper in the switch uses a fast but suboptimal path search method to find available paths. The worst case time complexity of this new routing method is O(NlogN) using a single processor, which matches the best known results reported in the literature. Disruption of the ongoing communications in this class of networks during rearrangements is an open issue. In this thesis I explored a modification of the topology of these networks which gives rise to what is termed as repackable networks. A repackable topology allows rearrangements of paths without intermittently losing connectivity by breaking the existing communication paths momentarily. The repackable network structure proposed in this thesis is efficient in its use of hardware when compared to other proposals in the literature. As most of the deterministic algorithms designed for Benes networks implement a permutation of all inputs to find the routing tags for the requested inputoutput pairs, I proposed a new algorithm that can work for partial permutations. If the network load is defined as ρ, the mean number of active inputs in a partial permutation is, m = ρN, where N is the network size. This new method is based on mapping the network stages into a set of sub-matrices and then determines the routing tags for each pair of requests by populating the cells of the sub-matrices without creating a blocking state. Overall the serial time complexity of this method is O(NlogN) and O(mlogN) where all N inputs are active and with m < N active inputs respectively. With minor modification to the serial algorithm this method can be made to work in the parallel domain. The time complexity of this routing algorithm in a parallel machine with N completely connected processors is O(log^2 N). With m active requests the time complexity goes down to (logmlogN), which is better than the O(log^2 m + logN), reported in the literature for 2^0.5((log^2 -4logN)^0.5-logN)<= ρ <= 1. I also designed multistage symmetric rearrangeable networks using larger switching elements and implement a new routing algorithm for these classes of networks. The network topology and routing algorithms presented in this thesis should allow large scale networks of modest cost, with low setup times and moderate blocking rates, to be constructed. Such switching networks will be required to meet the bandwidth requirements of future communication networks

Multirate Rearrangeable Clos Networks and a Generalized Edge Coloring Problem on Bipartite Graphs

... that the minimum number m(n, r) of middle-state switches for the symmetric 3-stage Clos network C(n, m(n, r), r) to be rearrangeable in the multirate enviroment is at most 2n − 1. This problem is equivalent to a generalized version of the bipartite graph edge coloring problem. The best bounds known so far on the function m(n, r) is 11n/9 ≤ m(n, r) ≤ 41n/16 + O(1), for n, r ≥ 2, derived by Du-Gao-Hwang-Kim (SIAM J. Comput., 28, 1999). In this paper, we make several contributions. Firstly, we give evidence to show that even a stronger result might hold. In particular, we give a coloring algorithm to show that m(n, r) ≤ ⌈(r + 1)n/2⌉, which implies m(n, 2) ≤ ⌈3n/2⌉- stronger than the conjectured value of 2n − 1. Secondly, we derive that m(2, r) = 3 by an elegant argument. Lastly, we improve both the best upper and lower bounds given above: ⌈5n/4 ⌉ ≤ m(n, r) ≤ 2n − 1 + ⌈(r − 1)/2⌉, where the upper bound is an improvement over 41n/16 when r is relatively small compared to n. We also conjecture that m(n, r) ≤ ⌊2n(1 − 1/2 r)⌋