Performance evaluation of deterministic wormhole routing in k-ary n-cubes

Abstract

We present a new analytical approach for the performance evaluation of deterministic wormhole routing in k-ary n-cubes. Our methodology achieves closed formulas for average time values through the analysis of network flows. The comparison with simulation models demonstrates that our methodology gives accurate results for both low and high traffic conditions. Another important quality is the flexibility of our approach. We demonstrate that it can be used to model dimension-ordered-routing in several k-ary n-cubes such as hypercubes, 3D symmetric and asymmetric tori, architectures with uni- and bi-directional channels.

Introduction

Wormhole routing is an efficient switching technique which is adopted by a variety of parallel machines with distributed memory architectures, such as Intel Paragon, nCUBE-2, MIT J-machine, Cray T3D. Most multicomputers employ wormhole in combination with deterministic routing policies: the route between two nodes is determined in a static way and, in case of link conflict, the transmission waits until the link becomes free. Although the deterministic policies do not have the ability to respond to dynamic network conditions, their implementation is quite simple. Conversely, adaptive wormhole routing requires more sophisticated hardware and more complex algorithms for preventing deadlocks and livelocks. An interesting survey of wormhole routing is contained in [14]. Commercial multicomputers adopted deterministic routing in three types of k-ary n-cubes: uni-directional hypercube, uni-directional and bi-directional torus, bi-directional mesh [13]. In uni-directional systems, there is only one link between adjacent nodes, while in a bi-directional system, each connection between two nodes consists of one link for each direction. For bi-directional networks, the wrap-around links between the edge nodes are not essential, while in uni-directional systems without wrap-around, a communication may require k-1 hops even between adjacent nodes.

The goal of this paper is to propose an approach that achieves accurate closed formulas for the mean latency time of dimension-ordered wormhole routing in k-ary n-cubes. The error is below 10% in most instances, and much less for low and average traffic conditions. An important contribution of our results is that the same approach is immediately applicable to any k-ary n-cube with wrap-around connections for k < 5. For higher values of k, it is impossible to construct a deadlock-free minimal deterministic routing algorithm [5]. We present the methodology for hypercubes, an asymmetric (k₀×k₁×k₂) and symmetric (k×k×k) torus with uni-directional and bi-directional links, respectively.

Performance analysis of wormhole routing in k-ary n-cubes topologies has been investigated by several authors. The majority of them adopted simulation models 2, 3, 4, 7, 8, 12, 15, 16, while few results were obtained analytically 6, 11, 1. Adve and Vernon [1] use approximate Mean Value Analysis to analyze the performance of wormhole routing in a k-ary n-cube, where the system is modeled as a closed queuing network. This choice makes their model realistic, although complex to the extent that it can be solved only through iterative algorithms. Dally [6] proposes a general model for k-ary n-cubes, that he adopts to demonstrate that low k-dimensional topologies provide best performance. The results of this paper are a very important contribution in this area, however the Dally model solves only symmetric topologies with uni-directional links. Kim and Das [11] specialize this model for the hypercube topology. They obtain accurate results, although limited to the hypercube that is a perfect symmetric and balanced topology. Our approach is more general than the models proposed in 6, 11, because it is applicable to several topologies such as hypercubes, symmetric and asymmetric tori with both uni-directional and bi-directional links. The contribution of our model is outlined in Table 1.

The paper is organized as follows. In Section 2, we describe the operational features of the interconnection network and motivate the assumptions that make the model analytically tractable. In Section 3, we present the approach for the message latency time evaluation focusing on a 3D asymmetric torus with uni-directional links. In 4 The model for torus with bi-directional links, 5 The model for hypercube, we demonstrate the flexibility of our approach by extending the analysis to a 3D symmetric torus with bi-directional links and a hypercube, respectively. In Section 6, we validate all models by comparing the analytical results against time values which are obtained from our simulations and other published results. In Section 7, we use our models to evaluate the performance of wormhole routing in several network dimensions and traffic loads. Section 8concludes the paper with some final remarks.