A Performance Model for k-Ary n-Cube Networks with Self-Similar Traffic Geyong Min and Mohamed Ould-Khaoua Department of Computing Science University of Glasgow Glasgow, G12 8RZ, U.K. Email: {geyong, mohamed}@dcs.gla.ac.uk Abstract Recently a number of studies have indicated that network traffic exhibits noticeable self-similar behaviour, i.e., traffic is bursty over a wide range of time scales. This fractal-like nature of traffic has received significant attention in the networking community as it has a considerable impact on queueing performance. Thus, it is very necessary to examine the performance properties of interconnection networks in the presence of self-similar traffic before practical implementations show their potential faults. However, adopting the simulation approach to evaluate system performance under self- similar workloads may be very costly and time-consuming because the convergence of simulations to a steady state is often very slow as burstiness appears over many time scales. This paper proposes the first analytical performance model for k-ary n-cubes with self-similar traffic. The validity of the model is demonstrated by comparing analytical results to those obtained through simulation experiments of the actual system. 1. Introduction Many recent network studies [3,8,13] by means of high quality, high time-resolution measurements have convincingly revealed that realistic network traffic exhibits self-similar nature and that the traditionally assumed models (e.g., the Poisson process) fail to capture the actual traffic properties. The traffic following a Poisson or Markovian arrival process has a characteristic burst length which tends to be smoothed by averaging over a long enough time scale. Rather, measurements of actual traffic indicate that noticeable traffic bursts are present over a wide range of time scales—typically at least four or five orders of magnitude [13]. The phenomenon of traffic self- similarity has a considerable impact on the queueing performance and has received significant attention in the networking research community. It has been suggested in [8,18] that many existing theoretical protocols and systems need to be reevaluated under this different type of traffic. Some researchers [3,18] have argued that a significant reason of self-similarity in network traffic is attributed essentially to the explosive growth in multimedia applications, typically exemplified by Variable Bit Rate (VBR) video. In order to efficiently support the emerging multimedia applications, multicomputer networks have to cope with self-similar traffic. Unfortunately, networks used in current multicomputers have been mainly designed and analysed [2,4~6,10,14,17,20] under the assumption of traditional Poisson process, which is unable to model the self-similarity nature of real network traffic [13]. Therefore, it is necessary to re-examine the performance properties of multicomputer networks in the context of more realistic traffic models before practical implementations show their potential faults. It is worth noting that more recent studies, e.g. [19], have indicated that even the workloads generated by a variety of scientific as well as engineering computations on multicomputers have been found to exhibit self-similar behaviour. This further makes it critical to reevaluate the properties of existing multicomputer networks. However, it is very costly and time-consuming to adopt the simulation approach to evaluate system performance under self-similar workloads [18]. This is because the convergence of simulations to a steady state is very slow as burstiness appears over a wide range of time scales. As a result, analytical models are viable alternative to simulations as cost-effective and versatile tools that can help designers to evaluate system performance under self- similar workload. Although a number of analytical models for multicomputer networks have been presented [2,4,5,14, 15,17,20], all these performance models have assumed that network traffic follows a Poisson or Markovian arrival process. There has not yet been any analytical model reported in the literature for multicomputer networks under self-similar traffic. As a step towards filling this gap, this paper proposes the first analytical model for k-ary n-cubes when subjected to self-similar traffic. The model is discussed in the context of pipelined circuit switching (or PCS for short) because PCS has recently been suggested as an efficient switching method that is well suited to support multimedia communication in k-ary n-cubes [7]. Proceedings of the International Parallel and Distributed Processing Symposium (IPDPS02) 1530-2075/02 $17.00 ' 2002 IEEE