A Random Graph Approach for Multicast Scheduling and Performance Analysis Guowen Han and Yuanyuan Yang Department of Electrical & Computer Engineering, State University of New York, Stony Brook, NY 11794, USA Abstract— In this paper, we consider scheduling in multicast switching networks, which aims to minimize the multicast latency for a set of multicast requests. Such a problem has been proved to be NP -Complete. We propose a simple, fast greedy multicast scheduling algorithm and derive a lower bound and an upper bound on the performance of the algorithm. As can be seen, while a lower bound is fairly straightforward, the upper bound is much more difficult to obtain. By translating the multicast scheduling problem into a graph theory problem and employing a random graph approach, we are able to obtain a probabilistic upper bound on the performance of the multicast scheduling algorithm. Our analytical and simulation results show that the performance of the proposed multicast scheduling algorithm is quite close to the lower bound and is statistically guaranteed by the probabilistic upper bound. I. I NTRODUCTION AND PREVIOUS WORK Multicast is an important operation in modern high- performance networks. Many emerging computing/networking applications exhibit the need of multicast communication patterns. For example, audio and video multimedia confer- encing, distance education and video-on-demand services in a communication network, matrix multiplication and barrier syn- chronization in a parallel and distributed computing system, and database and software updates in a distributed system. Many of these applications require not only multicast capabil- ity but also predictable communication performance, such as guaranteed multicast latency and bandwidth, called quality-of- service (QoS). The combination of the non-uniform nature of multicast traffic and the requirement of QoS guarantees makes the problem very challenging. The performance of multicast communication is mainly measured in terms of its latency in delivering a message to all its destinations. Most of related work in this area aims to optimize the multicast latency and design deadlock-free multicast routing algorithms, see, for example, [11], [12]. The switching networks which consist of one or more stages of switches can easily have deadlock-free routing and an equal communication latency between any sources and destinations. These features make them a good candidate for a QoS capable multicast architecture. To provide better QoS functions, researchers [5], [6] have considered the switching networks which can realize a well- defined type of multicast communication pattern referred to as multicast assignment, which is a mapping from a subset of source nodes to a subset of destination nodes with no over- lapping allowed among the destinations of different sources (for example, Fig. 1(a) is a multicast assignment, while Fig. 1(b) is not). Regardless of the current network state, multicast connections from any idle sources to any idle destinations in these networks are guaranteed to be always realizable in a constant latency in a nonblocking fashion [6]. Due to its sources destinations x1 x2 x3 x4 x5 x6 x7 x8 y1 y2 y3 y4 y5 y6 y7 y8 (a) sources destinations x1 x2 x3 x4 x5 x6 x7 x8 y1 y2 y3 y4 y5 y6 y7 y8 (b) Fig. 1. Examples of multicast requests in an 8 × 8 multicast switching network. predictable multicast latency for any multicast assignments, certainly this type of network can provide better QoS functions for multicast communication. However, in reality, there are usually multiple multicast applications running on the same network. In such a case, what we can observe at the physical layer of the network is that many independent multicast connections are being routed in the network, which leads to the situation that the combined multicast traffic in the network is not necessarily a multicast assignment and overlapping among destinations of different multicast connections is quite possible. To better meet the QoS requirements from real-world ap- plications, more powerful switching networks have been pro- posed to improve the bandwidth of the multicast communica- tion, such as multirate switching networks [7], [8], wavelength division multiplexing (WDM) optical switching networks [9] and the recently proposed k-fold multicast switching network- s [10]. In a multirate network, each link is assumed to have a bandwidth capacity one, and every connection has a bandwidth requirement no more than one. Thus, each link in such a network can carry multiple connections simultaneously as long as the sum of their bandwidth requirements does not exceed one. WDM is basically frequency-division multiplexing in the optical frequency domain, where on a single optical fiber there are multiple independent communication channels at different wavelengths, and each channel can be operated at the peak electronic data rate. In a k-fold multicast switching network, any destination can be involved in multicast connections from up to k different sources at a time. In fact, the basic idea behind these network models is essentially very similar, which is to allow one destination to be involved in multiple multicast connections. However, due to the limitation of the network architectures, none of these networks can guarantee to realize a set of arbitrary multicast connections in a single pass (round), and the contention (or overlapping) among destination nodes is still possible. When such contention occurs, a scheduling algorithm is needed to select a group of contention-free multi- cast connections for each round. Of course, a good scheduling algorithm should be fast and able to schedule all multicast 270 0-7803-7945-4/03/$17.00 (C) 2003 IEEE Authorized licensed use limited to: SUNY AT STONY BROOK. Downloaded on November 13, 2008 at 22:47 from IEEE Xplore. Restrictions apply.