# Link Load Correlation-Based Blocking Performance Analysis for Tree-Type Data Center Networks
###### Abstract
When full-wavelength converters are adopted, one transmission path may utilize different wavelengths in different links. Thus, an optical switching network can be treated as an equivalent multi-rate circuit switching network. On the basis of the link load independence assumption, the authors in [14] adopted the Erlang Fixed Point Approximation (EFPA) method to evaluate the blocking performance of optical switching networks with full-wavelength converters and static routing. In the EFPA method, the traffic at different links are considered to obey independent Poisson processes, the effective load on a link is equal to the effective accumulated loads from all of the connections passing through this link, and the blocking rates of links and connections are iteratively updated until convergence. In [14], the theoretical and simulation results of blocking probabilities have been separately compared in the star network, the ring network, and the mesh-torus network, where the theoretical and simulation results were found to match well, except for the ring network, due to the high load correlation between adjacent links in the ring network. Therefore, in this case, the link load independence assumption is not valid. The EFPA method has been widely used for blocking rate analysis in circuit switching and optical switching networks with general topologies [15; 16; 17; 18; 19]. In view of the high computational complexity of the EFPA method, [PERSON] et al. have proposed a simplified EFPA method, which is suitable for networks with large channel numbers in each link [20]. Besides theoretical analysis, the authors in [21] built a routing and spectrum assignment simulation framework to evaluate the blocking probabilities of existing spectrum assignment (SA) strategies under different network load conditions. In addition to the link load independence assumption, some works have proposed blocking rate analysis methods based on the object independence assumption [22; 23]. In the object independence assumption model, the correlation of links belonging to the same multi-hop connection is considered, and the individual free links and multi-hop connections are treated as independent objects. The object independence assumption-based analysis method was first proposed for the ring network, following which [PERSON] extended it to generic topologies, including linear, ring, tree, and mesh topologies [23]. Although the object independence assumption-based method can obtain higher approximation accuracy than that based on the link load independence assumption, it is only applicable to single-channel networks. Compared with other networks, the DCN is characterized by topological symmetry and traffic aggregation. The over-subscription ratio of a DCN is defined as the ratio of the aggregated bandwidth demand from end servers to the aggregated capacity of core switch links [24]. The over-subscription ratio can be set to be more than one to reduce the numbers of aggregation switches and core switches, saving capital investment. With a high over-subscription ratio, the core layer links tend to be the bottleneck, and the aggregated traffic results in considerable load correlation among links in adjacent layers. The authors in [25; 26] have analyzed the multi-cast blocking rates in fat-tree DCNs and proposed new multi-cast scheduling strategies to reduce blocking, based on the link load independence assumption. In [27], the blocking probabilities of different DCN architectures were compared through simulation and some congestion avoidance approaches were discussed, but a theoretical analysis was not carried out. [PERSON] et al. discussed how to minimize the occupied wavelengths by solving a optimization problem while obtaining the target blocking performance, with different wavelength converter distributions and buffer sizes [28]. The blocking performance analyses in [27; 28] are all based on simulation rather than theoretical derivation.