# Rearrangeable Nonblocking Conditions for Four Elastic Optical Data Center Networks
## 1 Introduction  
Recently, high transmission speed between the servers in data centers [1] has become an increasing requirement to meet the needs of current applications such as cloud computing and data mining. To support such high transmission speed, various data center network (DCN) architectures have been proposed [1, 2, 3, 4]. One among them integrates an electronic packet switching (EPS) network and an optical circuit switching (OCS) network [1, 2, 4]. In such a hybrid electrical/optical architecture, both the EPS and OCS networks connect to each top-of-rack (ToR) switch simultaneously, where EPS serves small flows and OCS serves big flows. It has been shown that such a hybrid electrical/optical architecture reduces the power consumption and the operating expense [5, 6].  
An elastic optical network (EON) [7, 8, 9, 10, 11] is a candidate for being the OCS part of a DCN [2]. In EONs, flexible frequency grids proposed by ITU-T [12] are used, and a different number, say \(m\), of _adjacent_ frequency slot units (FSUs) are assigned to an optical connection, where \(m\) is usually upper bounded by a value, say \(m_{\text{max}}\). The bandwidth of an FSU is 12.5 GHz [12], and a connection is called an \(m\)-slot connection if it is assigned \(m\) adjacent FSUs. Four variants of optical DCN architectures based on elastic optical switches, called DCN1, DCN2, DCN3 and DCN4, were proposed in [13]. The four DCN architectures are similar to wavelength-space-wavelength (W-S-W) networks [8, 9, 10], which are Clos-like architectures, but they do not adopt costly tunable wavelength converters as W-S-W networks do. In addition, the maximum number of connections generated from each input fiber in the four DCNs and W-S-W networks is limited due to the different components used. This leads to the nonblocking conditions derived for these four DCNs being different from those derived for W-S-W networks.  
When a network is called nonblocking, it is in reference to the nonblocking traffic assigned to the network [14]. To prevent excessive blocking of connections, the network should be nonblocking. A network is called strictly nonblocking (SNB) if a connection will never be blocked by existingconnections, and a network is called rearrangeable nonblocking (RNB) if a new connection can be accommodated by rearranging some existing connections [14]. An RNB network is also defined as one where any set (or frame) of connections can be routed simultaneously. The necessary and sufficient number of FSUs per fiber required for these four DCNs in the sense of their being SNB were given in [13], but no results in the sense of being RNB were proposed.  
The four DCNs usually require a great number of FSUs per fiber to be SNB, especially when \(m_{\max}\) is growing. However, the resource of FSUs per fiber in practical systems is limited since for the EON switches, and the C-band has only around 350 available FSUs (1530-1565 nm). Reducing the value of FSUs per fiber is a challenging task, and this issue has been studied in various research on EONs [8; 9; 10; 11]. In order to reduce the value of FSUs per fiber to realize nonblocking optical DCN architectures in practice, we studied the four DCNs in the sense of being RNB in this paper, and derived the sufficient number of FSUs per fiber by adopting two multigraph approaches. Our results show that two of the proposed RNB conditions reduced the SNB results significantly.  
The rest of the paper is organized as follows: In Section 2, we give a brief review of the four DCN architectures and introduce the notations used in the paper. In Section 3, we prove the RNB conditions for the DCN1 and DCN3 networks. In Section 4, we prove the RNB conditions for the DCN2 and DCN4 networks. Section 5 concludes the paper.