# Rearrangeable and Repackable S-W-S Elastic Optical Networks for Connections with Limited Bandwidths
## 1 Introduction  
Elastic optical networks (EONs) [1; 2; 3] are a technique introduced to support future high-speed optical networking and rapid growth of Internet traffic. EONs provide superior flexibility and scalability in spectrum allocation. To improve spectrum allocation efficiency, EONs use the flexible frequency grid proposed by the International Telecommunication Union [4]. In EONs, the spectrum is currently divided into several 12.5 GHz fixed-width frequency slots called frequency slot units (FSUs) [4]. An optical connection that requires bandwidths greater than 12.5 GHz can spread over a number of, say \(m\), _adjacent_ FSUs, and it is called an \(m\)-slot connection. Usually, \(m\) is upper bounded by a value, denoted by \(m_{\max}\). Various switching fabrics for EONs have been proposed [5; 6; 7; 8]. One of them is the three-stage wavelength-space-wavelength (W-S-W) architecture, which adopts spectrum converters in the first and last stages and space switching in the middle stage [8]. A demonstration of a \(16\times 16\) W-S-W optical switching test bed for EONs has been presented in [6]. This architecture with different types of nonblockingness was considered in [8; 9; 10; 11; 12]. Another three-stage switching fabric for EONs is the space-wavelength-space (S-W-S) architecture, which has only spectrum converters in the middle stage [7]. Both W-S-W and S-W-S architectures can be used as the optical circuit switching part, which serves elephant traffic between servers, of a datacenter network [7; 13]. The S-W-S architecture generally consumes fewer middle switches than W-S-W architectures [8; 9], especially when \(m_{\max}\) is growing. Moreover, unlike the W-S-W architecture, the S-W-S architecture can be practically realized for switching fabrics when a larger \(m_{\max}\) and a greater number of FSUs are used in one input or output. In this paper, we consider one S-W-S architecture, called an SWS1 switching fabric. In EONs, an \(m\)-slot connection is set up between \(m\) free FSUs in the input and output fibers. Thus, the switching fabrics can never be blocked because of a lack of free FSUs at the input and output fibers. Switching fabrics can be of various types of nonblockingness, three of which are strictly nonblocking(SNB) [14,15], rearrangeable nonblocking (RNB) [14,15], and repackable nonblocking (RPNB) [16, 17, 18, 19]. In an SNB switching fabric, a connection will never be blocked by existing connections. In an RNB switching fabric, the nonblocking state can be kept by rearranging existing connections upon the arrival of a connection, while in an RPNB switching fabric, rearranging existing connections is done upon the departure of a connection. It is worth noting that time is less critical for connection rearrangement during departures than during arrivals. An alternate definition of an RNB switching fabric is that a set (or frame) of connections can be routed simultaneously. Sufficient and necessary conditions for an SNB SWS1 switching fabric were derived and proved in [7]. In this paper, we use two different approaches to generate RNB SWS1 switching fabrics for \(m\)-slot connections for \(1\leq m\leq m_{\max}\) with \(m_{\max}=1\) and \(2\leq m_{\max}\leq n\), respectively, where \(n\) is the number of FSUs in each link of the SWS1 switching fabric. A similar approach to that for an RNB SWS1 switching fabric with \(2\leq m_{\max}\leq n\) is also applied to generate an RPNB SWS1 switching fabric with \(2\leq m_{\max}\leq n\). The proposed RNB and RPNB SWS1 switching fabrics for connections with a limited \(m_{\max}\) consume around half the number of middle wavelength switches than the SNB SWS1 switching fabric. The remainder of the paper is organized as follows. In Section 2, we introduce the SWS1 switching fabric and earlier results. The sufficient and necessary conditions under which an SWS1 switching fabric with \(m_{\max}=1\) will be RNB are derived in Section 3.