Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 1575828, 14 pages

https://doi.org/10.1155/2017/1575828

## The Strict-Sense Nonblocking Multirate Switching Network

Faculty of Electronics and Telecommunications, Chair of Communication and Computer Networks, Poznań University of Technology, Ul. Polanka 3, 60-965 Poznań, Poland

Correspondence should be addressed to Remigiusz Rajewski

Received 18 May 2016; Revised 7 November 2016; Accepted 14 November 2016; Published 7 February 2017

Academic Editor: Kyandoghere Kyamakya

Copyright © 2017 Wojciech Kabaciński and Remigiusz Rajewski. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper considers the nonblocking conditions for a multirate switching network at the connection level. The necessary and sufficient conditions for the discrete bandwidth model, as well as sufficient and, in particular cases, also necessary conditions for the continuous bandwidth model, were given. The results given for in the discrete bandwidth model are the same as those proposed by Hwang et al. (2005); however, in this paper, these results were extended to other values of , , and . In the continuous bandwidth model for , the results given in this paper are also the same as those by Hwang et al. (2005); however, for , it was proved that a smaller number of vertically stacked switching networks are needed.

#### 1. Introduction

A multirate switching network is a network in which any connection is associated with weight . Such a weight represents a certain bandwidth of input and output ports and interstage links connecting these input and output ports. The capacity of the links is normalized and is usually equal to 1 for interstage links. Input and output ports’ capacity is in many cases lower than the interstage links’ capacity and is denoted by , where . Weight is also limited by range , , where .

Depending on the possible values of , two models of multirate connections are considered in the literature: the discrete bandwidth model and the continuous bandwidth model. In the discrete bandwidth model, it is assumed that there are a finite number of distinct rates and the smallest rate divides all other rates , where . Denote and . The smallest rate is often called a channel. In this paper, we assumed that each internal link has channels and each input or output link has channels, where , , and . Every new connection is associated with a positive integer , where and is the maximal number of channels that one request can demand. In the continuous bandwidth model, connections may occupy any fraction of a link’s transmission capacity from interval . Both models are considered in this paper.

One of the best known multistage switching networks is the 3-stage Clos network [1]. Nonblocking conditions for single-rate and multicast connections were considered by many authors [2–5]. The first upper bound of nonblocking conditions in the case of the continuous bandwidth model was proposed by Melen and Turner in [6]. This upper bound was later improved by Chung and Ross in [7]. In turn, asymmetrical switch configurations were considered in [8]. More generalized 3-stage Clos switching fabrics were considered by Liotopoulos and Chalasani [9]. The results derived in those papers were limited to or . Both sufficient and necessary nonblocking conditions for any and were proved in [10, 11] in the case of symmetrical and asymmetrical 3-stage Clos switching networks, respectively. In some papers, the blocking probability at the connection level was also considered [12–14].

Another switching network considered in the literature is the vertically stacked Banyan type switching network [15–18]. Multirate switching fabrics were considered in [19, 20], where necessary and sufficient conditions were given for the discrete bandwidth model when is an integer, and a sufficient condition, as well as a necessary condition for , for the continuous bandwidth model was proved. Better upper bounds, which in some cases are also lower bounds, for switching networks were given in [21, 22]. Multirate switching fabrics with multicast connections were considered in turn in [23]. Some architectures, which may be considered as special cases of networks, like extended delta and Cantor switching fabrics, were considered earlier in [6, 7, 24].

The results presented in [19, 20] have been improved by Hwang et al. in [25]. They proved both sufficient and necessary conditions for the strict-sense nonblocking operation of networks for the discrete bandwidth model, but only where . They also proved sufficient and necessary conditions for the continuous bandwidth model when and sufficient conditions when (the conditions are also necessary in case ).

In this paper, we also consider a multirate switching network, however, only for . The results for are under study. First, we extended the results given in [25] for the discrete bandwidth model to the general case. Then, we also introduce sufficient condition for the continuous bandwidth model when . In most cases, these sufficient conditions are also necessary.

It should be noted that nowadays multirate switching networks [6–14, 16–19, 23, 24] are getting more popular each day [26–30]. A special kind of such a multirate network is an elastic optical network [31–36] which constitutes a “hot topic” in optical networking and switching. In the future, elastic optical networks will replace the current optical networks used by network operators and will also probably be used in data centers. In the elastic optical network, an optical path may occupy a bandwidth which is a multiple of the so-called frequency slot unit. This frequency slot unit occupies 12.5 GHz of the bandwidth and adjacent frequency slot units may be assigned to one optical path. This may be described by using the discrete bandwidth model with denoting the frequency slot unit and denoting the number of such units in one connection.

Multirate type of structure can be used, for example, in data center networks [37–41] or in multiprocessor systems. By using multirate structures, it is also possible to handle network traffic in telecommunication and computer networks generated by many services handled by network providers or companies. Using a new type of switching network structures in data centers or multiprocessor systems allows building more energy-efficient and cheaper architectures, where the cost could be understood as the number of cross points [3, 4] or as the number of active and passive optical switching elements [42]. The topic of energy efficiency is not considered in this paper; however, this paper could be a starting point for such an investigation.

In turn, the classification of different types of services enables the proper management of resources and appropriate performance for each of these services especially for the 4G/5G networks. Each service requires different resources, expressed very often in basic bandwidth units or in the number of channels. And for the 4G/5G networks bandwidth is a very crucial aspect. If sufficient resources are available, each service considered can be realized in such a switching network. It was assumed that one connection represents some service and each service requires a different number of channels , where the maximal number of channels one service could demand is , and .

Motivation of this paper was to improve already known best results for the strict-sense nonblocking multirate switching network [25]. In the next sections, it is described in detail how better results were achieved. The switching network constitutes nowadays a quite interesting architecture which could be used in optical network nodes especially for , where physical implementation is much easier than that for , where denotes the size of one switching element.

This paper is organized as follows. In Section 2, the model used in this paper is described. In the next section, the nonblocking operation of the considered network is discussed for the discrete bandwidth model. In Section 4, strict-sense nonblocking conditions for the continuous bandwidth model are considered. In the next section, a few numerical examples for both discrete and continuous bandwidth models are presented. In the same section, the results obtained in this paper are compared with the already known upper bounds. The last section includes conclusions.

#### 2. Model Description

The switching network was proposed in [16]. Such a network is constructed by vertically stacking copies of networks (the main idea of stacking planes is shown in Figure 1). This architecture was later extended to the switching network in [17]. Such an extended network is obtained by adding extra stages to the network and vertically stacking copies of the network. The switching network is a particular case of the switching network which consists of symmetrical switching elements. The switching network is nonblocking in the strict sense when there are a sufficient number of vertically stacked planes , so any connection between a free input and a free output can be realized regardless of the routing algorithm used.