Elsevier

Performance Evaluation

Volume 47, Issue 1, January 2002, Pages 43-71
Performance Evaluation

Blocking probabilities in circuit-switched wavelength division multiplexing networks under multicast service

https://doi.org/10.1016/S0166-5316(01)00056-6Get rights and content

Abstract

We evaluate the call-blocking probabilities in optical networks which support multicast (or multipoint) service. Due to the difficulty of the problem, we restrict our attention to the case of completely connected networks, for which the blocking probabilities act as a lower bound for other topologies. Nodes may, or may not be equipped with wavelength converters. In both cases, routing can be done either directly, using one hop, or indirectly, and even in the latter case the number of hops can be either restricted to two hops, or unrestricted. We evaluate the blocking probability by first estimating the load offered to each link, and then use the iterative-reduced load approximation method to refine this estimate. Using this estimate, we then obtain the probabilities of blocking and success. Numerical examples are presented, and are compared to simulation.

Introduction

The advent of wavelength division multiplexing (WDM) [1], [2] provided a solution to the problem of speed mismatch between the fiber bandwidth and the electronics speed, which is better known as the electronic bottleneck problem [3]. As such, an aggregate bandwidth in the tens of terabits per second range can be achieved by supporting multiple simultaneous transmissions, each utilizing a different wavelength, or optical frequency. This can provide a means to serve those applications which individually or collectively require a large amount of bandwidth. Such applications include the class of service with multicast, or multipoint traffic which requires the delivery of data from the source to a group of destinations [4]. Such applications include video distribution, network news distribution, multi-party conferences, etc. The nature of the optical networks requires that the support of multicast traffic either creates an independent channel from the source to each destination, or that destinations cooperate in relaying the traffic they receive to other destinations. Although the first approach is easy to implement, its bandwidth requirements are very large due to the creation of multiple channels. The bandwidth requirements of the second approach are modest compared to those of the first one, but they require some special provision to implement. The problem is that in optical networks, the optical layer will establish a connection such that it starts at the source, and terminates at one of the destinations. In this paper, we assume that there is a provision by which that destination will relay the same traffic, after reading it, to other destinations. This can be implemented easily by converting the optical signal to the electric domain, and then converting it back again to the optical domain, but in this case optical transparency may not be achievable. Alternatively, it can also be implemented in the optical domain directly through the use of active optical-switching elements that control the fractions of the optical signals that can be diverted to each of its two outputs [5].

The purpose of this paper is to derive expressions for the call-blocking probabilities in optical networks using WDM and carrying multicast circuit-switched traffic. Several studies have derived the call-blocking probabilities for unicast circuit-switched applications, but to the best of our knowledge, this is the first study to obtain such blocking probability for the multicast service.

A few studies dealt with multicast routing in optical networks (e.g. [6], [7], [8]). However, those studies were concerned with packet-switched communication, and were concerned with either the establishment of the routing tree, or the derivation of the packet delay. The recent study in [9] considered the performance of unscheduled packet multicasting in broadcast-and-select networks, and found that randomizing the transmissions is better than the persistent approach. More recently, the work in [10] evaluated the distribution of the receiver busy times in a WDM star network under multicast packet switching. Qiao et al. [11], [12], [13] considered WDM support for IP multicasting. The work in [14] (also in [5], pp. 516–527) is probably the only study dealing with multicast services in circuit-switched service over linear lightwave networks, and presents algorithms for the construction of multicast connections. A simulation study of the call-blocking probability is also presented.

A few studies have also dealt with the derivation of the blocking probabilities under unicast service and with circuit-switched traffic. Girard [15] considers blocking probabilities in circuit-switched networks. However, most of the results in that manuscript are not directly applicable to WDM, especially with multicast traffic. The study in [16] derived the blocking probabilities under circuit-switched service in WDM networks, with and without wavelength conversion, and assuming a given wavelength utilization. The effects of path length, switch size, and interference length on the blocking probabilities were evaluated. This study was extended in [17] when multiple fiber links are used. A similar study in [18] used Erlang’s fixed point iterative method to arrive at the blocking probabilities. In [19], the blocking probabilities with and without wavelength conversion were also derived, and the solution was obtained iteratively using the reduced load approximation technique [20], [21], which is a generalization of the Erlang’s fixed point method. Alternate routing with fixed, and least lightly loaded strategies was considered. Subramaniam et al. [22] derived the blocking probabilities when only a group of nodes are capable of wavelength conversion. Erlang’s fixed point approach was not used on the premise that its use enhances the results only in a minor way when the blocking probabilities are small. The same authors derived the blocking probabilities when the arriving traffic is non-Poisson in [23]. They used the moment-matching approach, and the Bernoulli–Poisson–Pascal approximation. In [24], the blocking probabilities in all-optical networks employing limited wavelength translation were derived in terms of the link utilization. It was shown that the benefits of full-range wavelength translation can be almost achieved with a limited-range wavelength translation. Sharma and Varvarigos [25] considered the same problem in a mesh network. Harai et al. [26] obtained the blocking probabilities in all-optical switching networks when alternate routing is used, and found that alternate routing enhances the system performance. In [27], the same authors introduced a general model using the layered graph approach [28]. The model can be used for any routing or wavelength selection strategies. However, the complexity of the model limits its applicability to networks with a few nodes, and a few wavelength channels. Li and Somani [29] used the same model in [23] in order to study the performance of different routing strategies they proposed. In [30], a model based on circuit-switched alternate routing [15] is used to evaluate the probability of blocking when adaptive, joint-route selection and wavelength selection is used, and without wavelength conversion. Karasan and Ayanoglu [31] has provided analysis of the fixed-route, first-fit wavelength selection algorithm under no wavelength conversion. In [32], an exact method to evaluate the blocking probabilities in wavelength-routed networks under fixed path routing, and random wavelength selection was presented. However, in order to obtain a closed-form expression for the probability distributions, an approximation was presented to convert the process into a time reversible one. In addition, a path decomposition technique was also presented to reduce the complexity of the problem.

In this paper, we derive the blocking probabilities under multicast service when the topology forms a completely connected graph. We have chosen this topology for two reasons. First, it is a simple topology that avoids all the routing decisions which must be made by other special topologies which will affect (and in most cases complicate) the derivation of the blocking probabilities. Second, since a link exists between each pair of nodes in the completely connected topology, and since this number of routes from a source to any group of destinations is at least equal to the number of routes with any other topology, the blocking probabilities derived in this study serve as lower bounds on the blocking probabilities in other topologies. Moreover, some networks, such as the vBNS [33], [34], implement a virtually completely connected topology using ATM virtual paths. The same modeling techniques presented in this paper can be extended to model those networks, while taking the virtual paths and the type of traffic into account.

We derive the blocking probabilities with and without wavelength conversion. We also consider three cases for the maximum number of hops allowed: a single hop which corresponds to direct routing, a two-hop indirect routing,2 and an unrestricted indirect routing strategy. As expected, when the limit on the number of hops increases, the blocking probabilities improve. However, increasing such a limit requires a more complicated control mechanism, and as will be shown in Section 5, this enhances the blocking probabilities only marginally over the two-hop case. It should be noted that in the case of no wavelength conversion, as the limit on the number of hops increases, the problem becomes more intractable. Therefore, in that case we only evaluate the blocking probabilities in the cases of direct routing, and indirect routing with a limit of two hops only, similar to the model in [19], [24]. With the unrestricted indirect routing strategy, we rely on simulation to assess the effectiveness of that method.

The approximate approach we follow in this paper is to estimate the successful call arrival rate to a certain link given the number of available channels (wavelengths) on that link and on an adjacent link. This joint availability condition is necessary to capture the strong correlation that arises with multicast traffic. This estimated load is in fact the reduced load which is used to calculate the probabilities of wavelength availabilities. The procedure is iterative, and in most cases converges.3

In Section 2, we introduce the network model. In Section 3, we derive the blocking probabilities when wavelength conversion is used, while the probabilities of blocking without wavelength conversion are obtained in Section 4. Section 5 presents some numerical examples and discussions, and Section 6 concludes with some remarks.

Section snippets

Network model

We consider a completely connected network with N nodes, such that each pair of nodes is connected by two fibers for communication in both directions. Each fiber (link) carries W wavelength channels. Calls are assumed to arrive at a node according to a Poisson process with rate a, and the call-holding time is exponentially distributed with a mean of one time unit. Each of the arriving calls is directed to k destinations with probability rk, and the destination nodes are uniformly chosen from

Networks with wavelength conversion

This section derives the session-blocking probabilities under the assumption that nodes are capable of full wavelength conversion. We will consider the three cases in which routing is either direct, indirect with a maximum of two hops, and indirect with no restriction on the number of hops.

Networks without wavelength conversion

In this section, we assume that the switches do not implement wavelength conversion. We also assume that to establish a call, all routes and wavelengths are exhaustively searched until either the call can be established, or all the routes and wavelengths have been exhausted. The outgoing links from the source may use the same or different wavelength channels.

The derivation of the reduced load equations in this section is more involved since one has to keep track of the number of commonly

Numerical results

The expressions for the reduced load derived in the previous section assume the knowledge of the channel availability distribution. Since these probabilities depend in turn on the knowledge of the reduced loads, through the matrix P, an iterative solution procedure must be used. In this procedure, we start by assuming arbitrary values of λ(m,n), and we use them to solve for the steady-state probabilities and q(m). Using the computed values of q(m), we recalculate λ(m,n), and solve again for the

Conclusions

In this paper, we derived expressions for the call-blocking probabilities in optical networks under multicast service in a completely connected network topology. We considered the case in which full wavelength conversion is allowed, and the case in which no wavelength conversion is used. For both cases, we considered the direct routing method, and the indirect routing method in which the maximum number of hops can be restricted to two hops, or can be unrestricted. This last case under no

Acknowledgements

The authors wish to acknowledge the comments made by the referees, which helped improve the paper presentation.

Ahmed E. Kamal was born in Giza, Egypt. He received his B.Sc. (distinction with honors) and M.Sc. both from Cairo University, Giza, Egypt, and M.A.Sc. and Ph.D. both from the University of Toronto, Toronto, Canada, all in electrical engineering in 1978, 1980, 1982 and 1986, respectively. From 1978 to 1980, he was an instructor in the Department of Electrical Engineering, Cairo University, and from 1980 to 1985, he was a teaching and a research assistant in the Department of Electrical

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    Ahmed E. Kamal was born in Giza, Egypt. He received his B.Sc. (distinction with honors) and M.Sc. both from Cairo University, Giza, Egypt, and M.A.Sc. and Ph.D. both from the University of Toronto, Toronto, Canada, all in electrical engineering in 1978, 1980, 1982 and 1986, respectively. From 1978 to 1980, he was an instructor in the Department of Electrical Engineering, Cairo University, and from 1980 to 1985, he was a teaching and a research assistant in the Department of Electrical Engineering, University of Toronto. He was with the Department of Computing Science at the University of Alberta, Edmonton, Canada, from January 1986 to June 1991 as an Assistant Professor, and from July 1991 until August 1994 as an Associate Professor. He was also an Adjunct Professor at the Telecommunications Research Labs, Edmonton, Alberta, from 1988 until 1994. In September 1992, he joined the Department of Electrical and Computer Engineering at Kuwait University, Kuwait, as an Associate Professor and became a Professor in the same department in 1997. He is currently a Professor of Electrical and Computer Engineering at Iowa State University. His teaching and research interests include high performance networks, B-ISDN and ATM networks, performance evaluation, optical networks, and quality of service. He is a senior member of the IEEE, a member of the Association of Computing Machinery, and a registered professional engineer in the province of Alberta, Canada. He was the co-recipient of the 1993 IEE Hartree Premium for papers published in Computers and Control in IEE Proceedings for his paper entitled ‘Study of the Behavior of Hubnet’.

    Anwar K. Al-Yatama received his B.S. degree in electrical engineering from Kuwait University in 1985, M.S. degree in computer engineering from Florida Institute of Technology in 1987, and Ph.D. in electrical engineering from Georgia Institute of Technology in 1992. He is now an Assistant Professor of Computer Engineering, and the Assistant Vice President for Academic Services at Kuwait University. His research interests include wireless and broadband networks, queuing analysis and simulation. He is a member of IEEE.

    1

    This work was performed when A.E. Kamal was with Kuwait University, and was sponsored by grant EE-102 from the Kuwait University Research Administration.

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