Blocking in Multiwavelength TDM Networks - Semantic Scholar

Telecommunication Systems 6 (1998) 1{22

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Blocking in Multiwavelength TDM Networks Jennifer Yates a , Jonathan Lacey b and David Everitt c

Software Engineering Research Centre, Royal Melbourne Institute of Technology University, Melbourne, Australia E-mail: [email protected] b Australian Photonics Cooperative Research Centre, Department of Electrical and Electronic Engineering, The University of Melbourne, Parkville, Vic. 3052, Australia E-mail: [email protected] c Department of Electrical and Electronic Engineering, The University of Melbourne, Parkville, Vic. 3052, Australia E-mail: [email protected] a

This paper examines the relative importance of wavelength conversion and timeslot interchange in improving the performance of multiwavelength time-division multiplexed networks. It is shown that, in networks with a small number of wavelengths, each carrying a large number of time-division multiplexed channels, signi cant performance improvements are achieved by the introduction of time-slot interchange alone, without wavelength conversion. However, some performance improvements are also achieved by the introduction of wavelength conversion alone. Keywords: Optical Networks, Wavelength Division Multiplexing, Wavelength Converters, Teletrac.

1. Introduction Wavelength Division Multiplexing (WDM) is a promising technique for providing high aggregate transmission rates on a single optical bre. In WDM transmission, data from dierent users is modulated onto several optical carriers with dierent wavelengths. The optical carriers are then combined and transmitted on a single bre. The transmission bandwidth available in WDM networks is limited by the gain bandwidth of optical ampli ers, and by the transmission bandwidth of the bre itself. Also, WDM networks with very closely-spaced wavelengths require very precise source and lter stabilisation [1], and are prone to crosstalk [2]. There is thus a limit on the number of wavelengths which may be used on each link in a WDM network. To provide large numbers of connections on each link in a WDM network, several connections can be multiplexed in the time domain and transmitted on the same wavelength. Thus, each wavelength on each link can carry more than one connection. In this paper, networks which use this approach will be referred

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J. Yates et. al. / Multiwavelength TDM Networks

to as multiwavelength time-division multiplexed (TDM) networks. One way to build a multiwavelength TDM network is to establish dedicated channels between each source-destination pair. Connections between the same source and destination may then be multiplexed in the time domain onto these channels. Another alternative is to use a multi-hop approach, in which connections between several source-destination pairs may share common channels, with some connections established over more than one channel. Networks which dedicate resources to speci c source-destination pairs may either dedicate entire wavelengths, or speci c time slots within a wavelength. In networks which dedicate entire wavelength channels to each source-destination pair, e.g. [3], connections between a source and a destination are multiplexed in the time domain onto the appropriate wavelength(s), and are transmitted together from the source to the destination. Connections between dierent sourcedestination pairs do not share common wavelength channels. No switching is performed in the time domain between the source node and the destination node. Sabry and Midwinter [4] have proposed a network which only dedicates time slots within a wavelength, rather than entire wavelengths, to each sourcedestination pair. In their proposed network, each wavelength channel is divided in the time domain into repeating frames, as in a standard single-wavelength TDM system. Each frame is then divided into time slots. A channel is de ned by choosing a wavelength, and a time slot on that wavelength, on each link within a path. This channel is then dedicated to a speci c source-destination pair. This is illustrated schematically in Figure 1, which shows time on the horizontal axis and wavelength on the vertical axis. Each pixel in Figure 1 represents a single channel. Switching at each node in these networks is performed in wavelength, space and time, so that one slot may be switched to one particular output link, whilst an adjacent slot is switched to a dierent link. Thus, in contrast with the networks which dedicate an entire wavelength channel to a source-destination pair, dierent source-destination pairs may share a common wavelength on a particular link by using dierent time slots. In this paper, we will refer to networks of the kind proposed by Sabry and Midwinter [4] as shared-wavelength TDM networks. Such networks make more ecient use of the bre transmission bandwidth than networks which dedicate an entire wavelength channel to a speci c source-destination pair, at the expense of increased switch complexity. This paper presents an analysis of the performance of shared-wavelength TDM networks, in terms of their blocking probability and network utilisation. Speci cally, we investigate the relative importance of wavelength conversion and time-slot interchange in improving the performance of these networks. There are many issues concerning possible hardware implementations of such networks, and we do not attempt to fully address these issues in this paper. Instead, our aim is to establish whether such networks could potentially bene t, in terms of improved trac performance, from the use of wavelength conversion, or time slot interchange, or both.


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Channel resulting from time- and wavelengthdivision multiplexing

Wavelength Λ

.. .

2 1 1

. . .

N

1

.

.

.

Time

Frame 2

Frame 1

Figure 1. Division of the wavelength and time domains into channels in a shared-wavelength TDM system.

1.1. Blocking in shared-wavelength TDM networks

We will begin by discussing how blocking happens in shared-wavelength TDM networks, and then discuss techniques which can be used to reduce blocking. In simple WDM networks, a connection between two nodes along a path must use the same wavelength on all hops, or links, within the path. This requirement is referred to as the wavelength continuity constraint. For instance, consider the two-hop path shown in Figure 2. A connection between nodes 1 and 3 can only be established if the same wavelength is available on both links. If only wavelength 1 is available on link 1, and only wavelength 2 is available on link 2, the connection cannot be established. Node 1

Node 3

Node 2

λ1 λ2 λ3 λ4

λ1 λ2 λ3 λ4 Link 1

Link 2

Figure 2. Blocking in WDM networks. Bold lines represent wavelengths which are not available.

The restriction imposed by the wavelength continuity constraint can be avoided using wavelength conversion. A wavelength converter is a device which takes an optical carrier, modulated with a particular data signal, and converts it to an optical carrier at a dierent wavelength, modulated with the same data signal. If wavelength converters are used in WDM networks, connections can be established without the need to nd an unoccupied wavelength which is the

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same on all the hops making up the path. Wavelength converters can thus reduce network blocking probability. A number of studies have analysed the eect of wavelength conversion on blocking probabilities in WDM networks [5{10]. This work has shown that the eect of wavelength conversion depends on network topology, but that in many networks the introduction of wavelength conversion results in signi cant improvements in network performance. Blocking probabilities in shared-wavelength TDM networks can be further reduced by using time-slot interchange (TSI). A time-slot interchanger is a device which can rearrange the order of the time slots passing through it. TSI reduces blocking in TDM networks in the same way as wavelength converters reduce blocking in WDM networks. In networks without TSI, a connection along a path must use the same time slot on each of the links which make up the path. WDM networks with TDM may be implemented using either electronic switching or all-optical switching. Electronic techniques for the manipulation of data streams in the time domain are currently very advanced compared with their optical equivalents [11]. Therefore it is currently attractive to build sharedwavelength TDM networks using optical transmission between nodes, and electronic time-division multiplexing, demultiplexing, add-drop multiplexing, switching, TSI, and wavelength conversion within nodes. In the future, it may be possible to build all-optical shared-wavelength TDM networks, in which the signal remains in an optical form from source to destination. Such all-optical sharedwavelength TDM networks may require all-optical time-slot interchangers and all-optical wavelength converters. Many techniques for all-optical wavelength conversion have been successfully demonstrated in research laboratories [12{14], but demonstrations of all-optical time-slot interchangers are considerably less advanced [15,16]. Although time-slot interchangers and wavelength converters can reduce the blocking probability in shared-wavelength TDM networks, the use of either or both of these devices, whether they are electronic or optical, introduces an increase in hardware cost and switch complexity. It is therefore important to establish precisely what advantages wavelength conversion and TSI oer to network performance. In this paper, we examine the eectiveness of wavelength conversion and TSI in improving the performance of shared-wavelength TDM networks. In particular, we examine the relative importance of wavelength conversion and TSI in reducing blocking and increasing network utilisation. In Section 2, we present an analytical description of the blocking probability in shared-wavelength TDM networks. Our analysis is an extension of the analysis of Kovacevic and Acampora [8,9], and assumes that connections are randomly assigned to both wavelengths and time slots. Section 3 presents results from this analysis for a multiple-hop path, and Section 4 presents results from simulations of shared-wavelength TDM networks, with mesh-torus, unidirectional ring, and ARPA2 topologies, using a rst- t algorithm for assigning wavelengths and time slots.


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Wauters and Demeester [17] have shown that shared-wavelength TDM networks are logically equivalent to WDM networks with multiple bres on each link and no TDM. The arguments of [17] are based directly on those in the conference version [18] of this paper. Subramaniam and Barry [19] have also recently published an analytical model for multi bre WDM networks which is equivalent to that in [18,20]. We also point out that, while the multi bre WDM network and the WDM network with TDM are logically equivalent, there is some dierence in the interpretation of results { in the multi bre WDM network, the question is whether to use wavelength conversion or not (since the analogue of TSI is that of switching between the same wavelength on dierent bres), while in the WDM network with TDM, we are interested in whether to use TSI and/or wavelength conversion. Our results show, in summary, that, in a shared-wavelength TDM system, any exibility in allocating paths, by either using time-slot interchange, or wavelength conversion, or both, may result in signi cant reduction in network blocking probability. Furthermore, if the number of wavelengths is small compared to the number of time slots per wavelength, then time-slot interchange alone will provide almost all the performance improvement introduced by using both time-slot interchangers and wavelength converters simultaneously. Similarly, if the number of time slots per wavelength is small compared to the number of wavelengths, then wavelength conversion alone is sucient. These results are consistent with one's intuitive expectation { that the dimension which provides the greatest amount of exibility in terms of either the number of time slot choices or the number of wavelength choices, is sucient. This is similar to the use of limited availability in circuit-switched networks [21] { here the limits on availability are due to limits on wavelength converters and time-slot interchangers. This is in some sense similar in concept to the notion of trunking eciency in loss networks where large trunk groups are more ecient (in terms of higher percentage of carried trac) e.g. [22{24], or to a queueing network where an M/M/1 queue with service rate is more ecient (in terms of reduced queueing delay) than n individual M/M/1 queues, each with service rate =n e.g. [23,25].

2. Analytic Models In this section, we extend an existing model for blocking in multiwavelength systems without TDM [8,9] to describe blocking in shared-wavelength TDM systems. 2.1. Existing analysis

Kovacevic and Acampora [8,9] have analysed blocking probabilities in multiple-hop paths for a wavelength-continuous network (i.e. a network with-

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out wavelength conversion). This analysis is based on the following assumptions: each connection uses an entire wavelength channel - that is, there is no timedivision multiplexing of multiple connections onto wavelength channels; the system uses a total of wavelengths, and all of these wavelengths are available on each link in the system; point-to-point trac (i.e. no broadcast or multicast trac); Poisson arrivals, negative exponential service times; blocked connections are discarded; loads and wavelength allocations on dierent links are independent; static routing of connections and dynamic allocation of connections to wavelengths. Thus connections between a particular source-destination pair always use the same xed path, but may use any available wavelength on that path; random wavelength assignment is used, in which connections are assigned to a wavelength which is chosen randomly from among those available. To analyse the blocking experienced along an n-hop path with no wavelength conversion, Kovacevic and Acampora begin by examining two links, with n busy wavelengths on the rst link and n busy on the second. The conditional probability that exactly k wavelength-continuous channels can be established across both links is [8,9] ?n n k ? n ? k (2.1) R(kjn ; n ) = a

b

a

a

b

a

b

n

b

for ? n ? n k min( ? n ; ? n ) and 0 otherwise. The probability that k wavelengths are busy on at least one link in an n-hop path is then given recursively by X X ( ) q (k) = R( ? kji; j )q( ?1) (i)p (j ) (2.2) a

a

b

b

n

n

n

=0 =0

i

j

where p (j ) is the probability that j wavelengths are busy on link n. For a Poisson distribution of connection requests, the probability p (j ) is given by the truncated Poisson distribution !?1 X (2.3) A =i! Aj ! p (j ) = =0 where A is the total oered trac for link n. The blocking probability for a wavelength-continuous n-hop connection is nally given by [8,9] P = q( ) () (2.4) n

n

i n

n

i

n

n

n

j n


7

In a network with wavelength converters, the blocking probability is [8,9] Y P = 1 ? (1 ? p ()) (2.5) n

n

=1

i

i

2.2. Analysis of shared-wavelength TDM systems

We now extend Kovacevic's and Acampora's analysis [8,9] to examine blocking probabilities for connections multiplexed in the time domain onto shared wavelength channels. Assumptions similar to those used in [8,9] and Section 2.1 are used here, but now each connection is assumed to use an entire time slot on a single wavelength on each link within its path. As previously discussed, to establish a new connection in a wavelengthcontinuous network which does not use TDM, it is necessary to nd a single wavelength which is unused on all the links of the desired path. Now consider a shared-wavelength TDM network, which has connections multiplexed in the time domain onto shared wavelength channels. To establish a new connection in such a network which has no wavelength conversion and no TSI, it is necessary to nd a single wavelength and a single time slot which is unused on every link of the desired path. Therefore blocking in these networks occurs in the same way as blocking in wavelength-continuous WDM networks which do not use TDM, and the same analysis can be used to describe both networks. A shared-wavelength TDM network with wavelengths and N time slots per frame on each wavelength has N: total channels. Therefore a shared-wavelength TDM network with wavelengths and N time slots per frame on each wavelength, and no TSI or wavelength conversion, is exactly equivalent to a wavelength-continuous WDM network which does not use TDM and has :N wavelengths. Similarly, a shared-wavelength TDM network which uses both TSI and wavelength conversion, and has wavelengths and N time slots per wavelength, is exactly equivalent to a WDM network with wavelength conversion which does not use TDM and has :N wavelengths. However, shared-wavelength TDM networks which use only wavelength conversion, and not TSI, require new analysis. To establish a connection in such a network, it is necessary to nd a single time slot which is unused on every link in the path. However, dierent wavelengths may be used on each link. Sharedwavelength TDM networks which use only TSI, and do not use wavelength conversion, also require new analysis. To establish a connection in these networks, it is necessary to nd a single wavelength which is unused on every link in the path. However, dierent time slots may be used on each link. We will now extend Kovacevic's and Acampora's analysis [8,9] to consider shared-wavelength TDM networks which have wavelength conversion but not TSI, and vice versa. We begin by analysing paths with wavelength converters but no TSI. In performing this analysis, we assume that connections are randomly assigned to

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both wavelengths and time slots. De ne to be the number of wavelengths per link, N to be the number of time slots per frame on each wavelength, and n to be the number of connections already established on a link. A given time slot on this link is \full" if that time slot is used on all wavelengths on the link. The event that j time slots are \full" on a link is denoted F (j ). The probability of this event occurring on link i, p (F (j )), is given by X p (F (j )) = p(F (j ) j n ) : p (n ) (2.6) a

i

:N

i

a

a =0

i

a

n

To evaluate Equation (2.6) we must calculate N :n(F (N ? j ) j n ? j:) p(F (j ) j n ) = j :N a

a

n

(2.7)

a

where F (a) denotes the event that a time slots are not \full" and n(F (a) j b) is the number of ways in which we can have a time slots which are not \full" given that there are a total of b connections established on the link. This is de ned recursively as being 0 if b < 0 and

n(F (a) j b) = when b 0 and where

min( X?1 ) ;b

=0

i

: n(F (a ? 1) j b ? i) i

8 < n(F (1) j b) = : b if 0 b < 0 otherwise

(2.8)

(2.9)

The probability p (n ) of having n established connections on link i is evaluated using Equation (2.3), with replaced by :N . On a path with wavelength conversion but no TSI, a connection is blocked if none of the N time slots are empty on at least one wavelength on each of the links making up the path. Thus, the blocking probability for paths with wavelength conversion but no TSI is calculated using Equation (2.4), with substitutions from Equations (2.1) and (2.2) as before. However p (F (k)), given by Equation (2.6), is used in Equation (2.2) in place of p (k). The model developed in this section was derived for paths with wavelength converters but no TSI. On paths with TSI but no wavelength converters, on the other hand, blocking occurs in exactly the same way, but with the roles of time slots and wavelengths interchanged. Therefore the result derived in this section may be reinterpreted to describe paths with TSI but no wavelength conversion i

a

a

i

i


9

by interchanging and N in the above expressions. This is an important point, which will be discussed further below.

3. Analytic Results We now use Equations (2.4) and (2.6) to analyse the relative importance of wavelength conversion and TSI in improving the performance of sharedwavelength TDM systems. 3.1. Two-hop path

We consider a two-hop shared-wavelength TDM path with 2 wavelengths and 16 time slots per wavelength. Figure 3 plots blocking probability versus oered trac per link for this path, when it is assumed that link loads and wavelength allocations on dierent links are independent. The uppermost curve shows the blocking probability experienced in a path with no wavelength converters (wavelength-continuous) and no TSI. The lowest curve in Figure 3 shows blocking probability if both wavelength converters and TSI are used. This curve shows that, for all values of oered trac, blocking probability is reduced when both wavelength converters and TSI are introduced. Figure 3 also shows blocking probability if wavelength conversion, but no TSI, is used. The blocking probability is reduced for all values of oered trac, but this reduction is only slight at high oered trac. However, Figure 3 also shows that the introduction of TSI, but no wavelength conversion, reduces blocking probability almost as much as the introduction of both wavelength conversion and TSI. This reduction occurs even though the introduction of TSI alone at the central node allows any incoming channel to access only half of the total time-wavelength slots on the outgoing link. An alternative approach to examining the performance improvements oered by wavelength converters and time-slot interchangers is to examine the increase in carried trac for a given blocking probability [7]. We de ne the utilisation gain for a particular blocking probability achieved by introducing TSI, say, as the ratio of the carried trac with TSI to the carried trac without TSI, for the given blocking probability. Similarly, it is possible to calculate the utilisation gain achieved by introducing wavelength conversion only, and the utilisation gain achieved by introducing both wavelength conversion and TSI. Utilisation gain is of particular interest to telecommunications providers, as it quanti es the possible increase in revenue for a given quality of service. Figure 4 plots the utilisation gain versus the number of time slots per wavelength for a two-hop path with = 2 wavelengths and a blocking probability of 10?3 . The most notable feature of Figure 4 is that all of the utilisation gains shown are small. In the best case, the introduction of both wavelength conversion and TSI allows the carried trac to be increased by a factor of only 1.26 if

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J. Yates et. al. / Multiwavelength TDM Networks 1

Blocking probablity

0.1

0.01

No conversion, no TSI Conversion, no TSI No conversion, TSI Conversion, TSI

0.001

0.0001 10

15

20 25 Offered traffic (Erlangs)

30

35

Figure 3. Blocking probability vs. oered trac for a two-hop path with 2 wavelengths per link and 16 time slots per wavelength. 1.3 Conversion, TSI No conversion, TSI Conversion, no TSI

Utilisation gain

1.25 1.2 1.15 1.1 1.05 1 2

4

6 8 10 12 Number of time slots (N)

14

16

Figure 4. Utilisation gain versus number of time slots for a two-hop path with = 2 and a blocking probability of 10?3 .

a blocking probability of 10?3 is to be maintained. These small gains occur because Figure 4 only considers a path with two hops. As we will see in Section 3.2, the introduction of wavelength conversion and / or TSI in paths with more hops generally results in more signi cant performance improvements. However, we can obtain important insights by considering the simple case illustrated in Figure 4. The introduction of both wavelength conversion and TSI result in a utilisation gain which has a maximum when N , the number of time slots per wavelength, is 2. The utilisation gain resulting from the introduction of both wavelength


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conversion and TSI then decreases as the number of time slots is increased, approaching 1 as N ! 1. The same eect has been observed when wavelength conversion is introduced in multiwavelength networks without TDM and the number of wavelengths becomes large [7], and occurs simply because large trunk groups are more ecient than small trunk groups [7]. Figure 4 shows that, when N = 2, the introduction of either TSI or wavelength conversion alone achieves the same utilisation gain, as either allows an incoming channel to access half the total time-wavelength slots on the outgoing links. As the number of time slots per wavelength increases, the utilisation gain achieved with TSI alone approaches the utilisation gain achieved using both wavelength conversion and TSI. This result agrees with the trend observed in Figure 3 where, for 16 time slots and a blocking probability of 10?3 , the curves corresponding to TSI only and to both wavelength conversion and TSI are indistinguishable from one another. However, Figure 4 predicts that this conclusion is also applicable for fewer than 16 time slots. Figure 4 also predicts that wavelength conversion alone can provide some utilisation gain, even for paths with a relatively small number of wavelengths compared with the number of time slots. Similar predictions regarding performance improvements at low blocking probabilities can be obtained from Figure 3. 3.2. Multiple-hop paths

Studies of multiwavelength paths with no wavelength conversion have shown a dramatic increase in the blocking probability as the number of hops is increased [5]. This is because, as the number of hops increases, it becomes more dicult to locate a single wavelength which is unused on all hops. However, it has also been shown that the increase in blocking probability with number of hops is signi cantly reduced if wavelength conversion is allowed [5]. This reduction is important, as it not only reduces the overall blocking probability, but makes networks with paths of varying lengths more \fair", as blocking probability is no longer a signi cant function of path length. In shared-wavelength TDM networks, TSI should similarly improve the blocking probability and \fairness" of the networks. It is thus important to consider the eect of wavelength conversion and TSI on blocking probability in shared-wavelength TDM networks as the path length increases. Figure 5 plots blocking probability as a function of path length (number of hops) for shared-wavelength TDM paths with 2 wavelengths per link, 16 time slots per wavelength, and a constant oered load of 15.0 Erlangs per link, and where it is assumed that link loads and wavelength allocations on dierent links are independent. Blocking probabilities were calculated using Equations (2.4) and (2.6). Figure 5 shows that, with no wavelength conversion or TSI, blocking probability increases dramatically as path length increases. However, with both conversion and TSI, the increase in blocking with increasing path length is con-

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Blocking probability

0.1 No conversion, no TSI Conversion, no TSI No conversion, TSI Conversion, TSI

0.01

0.001

0.0001

1e-05 2

4

6

8 10 12 14 Number of hops (H)

16

18

20

Figure 5. Blocking probability as a function of path length (number of hops) for paths with 2 wavelengths, 16 time slots per wavelength, and an oered load of 15.0 Erlangs per link.

siderably less signi cant. Figure 5 also shows that the blocking probability in a path with wavelength conversion and no TSI is still a signi cant function of the path length. However, Figure 5 shows that the blocking probability can be significantly reduced by the introduction of TSI only, without wavelength conversion. In fact, in Figure 5, the curve for blocking probability with TSI only is almost indistinguishable from the curve for blocking probability with both wavelength conversion and TSI. These conclusions are based on assumptions of independence of loads and of wavelength and time slot allocations between dierent links, and of random wavelength and time slot assignment. However, these assumptions both tend to overestimate the potential bene ts of wavelength converters and TSI. If the loads and wavelength / time slot allocations on dierent links are not independent, or if an improved wavelength / time slot assignment scheme is used, we would expect to see wavelength converters and TSI oering less signi cant performance advantages. This is discussed further in Section 4.2.1. We can also examine the utilisation gains, de ned in Section 3.1, as the path length increases. Figure 6 plots the utilisation gains versus path length for 2 wavelengths, 16 time slots and a xed blocking probability of 10?3 . An outstanding feature of Figure 6 is that the utilisation gains shown are much larger than those seen in Figure 4. For instance, in a 20-hop path, the introduction of both wavelength conversion and TSI results in a utilisation gain of about 6. This value for the utilisation gain may be an overestimate due to the assumptions of link independence and random wavelength and time slot assignment, but nevertheless it illustrates the increasing importance of wavelength conversion and TSI as path length increases.

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J. Yates et. al. / Multiwavelength TDM Networks 6 Conversion, TSI No conversion, TSI Conversion, no TSI No conversion, no TSI

Utilisation gain

5 4 3 2 1 0 2

4

6

8 10 12 14 Number of hops (H)

16

18

20

Figure 6. Utilisation gain vs number of hops for a path with 2 wavelengths, 16 time slots and a blocking probability of 10?3 .

Figure 6 also shows again that, in a path with a small number of wavelengths and a relatively large number of time slots per wavelength, TSI alone can provide almost all of the utilisation gain provided by both wavelength conversion and TSI. This is illustrated by the fact that the corresponding curves in Figure 6 are almost indistinguishable from one another. However, moderate utilisation gain can also be achieved using only wavelength conversion. For example, in a 20-hop path, the use of both wavelength conversion and TSI provides a utilisation gain of 6.02, the use of only TSI gives a gain of 6.01, and the use of only wavelength conversion gives a utilisation gain of 2.8. Section 2.2 described the equivalence between wavelengths and time slots in shared-wavelength TDM systems. We can now use this equivalence, together with Figures 3{6, to draw conclusions about paths with a large number of wavelengths, and a relatively small number of time slots per wavelength. In these systems, we can conclude that the introduction of TSI alone, without wavelength conversion, provides moderate performance improvements. However, the introduction of wavelength conversion alone, without TSI, will provide almost all of the improvement in performance and in \fairness" introduced by both wavelength conversion and TSI. In such cases TSI may not be necessary.

4. Network Simulations The analysis presented in the previous sections predicts the relative importance of wavelength conversion and TSI in improving the performance of simple multiple-hop paths in shared-wavelength TDM systems. We now use network simulations to examine wavelength conversion and TSI in shared-wavelength

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TDM networks with more complicated topologies. Our simulations have assumed:

each time slot on each wavelength channel carries only a single connection; Poisson arrival of connection requests and negative exponential holding times; no queueing of blocked connection requests; static routing of connections and dynamic allocation of connections to wavelengths and time slots; and in each topology, each of the nodes generates equal trac, and the destination node is randomly and equiprobably chosen from the other nodes. In the analytical work described above, connections were randomly assigned to time slots and to wavelengths. However, improved performance may be achieved using other wavelength and time slot assignment schemes. In our network simulations, wavelength and time slot assignment is performed using a rst- t algorithm. When neither wavelength conversion nor TSI are used, the algorithm assigns a new connection to the wavelength with the smallest index from within the set of wavelengths which have available time slots. The time slot with the lowest index is then selected from the set of available time slots on the chosen wavelength. We now present simulation results for three dierent network topologies: a mesh-torus network, a unidirectional ring network, and the ARPA2 network [5]. Con dence intervals for all simulation results are shown at the 95% level. 4.1. Mesh-torus network

We now consider an 11 11 mesh-torus topology [8,9] with 2 wavelengths and 16 time slots per wavelength. Figure 7 plots blocking probability against total oered network trac for this topology. The uppermost curve depicts the blocking probability in the network without wavelength conversion or TSI, whilst the lowest curve shows blocking probability in a network with both conversion and TSI. The other two curves show blocking probability in networks with conversion but no TSI, and networks with TSI but no conversion. Similar conclusions can be drawn from the network simulations shown in Figure 7 as from the single path analyses in Figures 3 and 5. Figure 7 shows that, for the mesh-torus network [8,9] with a small number of wavelengths and a relatively large number of time slots per wavelength, signi cant improvement in blocking probability is achieved by introducing both wavelength conversion and TSI. Moderate improvement is achieved by introducing wavelength conversion alone, without TSI. However, almost all of the improvement oered by both wavelength conversion and TSI is achieved with TSI alone, without wavelength conversion.


15

1


0.1


0.01

0.001

0.0001

1e-05 700

800

900

1000 1100 1200 Offered traffic (Erlangs)

1300

1400

Figure 7. Blocking in an 11 11 mesh-torus network with 2 wavelengths and 16 time slots per wavelength. 95% con dence intervals are shown.

4.2. Unidirectional ring network

We now consider a 9-node unidirectional ring topology with 2 wavelengths and 16 time slots per wavelength. Figure 8 plots blocking probability versus total network oered load for this topology. For WDM ring networks without TDM, it is known [7] that wavelength conversion generally provides less signi cant performance improvements than in the more highly connected mesh-torus topology, due to the high correlation between the oered loads on adjacent links in a ring network. Figure 8 shows that this result can be extended to shared-wavelength TDM networks, to say that wavelength conversion and TDM together provide less signi cant performance improvements in a ring network than in a mesh network. This is again due to the high correlations between loads on adjacent links. In terms of the relative performance improvements oered by wavelength converters and TSI, Figure 8 shows that TSI provides most of the performance improvement oered by both wavelength conversion and TSI, and that wavelength converters alone oer only very small performance improvements. We quantify this improvement by examining the extra trac which can be carried at a speci ed blocking probability. We de ne the extra trac carried as a result of, for example, wavelength conversion and TSI, to be the dierence between the trac carried in a network with both wavelength conversion and TSI, and the trac carried in a network without wavelength conversion or TSI. This is a dierent performance metric than the utilisation gain previously used. We will now compare the extra trac that could be supported with wavelength conversion but without TSI to the extra trac that could be supported using both wavelength conversion and TSI, at the 10?3 blocking level. From Fig-

16



0.1 0.01 0.001 0.0001


1e-05 1e-06 20

30

40 50 60 70 Offered traffic (Erlangs)

80

Figure 8. Blocking vs total network oered oered load in a 9-node uni-directional ring network with 2 wavelengths and 16 time slots per wavelength. 95% con dence intervals are shown.

ure 8, we can determine that the network without TSI supports only 28% of the extra trac that could be supported using both wavelength conversion and TSI. In contrast, for the 11 11 mesh-torus topology of section 4.1, the corresponding gure is 47%. We conclude that wavelength converters alone oer signi cantly smaller improvements in a ring topology compared with a mesh-torus topology, and we believe that this is due to the correlations between the wavelengths and time slots used on adjacent links in the ring topology. We now investigate directly the eect of these correlations. 4.2.1. Eect of correlated trac In order to investigate the eect of correlated trac, we use a simple 5hop topology with two dierent trac scenarios, rstly assuming independence between the loads, wavelengths and time slot allocations on adjacent links, and secondly assuming that 75% of the trac oered to each link is from a 5-hop trac stream requiring all 5 hops to establish a connection, while the remainder of the trac belongs to 1-hop streams. This means that there is a high correlation in the trac on each of the links along the path, as in the case of the unidirectional ring. The results for the independent scenario are calculated using the analytic results of Section 2.2, while the results for the correlated scenario are found by simulation, since the assumptions for the analysis no longer apply. In both the analysis and the simulation, random wavelength and time slot assignment is assumed. The results for the two scenarios are shown in Figures 9 and 10. We are interested here, not in comparing the absolute values of blocking probabilities

17


between the two scenarios, but in the relative dierences between the curves, in the two scenarios separately. 1


0.1 0.01 0.001 0.0001


1e-05 1e-06 5

10

15 20 25 Offered load (Erlangs)

30

35

Figure 9. Blocking vs oered load per link along a 5-hop path with 2 wavelengths and 16 time slots per wavelength, and with independence between the loads, wavelength, and time slot allocations on adjacent links. 1

Blocking probablity

0.1

0.01

0.001 No conversion, no TSI Conversion, no TSI No conversion, TSI Conversion, TSI

0.0001

1e-05 5

10

15 20 25 Offered traffic (Erlangs)

30

35

Figure 10. Blocking vs oered load per link along a 5-hop path with 2 wavelengths and 16 time slots per wavelength, and with 75% of the oered trac per link requiring all ve links to establish a connection. 95% con dence intervals are shown.

Comparing the network without either conversion or TSI with the network with both conversion and TSI (i.e. comparing the upper and lower curves in Fig-

18


ures 9 and 10): in the independent scenario we see that wavelength conversion and TSI produces signi cant reduction in the blocking probabilities compared to no conversion and no TSI; in contrast, in the correlated scenario, wavelength conversion and TSI has a much less marked impact in reducing the blocking probabilities. In other words, the curves are much closer together in the correlated scenario. We conclude that increased trac correlation reduces the relative improvement obtainable either by wavelength conversion or by TSI in WDM/TDM networks. For the network without TSI (the upper two curves in Figures 9 and 10): similarly to the situation discussed in the previous paragraph, these curves are much closer together in the correlated scenario than in the independent scenario. We conclude that increased trac correlation reduces the relative improvement obtainable by wavelength conversion in WDM/TDM networks without TSI. For the network with TSI (the lower two curves in Figures 9 and 10): in both scenarios we see that there is little dierence between the cases with and without wavelength conversion, irrespective of whether the trac is correlated or not, and we conclude that TSI alone oers almost all of the performance improvement achievable by using both wavelength conversion and TSI. 4.3. ARPA2 network

The ARPA2 network [5] shown in Figure 11 is an irregular topology consisting of 21 nodes and 26 bidirectional links. Figure 12 plots the blocking probability versus total network oered load, when there are 2 wavelengths and 16 time slots per wavelength. The conclusions to be drawn are the same as those for the unidirectional ring network; namely that TSI provides most of the performance improvement oered by both wavelength conversion and TSI, and that wavelength converters alone oer only very small performance improvements.

Figure 11. The 21-node, 26-bidirectional link ARPA2 topology.

5. Conclusion In this paper, we have examined the blocking performance of so-called shared-wavelength TDM systems [4]. Shared-wavelength TDM systems are multi-

19



0.1


0.01

0.001

0.0001

1e-05 100

150

200 250 Offered traffic (Erlangs)

300

Figure 12. Blocking in the ARPA-2 network with 2 wavelengths and 16 time slots per wavelength. 95% con dence intervals are shown.

wavelength systems in which connections are assigned on demand to a wavelength, and to a time slot on that wavelength. We have examined the relative importance of wavelength conversion and time-slot interchange in reducing blocking, or increasing utilisation, in these systems. We have developed an analytical description of blocking on multiple-hop paths in shared-wavelength TDM systems, and we have performed simulations of several shared-wavelength TDM network topologies. In a shared-wavelength TDM system with a small number of wavelengths and a relatively large number of time slots per wavelength, our results show, when there is low correlation between loads on dierent links, that TSI and wavelength conversion together can oer signi cant performance improvements, and that wavelength conversion alone can oer moderate performance improvements. However, when there is high correlation between loads on dierent links, our results show that TSI and wavelength conversion together can oer only minor performance improvements, and that wavelength conversion alone generally oers insigni cant performance improvements. It is very signi cant that, whether loads are correlated or not, TSI alone provides almost all the performance improvement that is achievable using both TSI and wavelength conversion. By contrast, in a network with a large number of wavelengths and a relatively small number of time slots per wavelength, the equivalence between wavelengths and time slots allows us to reinterpret the above results, everywhere replacing TSI by wavelength conversion, and vice versa.

20


Acknowledgements The authors wish to thank Steve Madden from Telstra Research Laboratories for helpful discussions. Jennifer Yates also thanks Telstra Research Laboratories for nancial support.

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