480 W D M NETWORK DESIGN Table 8.4 Number of wavelengths required to perform offline wavelength assign- ment as a function of the load L with and without wavelength converters.. As in th
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Table 8.4 Number of wavelengths required to perform offline wavelength assign- ment as a function of the load L with and without wavelength converters The fixed conversion result for arbitrary topologies applies only to one- and two-hop lightpaths
Arbitrary m i n [ ( L - 1)D + 1, L L
(2L - 1 ) x f M - L + 2]
Other topologies such as star networks and tree networks have also been consid- ered in the literature In star and tree networks, ~-L wavelengths are sufficient to do WA-NC [RU94] In star networks, L wavelengths are sufficient for WA-FC [RS97] The same result can be extended to arbitrary networks where lightpaths are at most two hops long Table 8.4 summarizes the results to date on this problem It is still a topic of intense research
Multifiber Rings
The wavelength assignment problem in multifiber rings is considered in [LS00] In a multifiber ring, each pair of adjacent nodes is connected by k > 1 fiber pairs: k > 1 fibers are used for each direction of transmission instead of i fiber Recall that we are considering undirected edges and lightpaths, and each edge represents a pair of fibers, one for each direction of transmission Thus, such a multifiber ring is represented by
k edges between pairs of adjacent nodes There is no wavelength conversion, but it
is assumed that the same wavelength can be switched from an incoming fiber to any
of the k outgoing fibers at each node The following results on multifiber rings are proved in [LS00]
T h e o r e m 8.6 [LS00] Given a set of lightpath requests and a routing on a k-fiber-pair ring with load L on each multifiber link, the number of wavelengths, summed over all the fibers, required to solve the wavelength assignment problem
is no more than [k~ !lL- 11
Trang 2Thus, for a dual-fiber-pair ring (k = 2), the number of wavelengths required is no than [3L - 1 ~ , which is a significant improvement over the bound of 2L - 1 more
! / for a single-fiber-pair ring
As in the case of the single-fiber-pair ring, you can come up with a set of lightpath
requests with load L for which this upper bound on the number of wavelengths is tight, for all values of the fiber multiplicity, k
We next consider the online wavelength assignment problem in rings Assume that the routing of the lightpaths is already given and that lightpaths are set up as well
as taken down, that is, the lightpaths are nonpermanent Here, it becomes much more difficult to come up with smart algorithms that maximize the load that can be supported for networks without full wavelength conversion (With full wavelength conversion at all the nodes, an algorithm that assigns an arbitrary free wavelength can support all lightpath requests with load up to W.) We describe an algorithm that provides efficient wavelength assignment for line and ring networks without wavelength conversion
Lemma 8.7 [GSKR99] Let W(N, L) denote the number of wavelengths re-
quired to support all online lightpath requests with load L in a network with N nodes without wavelength conversion In a line network, W(N, L) <_
L + W(N/2, L), when N is a power of 2
Proof Break the line network in the middle to realize two disjoint subline
networks, each with N/2 nodes Break the set of lightpath requests into two
groups: one group consisting of lightpaths that lie entirely within the subline networks, and the other group consisting of lightpaths that go across between the two subline networks The former group of lightpaths can be supported with
at most W(N/2, L) wavelengths (the same set of wavelengths can be used in
both subline networks) The latter group of lightpaths can have a load of at most L Dedicate L additional wavelengths to serving this group This proves the lemma I
The following theorem follows immediately from Lemma 8.7, with the added condition that W (1, L) = 0 (or W (2, L) = L)
Theorem 8.8 [GSKR99] In a line network with N nodes, all online lightpath
requests with load L can be supported using at most L [log 2 N] wavelengths without requiring wavelength conversion
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The algorithm implied by this theorem is quite efficient since it is possible to come up with lightpath traffic patterns for which any algorithm will require at least 0.5L log 2 N wavelengths [GSKR99]
Theorem 8.9 [GSKR99] In a ring network with N nodes, all online lightpath
requests with load L can be supported using at most L [log 2 N] + L wavelengths,
without requiring wavelength conversion
The proof of this theorem is left as an exercise (Problem 8.21)
When we have permanent lightpaths being set up, it is possible to obtain some- what better wavelength assignments, as given by the following theorem, the proof
of which is beyond the scope of this book
T h e o r e m 8.10 [GSKR99] In a ring network with N nodes, all online perma-
nent lightpath requests with load L can be supported using (a) at most 2L wave- lengths without wavelength conversion, and (b) with at most max(0, L - d) + L
wavelengths with degree-d (d >_ 2) limited wavelength conversion
Table 8.5 summarizes the results to date on the offline and online RWA problem for ring networks, with the traffic model characterized by the maximum link load For this model, observe that significant increases in the traffic load can be achieved
by having wavelength converters in the network For the offline case, very limited conversion provides almost as much benefit as full wavelength conversion For the online cases, the loads that can be supported are much less than the offline case The caveat is that, as illustrated in Figure 8.13, this model represents worst-case scenarios, and a majority of traffic patterns could perhaps be supported efficiently without requiring as many wavelengths or as many wavelength converters
Summary
We studied the design of wavelength-routing networks in this chapter We saw that there is a clear benefit to building wavelength-routing networks, as opposed to simple point-to-point WDM links The main benefit is that traffic that is not to be terminated within a node can be passed through by the node, resulting in significant savings in higher-layer terminating equipment
The design of these networks is more complicated than the design of traditional networks It includes the design of the higher-layer topology (IP or SONET), which is the lightpath topology design problem, and its realization in the optical layer, which
is the routing and wavelength assignment problem These problems may need to be
Trang 4Table 8.5 Bounds on the number of wavelengths required in rings to sup-
port all traffic patterns with maximum load L for different models, offline and
online, from [GRS97, GSKR99] d denotes the degree of wavelength conver-
sion The upper bound indicates the number of wavelengths that are sufficient
to accommodate all traffic patterns with maximum load L, using some RWA
algorithm The lower bound indicates that there is some traffic pattern with
maximum load L that requires this many wavelengths regardless of the RWA
algorithm that is employed For the online traffic model, we consider two
cases, one where lightpaths are set up over time but never taken down, and
another where lightpaths are both set up and taken down over time
Conversion Degree Lower Bound on W Upper Bound on W
Offline traffic model
No conversion 2 L - 1 2 L - 1
Fixed conversion L + 1 L + 1
m
Online model without lightpath terminations
Online model with lightpath terminations
No conversion 0.5L [log 2 N] L [log 2 N] + L
solved in conjunction if the carrier provides IP or SONET VTs over its own optical infrastructure However, this is difficult to do, and a practical approach may be to iteratively solve these problems
We then discussed the wavelength dimensioning problem The problem here is to provide sufficient capacity on the links of the wavelength-routing network to handle the expected demand for lightpaths This problem is solved today by periodically forecasting a traffic matrix and (re)designing the network to support the forecasted matrix Alternatively, you can employ statistical traffic demand models to estimate the required capacities, and we discussed two such models
The absence of wavelength conversion in the network can be overcome by pro- viding more wavelengths on the links In the last section, we studied this trade-off under various models
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Further Reading
The issue of how much cost savings is afforded by providing networking functions within the optical layer is only beginning to be understood For some more insights into this issue, see [RLB95, Ba196, GRS98, SGS99, CM00, BM00] The material in this chapter is based on [GRS98] See [Wi196, WW98, Ber96] for a discussion of the problem of setting up connections between all pairs of nodes in a WDM ring network
The lightpath topology design problem is discussed in [RS96, KS98, CMLF00, MBRM96, BG95, ZA95, JBM95, GW94, CGK93, LA91] Our discussion is based
on [RS96] This is an example of a network flow problem; these problems are dealt with in detail in [AMO93]
Several papers [ABC+94, RU94, RS95, CGK92, RS97, MKR95, KS97, KPEJ97, ACKP97] study the offline routing and wavelength assignment problem There is also
a vast body of literature describing routing and wavelength assignment heuristics See, for example, [CGK92, SBJS93, RS95, Bir96, WD96, SOW95]
The statistical blocking model for dimensioning is analyzed in [SS00, BK95, RS95, KA96, SAS96, YLES96, BH96]
The worst-case analysis of the maximum load model with online traffic is con- sidered in [GK97]
8.1
8.2
Problems
In general there are several valid design options even for a three-node network Consider the designs shown in Figure 8.1(c), but now assume that the number of dropped lightpaths is six instead of five as discussed in the text The advantage of this design is that it provides more flexibility in handling surges in A-B and B-C traffic For example, this design not only can handle the traffic requirement of 50 Gb/s between every pair of nodes, it can also handle a traffic requirement of 60 Gb/s between nodes A-B and B-C, and 40 Gb/s between nodes A-C This latter traffic pattern cannot be handled if only five lightpaths/wavelengths are dropped
Consider the design of Figure 8.1(c), and assume that x wavelengths are dropped
at node B and y wavelengths pass through Determine the range of traffic matrices that this design is capable of handling as a function of x and y
Consider the network design approach using fixed-wavelength routing in a four-node ring network with consecutive nodes A, B, C, and D Suppose the traffic requirements are as follows:
Trang 68.3
8 4
8 5
8 6
8.7
8 8
8 9
8.10
A
B
C
D
- 3 - 3
- 2 - 2
(a) Do a careful routing of traffic onto each wavelength so as to minimize the number of wavelengths needed
(b) How do you know that your solution uses the minimum possible number of wavelengths required to do this routing for any algorithm?
(c) How many ADMs are required at each node to support this traffic?
(d) How many ADMs are required at each node if instead of fixed-wavelength routing, you decided to use point-to-point WDM links and receive and re- transmit all the wavelengths at each node? How many ADMs does wave- length routing eliminate?
Derive (8.1) What is the value when N is odd?
Derive (8.5) What is the value when N is odd?
Derive (8.8) for the case where there is one full-duplex lightpath between each pair
of nodes Hint: Use induction Start with two nodes on the ring, and determine the
number of wavelengths required Add two more nodes so that they are diametrically opposite to each other on the ring and continue
Show that when N is odd, (8.8) is modified to
W = N - 1 8 "
Derive (8.9) What is the value when N is odd?
Develop other network designs besides the ones shown in Examples 8.2, 8.3, and 8.4, and compare the number of LTs and wavelengths required for these designs against these three examples
Consider the network shown in Figure 8.9(a) Assume that each undirected edge can be represented by a pair of directed edges as in Figure 8.9(c) Represent each undirected lightpath in Figure 8.9(a) by a pair of directed lightpaths with oppo- site directions Consider the RWA problem in the resulting network and show that two wavelengths are sufficient to support these directed lightpaths Note that three wavelengths were required to support the corresponding undirected lightpaths This problem illustrates the complexity of wavelength assignment in networks where the transmission is bidirectional over each fiber Consider the two networks shown
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rectional links
8.11
8.12
8.13
8.14
8.15
in Figure 8.22 In Figure 8.22(a), the network uses two fibers on each link, with two wavelengths on each fiber, with unidirectional transmission on each fiber In Figure 8.22(b), the network uses one fiber on each link, with four wavelengths Transmission is bidirectional on each fiber, with two wavelengths in one direction and two in the other No wavelength conversion is allowed in either network Both networks have the same nominal capacity (four wavelengths/link) Which network utilizes the capacity more efficiently?
Show that a network having P fiber pairs between nodes and W wavelengths on each fiber with no wavelength conversion is equivalent to a network with one fiber pair between nodes with P W wavelengths, and degree P wavelength conversion capability at the nodes
Generalize the example of Figure 8.13 to the case when the number of nodes is arbitrary, say, N Compare the number of wavelengths required in this general case
to the upper bound given by Theorem 8.1
In order to prove that W _< (2L - 1 ) x / M - L + 2 in Theorem 8.1, we supposed that there were K lightpaths of length > ff-M hops Instead suppose there are K(x) lightpaths of length >_ x hops, and derive an upper bound for W that holds for every
x Now, optimize x to get the least upper bound for W Compare this bound with the bound obtained in Theorem 8.1
Show that Algorithm 8.3 always does the wavelength assignment using L wave- lengths Hint: Use induction on the number of nodes
Consider the following modified version of Algorithm 8.3 In step 2, the algorithm
is permitted to assign any free wavelength from a fixed set of L wavelengths, instead
of the least numbered wavelength Show that this algorithm always succeeds in performing the wavelength assignment
Trang 88.16
8.17
8.18
8.19
8.20
8.21
8.22
8.23
8.24
Prove that Theorem 8.3 can be tight in some cases In other words, give an example
of a ring network and a set of lightpath requests and routing with load L that requires
2L - 1 wavelengths Hint: First, give an example that requires 2L - 2 wavelengths
and then modify it by adding an additional lightpath without increasing the load Note that the example in Figure 8.20 shows such an example for the case L = 2 Obtain an example for the case L > 2
Consider a ring network with a lightpath request set of one lightpath between each source-destination pair Compute the number of wavelengths sufficient to support this set with full wavelength conversion and without wavelength conversion What
do you conclude from this?
Give an example of a star network without wavelength conversion where 3 L wave- lengths are necessary to perform the wavelength assignment
Prove Theorem 8.4
Prove Theorem 8.8 Based on this proof, write pseudo-code for an algorithm to perform wavelength assignment
Prove Theorem 8.9
This problem relates to the wavelength assignment problem in networks without wavelength conversion Let us assume that the links in the network are duplex, that
is, consist of two unidirectional links in opposite directions A set of duplex lightpath requests and their routing is given In practice, each request between two nodes A and B is for a lightpath 1 from A to B and another lightpath 1 f from B to A, which
we will assume are both routed along the same path in the network
One wavelength assignment scheme (scheme 1) is to assign the same wavelength
to both 1 and l I Give an example to show that it is possible to do a better wavelength assignment (using fewer wavelengths) by assigning different wavelengths to 1 and 1 t (scheme 2) Show using this example that scheme I can need up to ~- W wavelengths,
where W is the number of wavelengths required for scheme 2 Hint: Consider a
representation of the path graph corresponding to directed lightpaths
Derive the expression (8.13) for the probability that a lightpath request is blocked when the network uses full wavelength conversion
Derive the approximate expressions for ~nc and ~fc given by (8.16) and (8.17) Plot these approximations and the exact values given by (8.14) versus W for Pb = 10 -3,
10 -4, and 10 -5, and H = 5, 10, and 20 hops to study the behavior of ~nc and ~fc, and to verify the range of accuracy of these approximations
8.25 Derive (8.18)
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8.26 Consider the five-node fiber topology shown in Figure 8.23 on which IP bandwidth
is to be routed between IP router node pairs over a WDM network The bandwidth demands are given for each node pair in the following table Assume that all demands are bidirectional, and both directions are routed along the same path using the same wavelengths in opposite directions
(a) Assuming OC-192c (10 Gb/s) trunks are used, complete an equivalent table for the required number of lightpaths (that is, wavelengths) between each pair of nodes
(b) Using the given physical topology, and assuming that there are no wave- length conversion capabilities contained within the optical crossconnects at the nodes, specify a reasonable wavelength-routing design for each light- path Clearly label each wavelength along its end-to-end path through the network
(c) What is the maximum load on any link in the network, and how does it compare with the number of wavelengths you are using in to- tal?
References
[ABC+94] A Aggarwal, A Bar-Noy, D Coppersmith, R Ramaswami, B Schieber, and
M Sudan Efficient routing and scheduling algorithms for optical networks In
Trang 10Proceedings of 5th Annual A CM-SIAM Symposium on Discrete Algorithms, pages
412-423, Jan 1994
[ACKP97] V Auletta, I Caragiannis, C Kaklamanis, and P Persiano Bandwidth allocation
algorithms on tree-shaped all-optical networks with wavelength converters In
Proceedings of the 4th International Colloquium on Structural Information and Communication Complexity, 1997
and Applications Prentice Hall, Englewood Cliffs, mJ, 1993
Engineers Conference, pages 163-174, 1996
[Ber96] J.-C Bermond et al Efficient collective communication in optical networks In
23rd International Colloquium on Automata, Languages and
Programming ICALP '96, Paderborn, Germany, pages 574-585, 1996
NJ, 1992
[BG95] D Bienstock and O Gunluk Computational experience with a difficult
68:213-237, 1995
[BH96] R.A Barry and P A Humblet Models of blocking probability in all-optical
Optical Networks, 14(5):858-867, June 1996
[Big90] N Biggs Some heuristics for graph colouring In R Nelson and R J Wilson,
87-96 Longman Scientific & Technical, Burnt Mill, Harlow, Essex, UK, 1990 [Bir96] A Birman Computing approximate blocking probabilities for a class of optical
June 1996
[BK95] A Birman and A Kershenbaum Routing and wavelength assignment methods in
pages 431-438, 1995
[BM00] R Berry and E Modiano Reducing electronic multiplexing costs in SONET/WDM
Communications, 18:1961-1971, 2000
Morgan Kaufmann, San Francisco, 1998