Optical Networks - _10_5 Maximum Load Dimensioning Models_121

# Fixed conversion result for arbitrary topologies

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fixed conversion result for arbitrary topologies applies only to one- and two-hop lightpaths. Network Conversion Type None Fixed Full Limited Arbitrary min[ (L 1 )D + 1 , L L ( 2 L 1 ) M L + 2] Ring 2 L 1 L + 1 L L Star 3 2 L L L Tree 3 2 L L L 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]. Theorem 10.6 [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 + 1 k L 1 . Thus, for a dual-fiber-pair ring ( k = 2 ), the number of wavelengths required is no more than 3 2 L 1 , which is a significant improvement over the bound of 2 L 1 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 . 10.5.2 Online RWA in Rings 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

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616 WDM Network Design that provides efficient wavelength assignment for line and ring networks without wavelength conversion. Lemma 10.7 [GSKR99] Let W(N, L) denote the number of wavelengths required 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. The following theorem follows immediately from Lemma 10.7, with the added condition that W( 1 , L) = 0 (or W( 2 , L) = L ). Theorem 10.8 [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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